ChemWiki - User contributions [en]

Rep:Mod:ETZ13TS

2016-01-29T11:20:43Z

Etz13: /* Comparison of bond lengths of exo and endo TS */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the pericyclic mechanism is now widely accepted. The debate about the contrasting mechanisms arose from the small energy differences between the diradical intermediate and transition state amid a lack of experimental data and conflicting theoretical calculations.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle. Note that the bond lengths are symmetrical about the centre of the molecule and so only half have been included.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second order derivatives of energy.
A negative sign in the second order derivative implies an energy maximum. Hence a negative eigenvalue will give an imaginary frequency, which is represented with a negative number with Gaussian. As a result the presence of negative frequencies indicates that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented in the table below. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, confirming the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the structures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kcal mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kcal mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref> Note that this study does not account for the pre- and postchemical processes, including the rotation of ''trans'' butadiene to the more stable ''cis'' conformer. A recent study revealed that charge transfer facilitates the formation of the ''cis'' butadiene conformer as well as the pyramidalization of the reacting carbon centers. <ref>T. Sexton, E. Kraka, and D. Cremer, The Extraordinary Mechanism of the Diels-Alder Reaction: Investigation of Stereochemistry, Charge Transfer, Charge Polarization, and Biradicaloid Formation, J. Phys. Chem. A. published online on 19 Jan 2016</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-C2, C3-C4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds indicate the extension and compression of bonds as the reactant transitions to the product and vice versa. Furthermore they are similar for both the exo and endo transition structures. Note that only half the bond lengths are used as the transition structures have an internal mirror plane of symmetry (meso).

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. In the meantime, the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and so the Frozen Coordinate method is a better stepwise alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively). The high accuracy of the hybrid DFT method using B3LYP/6-31G(d) is also demonstrated.

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T10:39:05Z

Etz13: /* Optimization of cis-butadiene and ethylene TS */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the pericyclic mechanism is now widely accepted. The debate about the contrasting mechanisms arose from the small energy differences between the diradical intermediate and transition state amid a lack of experimental data and conflicting theoretical calculations.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle. Note that the bond lengths are symmetrical about the centre of the molecule and so only half have been included.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second order derivatives of energy.
A negative sign in the second order derivative implies an energy maximum. Hence a negative eigenvalue will give an imaginary frequency, which is represented with a negative number with Gaussian. As a result the presence of negative frequencies indicates that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented in the table below. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, confirming the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the structures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kcal mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kcal mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref> Note that this study does not account for the pre- and postchemical processes, including the rotation of ''trans'' butadiene to the more stable ''cis'' conformer. A recent study revealed that charge transfer facilitates the formation of the ''cis'' butadiene conformer as well as the pyramidalization of the reacting carbon centers. <ref>T. Sexton, E. Kraka, and D. Cremer, The Extraordinary Mechanism of the Diels-Alder Reaction: Investigation of Stereochemistry, Charge Transfer, Charge Polarization, and Biradicaloid Formation, J. Phys. Chem. A. published online on 19 Jan 2016</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-C2, C3-C4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds indicate the extension and compression of bonds as the reactant transitions to the product and vice versa. Furthermore they are similar for both the exo and endo transition structures.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. In the meantime, the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and so the Frozen Coordinate method is a better stepwise alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively). The high accuracy of the hybrid DFT method using B3LYP/6-31G(d) is also demonstrated.

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T10:32:24Z

Etz13: /* Diels-Alder */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the pericyclic mechanism is now widely accepted. The debate about the contrasting mechanisms arose from the small energy differences between the diradical intermediate and transition state amid a lack of experimental data and conflicting theoretical calculations.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle. Note that the bond lengths are symmetrical about the centre of the molecule and so only half have been included.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second order derivatives of energy.
A negative sign in the second order derivative implies an energy maximum. Hence a negative eigenvalue will give an imaginary frequency, which is represented with a negative number with Gaussian. As a result the presence of negative frequencies indicates that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented in the table below. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, confirming the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the structures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kcal mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kcal mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref> Note that this study does not account for the pre- and postchemical processes, including the rotation of ''trans'' butadiene to the more stable ''cis'' conformer. A recent study revealed that charge transfer facilitates the formation of the ''cis'' butadiene conformer as well as the pyramidalization of the reacting carbon centers. <ref>T. Sexton, E. Kraka, and D. Cremer, The Extraordinary Mechanism of the Diels-Alder Reaction: Investigation of Stereochemistry, Charge Transfer, Charge Polarization, and Biradicaloid Formation, J. Phys. Chem. A. published online on 19 Jan 2016</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds indicate the extension and compression of bonds as the reactant transitions to the product and vice versa. Furthermore they are similar for both the exo and endo transition structures.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. In the meantime, the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and so the Frozen Coordinate method is a better stepwise alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively). The high accuracy of the hybrid DFT method using B3LYP/6-31G(d) is also demonstrated.

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T10:31:43Z

Etz13: /* Conclusion */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the pericyclic mechanism is now widely accepted. The debate about the contrasting mechanisms arose from the small energy differences between the diradical intermediate and transition state amid a lack of experimental data and conflicting theoretical calculations.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle. Note that the bond lengths are symmetrical about the centre of the molecule and so only half have been included.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second order derivatives of energy.
A negative sign in the second order derivative implies an energy maximum. Hence a negative eigenvalue will give an imaginary frequency, which is represented with a negative number with Gaussian. As a result the presence of negative frequencies indicates that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented in the table below. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, confirming the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the structures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kcal mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kcal mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

Note that this study does not account for the pre- and postchemical processes, including the rotation of ''trans'' butadiene to the more stable ''cis'' conformer. A recent study revealed that charge transfer facilitates the formation of the ''cis'' butadiene conformer as well as the pyramidalization of the reacting carbon centers. <ref>T. Sexton, E. Kraka, and D. Cremer, The Extraordinary Mechanism of the Diels-Alder Reaction: Investigation of Stereochemistry, Charge Transfer, Charge Polarization, and Biradicaloid Formation, J. Phys. Chem. A. published online on 19 Jan 2016</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
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<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
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</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds indicate the extension and compression of bonds as the reactant transitions to the product and vice versa. Furthermore they are similar for both the exo and endo transition structures.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. In the meantime, the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and so the Frozen Coordinate method is a better stepwise alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively). The high accuracy of the hybrid DFT method using B3LYP/6-31G(d) is also demonstrated.

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T10:30:15Z

Etz13: /* Comparison of bond lengths of exo and endo TS */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the pericyclic mechanism is now widely accepted. The debate about the contrasting mechanisms arose from the small energy differences between the diradical intermediate and transition state amid a lack of experimental data and conflicting theoretical calculations.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle. Note that the bond lengths are symmetrical about the centre of the molecule and so only half have been included.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second order derivatives of energy.
A negative sign in the second order derivative implies an energy maximum. Hence a negative eigenvalue will give an imaginary frequency, which is represented with a negative number with Gaussian. As a result the presence of negative frequencies indicates that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented in the table below. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, confirming the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the structures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kcal mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kcal mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

Note that this study does not account for the pre- and postchemical processes, including the rotation of ''trans'' butadiene to the more stable ''cis'' conformer. A recent study revealed that charge transfer facilitates the formation of the ''cis'' butadiene conformer as well as the pyramidalization of the reacting carbon centers. <ref>T. Sexton, E. Kraka, and D. Cremer, The Extraordinary Mechanism of the Diels-Alder Reaction: Investigation of Stereochemistry, Charge Transfer, Charge Polarization, and Biradicaloid Formation, J. Phys. Chem. A. published online on 19 Jan 2016</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds indicate the extension and compression of bonds as the reactant transitions to the product and vice versa. Furthermore they are similar for both the exo and endo transition structures.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. In the meantime, the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively). The high accuracy of the hybrid DFT method using B3LYP/6-31G(d) is also demonstrated.

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T10:28:59Z

Etz13: /* Comparison of different levels of theory */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the pericyclic mechanism is now widely accepted. The debate about the contrasting mechanisms arose from the small energy differences between the diradical intermediate and transition state amid a lack of experimental data and conflicting theoretical calculations.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle. Note that the bond lengths are symmetrical about the centre of the molecule and so only half have been included.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second order derivatives of energy.
A negative sign in the second order derivative implies an energy maximum. Hence a negative eigenvalue will give an imaginary frequency, which is represented with a negative number with Gaussian. As a result the presence of negative frequencies indicates that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented in the table below. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, confirming the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the structures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kcal mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kcal mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

Note that this study does not account for the pre- and postchemical processes, including the rotation of ''trans'' butadiene to the more stable ''cis'' conformer. A recent study revealed that charge transfer facilitates the formation of the ''cis'' butadiene conformer as well as the pyramidalization of the reacting carbon centers. <ref>T. Sexton, E. Kraka, and D. Cremer, The Extraordinary Mechanism of the Diels-Alder Reaction: Investigation of Stereochemistry, Charge Transfer, Charge Polarization, and Biradicaloid Formation, J. Phys. Chem. A. published online on 19 Jan 2016</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. In the meantime, the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively). The high accuracy of the hybrid DFT method using B3LYP/6-31G(d) is also demonstrated.

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T10:25:38Z

Etz13: /* Diels-Alder */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the pericyclic mechanism is now widely accepted. The debate about the contrasting mechanisms arose from the small energy differences between the diradical intermediate and transition state amid a lack of experimental data and conflicting theoretical calculations.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second order derivatives of energy.
A negative sign in the second order derivative implies an energy maximum. Hence a negative eigenvalue will give an imaginary frequency, which is represented with a negative number with Gaussian. As a result the presence of negative frequencies indicates that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented in the table below. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, confirming the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the structures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kcal mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kcal mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

Note that this study does not account for the pre- and postchemical processes, including the rotation of ''trans'' butadiene to the more stable ''cis'' conformer. A recent study revealed that charge transfer facilitates the formation of the ''cis'' butadiene conformer as well as the pyramidalization of the reacting carbon centers. <ref>T. Sexton, E. Kraka, and D. Cremer, The Extraordinary Mechanism of the Diels-Alder Reaction: Investigation of Stereochemistry, Charge Transfer, Charge Polarization, and Biradicaloid Formation, J. Phys. Chem. A. published online on 19 Jan 2016</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. In the meantime, the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively). The high accuracy of the hybrid DFT method using B3LYP/6-31G(d) is also demonstrated.

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T10:13:32Z

Etz13: /* Diels-Alder */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the pericyclic mechanism is now widely accepted. The debate about the contrasting mechanisms arose from the small energy differences between the diradical intermediate and transition state amid a lack of experimental data and conflicting theoretical calculations.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second order derivatives of energy.
A negative sign in the second order derivative implies an energy maximum. Hence a negative eigenvalue will give an imaginary frequency, which is represented with a negative number with Gaussian. As a result the presence of negative frequencies indicates that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented in the table below. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, confirming the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the structures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kcal mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kcal mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

It should be mentioned that classical simulation studies have neglected structural transformations for the Diels Alder cycloaddition during the transition state which can take up to 20 fs. A recent study showed that by taking into account the pre- and post- structural transformations, more accurate predictions on reaction mechanisms and product stereochemistry can be envisaged.<ref>T. Sexton, E. Kraka, and D. Cremer, The Extraordinary Mechanism of the Diels-Alder Reaction: Investigation of Stereochemistry, Charge Transfer, Charge Polarization, and Biradicaloid Formation, J. Phys. Chem. A. published online on 19 Jan 2016</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. In the meantime, the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively). The high accuracy of the hybrid DFT method using B3LYP/6-31G(d) is also demonstrated.

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T06:52:55Z

Etz13: /* Activation Energies */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the pericyclic mechanism is now widely accepted. The debate about the contrasting mechanisms arose from the small energy differences between the diradical intermediate and transition state amid a lack of experimental data and conflicting theoretical calculations.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second order derivatives of energy.
A negative sign in the second order derivative implies an energy maximum. Hence a negative eigenvalue will give an imaginary frequency, which is represented with a negative number with Gaussian. As a result the presence of negative frequencies indicates that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented in the table below. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, confirming the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the structures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kcal mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kcal mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. In the meantime, the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively). The high accuracy of the hybrid DFT method using B3LYP/6-31G(d) is also demonstrated.

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T06:51:04Z

Etz13: /* Conclusion */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the pericyclic mechanism is now widely accepted. The debate about the contrasting mechanisms arose from the small energy differences between the diradical intermediate and transition state amid a lack of experimental data and conflicting theoretical calculations.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second order derivatives of energy.
A negative sign in the second order derivative implies an energy maximum. Hence a negative eigenvalue will give an imaginary frequency, which is represented with a negative number with Gaussian. As a result the presence of negative frequencies indicates that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented in the table below. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, confirming the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the structures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. In the meantime, the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively). The high accuracy of the hybrid DFT method using B3LYP/6-31G(d) is also demonstrated.

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T06:48:11Z

Etz13: /* Background */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the pericyclic mechanism is now widely accepted. The debate about the contrasting mechanisms arose from the small energy differences between the diradical intermediate and transition state amid a lack of experimental data and conflicting theoretical calculations.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second order derivatives of energy.
A negative sign in the second order derivative implies an energy maximum. Hence a negative eigenvalue will give an imaginary frequency, which is represented with a negative number with Gaussian. As a result the presence of negative frequencies indicates that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented in the table below. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, confirming the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the structures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T06:44:55Z

Etz13: /* Optimization to Boat TS utilizing QST2 */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second order derivatives of energy.
A negative sign in the second order derivative implies an energy maximum. Hence a negative eigenvalue will give an imaginary frequency, which is represented with a negative number with Gaussian. As a result the presence of negative frequencies indicates that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented in the table below. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, confirming the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the structures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T06:43:23Z

Etz13: /* Optimization of Chair TS with Guess Method */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second order derivatives of energy.
A negative sign in the second order derivative implies an energy maximum. Hence a negative eigenvalue will give an imaginary frequency, which is represented with a negative number with Gaussian. As a result the presence of negative frequencies indicates that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented in the table below. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, confirming the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T06:42:40Z

Etz13: /* Thermochemistry */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second order derivatives of energy.
A negative sign in the second order derivative implies an energy maximum. Hence a negative eigenvalue will give an imaginary frequency, which is represented with a negative number with Gaussian. As a result the presence of negative frequencies indicates that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented in the table below. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T06:42:20Z

Etz13: /* Comparison of different levels of theory */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second order derivatives of energy.
A negative sign in the second order derivative implies an energy maximum. Hence a negative eigenvalue will give an imaginary frequency, which is represented with a negative number with Gaussian. As a result the presence of negative frequencies indicates that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T06:35:30Z

Etz13: /* Optimization of 1,5-hexadiene Conformers */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory using the Berny Algorithm (default) and are presented in the table below, along with the corresponding point group and energy. The Berny Algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T06:33:58Z

Etz13: /* Background */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The Berny algorithm is used for the minimalisation. It is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-29T00:06:03Z

Etz13: /* Comparison of the activation energies of the endo and exo TS */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The Berny algorithm is used for the minimalisation. It is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals steric congestion in the exo transition state.

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T22:58:29Z

Etz13: /* Optimization of Chair TS with Guess Method */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The Berny algorithm is used for the minimalisation. It is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm which was described previously. This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T22:57:16Z

Etz13: /* Optimization of 1,5-hexadiene Conformers */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The Berny algorithm is used for the minimalisation. It is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T22:51:15Z

Etz13: /* Activation Energies */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values. This is assumed to be because electron electron correlation plays a smaller role in the determination of bond lengths than energy.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T22:50:16Z

Etz13: /* Activation Energies */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below. As previously mentioned the chair transition structure provides a lower energy pathway and hence is favoured. Furthermore both activation barriers decrease with temperature. HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. The origin of the discrepancies have been described previously.

Interestingly, although HF/3-21G overestimates the energies by approximately 30%, the bond lengths and dihedral angles are slightly under experimental values.

{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}
{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T22:19:37Z

Etz13: /* Optimization of chair and boat TS with B3LYP/6-31G(d) */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had imaginary frequencies of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T22:18:52Z

Etz13: /* Intrinsic Reaction Coordinate of Chair TS */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.691579 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script. Note that there was a slight optimization due to the IRC having a higher threshold of defining the minimum energy structure.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had an imaginary frequency of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T22:16:29Z

Etz13: /* Optimization to Boat TS utilizing QST2 */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result, the midpoints of the two geometries do not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. Hence it is important to note that QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry to better reflect the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -840 cm-1 and a corresponding energy, -231.602802 Hartrees was found. The boat-like transition state energy is slightly higher than that calculated for the chair-like transition state and so is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.69157870 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had an imaginary frequency of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T22:14:12Z

Etz13: /* Optimization to Boat TS utilizing QST2 */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the reactant/product and the transition structure. As a result the midpoints of the two geometries does not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. As a result QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry reflecting the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -839.96cm-1 and a corresponding energy, -231.60280197 Hartrees is found. The transition state energy is slightly higher than that calculated for the chair-like transition state previously and so the boat-like transition state is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.69157870 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had an imaginary frequency of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T22:12:24Z

Etz13: /* Comparison of Guess and Frozen Coordinates method */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below (Gaussian has an accuracy to the nearest 2 decimal points). Hence both methods are perfectly viable for calculating the transition structure .

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the initial and final product with the transition structure. As a result the midpoints of the two geometries does not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. As a result QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry reflecting the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -839.96cm-1 and a corresponding energy, -231.60280197 Hartrees is found. The transition state energy is slightly higher than that calculated for the chair-like transition state previously and so the boat-like transition state is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.69157870 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had an imaginary frequency of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T22:11:55Z

Etz13: /* Comparison of Guess and Frozen Coordinates method */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1 (Gaussian has an accuracy to the nearest integer). Furthermore, the bond lengths are the same to 3 decimal points as shown below. Hence both methods are perfectly viable for calculating the transition structure (Gaussian has an accuracy to the nearest 2 decimal points).

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the initial and final product with the transition structure. As a result the midpoints of the two geometries does not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. As a result QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry reflecting the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -839.96cm-1 and a corresponding energy, -231.60280197 Hartrees is found. The transition state energy is slightly higher than that calculated for the chair-like transition state previously and so the boat-like transition state is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.69157870 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had an imaginary frequency of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T22:09:56Z

Etz13: /* Optimization of Chair TS with Guess Method */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and through calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below, a single imaginary frequency of -818 cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1. Furthermore, the bond lengths are the same to 3 decimal points as shown below. Hence both methods are perfectly viable for calculating the transition structure.

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the initial and final product with the transition structure. As a result the midpoints of the two geometries does not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. As a result QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry reflecting the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -839.96cm-1 and a corresponding energy, -231.60280197 Hartrees is found. The transition state energy is slightly higher than that calculated for the chair-like transition state previously and so the boat-like transition state is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.69157870 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had an imaginary frequency of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T22:08:35Z

Etz13: /* Optimization of Chair TS with Guess Method */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. force constant (from the Hessian).

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below a single imaginary frequency of -817.91cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1. Furthermore, the bond lengths are the same to 3 decimal points as shown below. Hence both methods are perfectly viable for calculating the transition structure.

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the initial and final product with the transition structure. As a result the midpoints of the two geometries does not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. As a result QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry reflecting the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -839.96cm-1 and a corresponding energy, -231.60280197 Hartrees is found. The transition state energy is slightly higher than that calculated for the chair-like transition state previously and so the boat-like transition state is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.69157870 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had an imaginary frequency of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T22:08:13Z

Etz13: /* Optimization of Chair TS with Guess Method */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy with respect to the nuclear displacement aka. Force constant (from the Hessian).

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below a single imaginary frequency of -817.91cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1. Furthermore, the bond lengths are the same to 3 decimal points as shown below. Hence both methods are perfectly viable for calculating the transition structure.

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the initial and final product with the transition structure. As a result the midpoints of the two geometries does not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. As a result QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry reflecting the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -839.96cm-1 and a corresponding energy, -231.60280197 Hartrees is found. The transition state energy is slightly higher than that calculated for the chair-like transition state previously and so the boat-like transition state is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.69157870 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had an imaginary frequency of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T22:05:38Z

Etz13: /* Thermochemistry */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
The thermochemical properties of the ''anti2'' conformer at 0 K and 298 K are presented. The 0 K calculation is as expected where the thermal energy is 0. Moreover the thermal energy is shown to be an insignificant factor in the determination of the energy of the structure at 298 K.

{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy (from the Hessian) with respect to the nuclear displacement aka. the force constant.

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below a single imaginary frequency of -817.91cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1. Furthermore, the bond lengths are the same to 3 decimal points as shown below. Hence both methods are perfectly viable for calculating the transition structure.

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the initial and final product with the transition structure. As a result the midpoints of the two geometries does not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. As a result QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry reflecting the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -839.96cm-1 and a corresponding energy, -231.60280197 Hartrees is found. The transition state energy is slightly higher than that calculated for the chair-like transition state previously and so the boat-like transition state is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.69157870 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had an imaginary frequency of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T22:00:26Z

Etz13: /* Thermochemistry */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469203
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy (from the Hessian) with respect to the nuclear displacement aka. the force constant.

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below a single imaginary frequency of -817.91cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1. Furthermore, the bond lengths are the same to 3 decimal points as shown below. Hence both methods are perfectly viable for calculating the transition structure.

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the initial and final product with the transition structure. As a result the midpoints of the two geometries does not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. As a result QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry reflecting the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -839.96cm-1 and a corresponding energy, -231.60280197 Hartrees is found. The transition state energy is slightly higher than that calculated for the chair-like transition state previously and so the boat-like transition state is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.69157870 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had an imaginary frequency of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T21:59:01Z

Etz13: /* Comparison of different levels of theory */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency, and so a minima has been achieved.

====Thermochemistry====
{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469204
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy (from the Hessian) with respect to the nuclear displacement aka. the force constant.

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below a single imaginary frequency of -817.91cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1. Furthermore, the bond lengths are the same to 3 decimal points as shown below. Hence both methods are perfectly viable for calculating the transition structure.

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the initial and final product with the transition structure. As a result the midpoints of the two geometries does not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. As a result QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry reflecting the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -839.96cm-1 and a corresponding energy, -231.60280197 Hartrees is found. The transition state energy is slightly higher than that calculated for the chair-like transition state previously and so the boat-like transition state is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.69157870 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had an imaginary frequency of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T21:56:32Z

Etz13: /* Optimization of 1,5-hexadiene Conformers */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and the presence of CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency as shown below, and so a minima has been achieved.

====Thermochemistry====
{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469204
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy (from the Hessian) with respect to the nuclear displacement aka. the force constant.

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below a single imaginary frequency of -817.91cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1. Furthermore, the bond lengths are the same to 3 decimal points as shown below. Hence both methods are perfectly viable for calculating the transition structure.

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the initial and final product with the transition structure. As a result the midpoints of the two geometries does not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. As a result QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry reflecting the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -839.96cm-1 and a corresponding energy, -231.60280197 Hartrees is found. The transition state energy is slightly higher than that calculated for the chair-like transition state previously and so the boat-like transition state is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.69157870 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had an imaginary frequency of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T21:54:34Z

Etz13: /* Optimization of 1,5-hexadiene Conformers */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] of the script.

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency as shown below, and so a minima has been achieved.

====Thermochemistry====
{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469204
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy (from the Hessian) with respect to the nuclear displacement aka. the force constant.

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below a single imaginary frequency of -817.91cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1. Furthermore, the bond lengths are the same to 3 decimal points as shown below. Hence both methods are perfectly viable for calculating the transition structure.

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the initial and final product with the transition structure. As a result the midpoints of the two geometries does not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. As a result QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry reflecting the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -839.96cm-1 and a corresponding energy, -231.60280197 Hartrees is found. The transition state energy is slightly higher than that calculated for the chair-like transition state previously and so the boat-like transition state is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.69157870 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had an imaginary frequency of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T21:53:27Z

Etz13: /* Background */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement.<ref>A. Almenningen, O. Bastiansen and M. Traetteberg, An Electron Diffraction Reinvestigation of the Molecular Structure of 1,3-Butadiene, Acta Chemica Scandinavica, 12 (1958) 1221-1225</ref> A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed.<ref>L. S. Bartell, R. A. Bonham, Molecular Structure of Ethylene, The Journal of Chemical Physics, 31 (1959) 400-404</ref> However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in the Appendix of the script, [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency as shown below, and so a minima has been achieved.

====Thermochemistry====
{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469204
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy (from the Hessian) with respect to the nuclear displacement aka. the force constant.

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below a single imaginary frequency of -817.91cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1. Furthermore, the bond lengths are the same to 3 decimal points as shown below. Hence both methods are perfectly viable for calculating the transition structure.

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the initial and final product with the transition structure. As a result the midpoints of the two geometries does not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. As a result QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry reflecting the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -839.96cm-1 and a corresponding energy, -231.60280197 Hartrees is found. The transition state energy is slightly higher than that calculated for the chair-like transition state previously and so the boat-like transition state is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.69157870 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had an imaginary frequency of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==

Rep:Mod:ETZ13TS

2016-01-28T21:48:50Z

Etz13: /* Background */

==Introduction==

In this study, the mechanisms of two pericyclic reactions, the Cope rearrangement and Diels-Alder (DA) cycloaddition, were investigated using computational methods. Specifically, the analysis of the transition states of the [3,3] sigmatropic rearrangement of 1,5-hexadiene, DA addition of butadiene and ethylene, and DA addition of maleic anhydride and cyclohexa-1,3-diene.

The mechanistic understanding of reaction schemes relies on the detailed knowledge of transition state structures. However, transition structures correspond to the energy maxima in a reaction, and cannot be isolated and observed. Hence computational calculations are an effective method for the prediction and analysis of transition states. Quantum computational methods were adopted using GaussView 5.0 and Gaussian 09W for this study; These included ''ab initio'' Hartree-Fock, Density Functional Theory B3LYP, and the semiempirical method, AM1. The calculations were performed on basis sets 3-21G, 6-31G(d) and STO-3G, respectively. Three optimisation algorithms were employed for the determination of the transition structures; The Berny Algorithm, Frozen Coordinate Method, and QST2 method.

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

The ''ab initio'' Hartree Fock (HF) method solves for the wavefunction and energy of a system with a few significant approximations to the all-electron Schrodinger equation. Firstly the Born Oppenheimer approximation assumes that the nuclei are frozen with respect to electron movement. Secondly it assumes that each electron behaves independently and moves in a repulsion field produced by the sum of all the other electrons in the system. Thirdly it assumes that the total wavefunction of the electrons is given by a single Slater determinant of the individual spin orbitals, ensuring the antisymmetric property of the wavefunction. Each spin orbital consists of one electron and is a product of the molecular orbital and a spin function. The molecular orbitals consist of a linear combination of atomic orbitals (LCAO), which depend on the basis set used for the approximation. Electron-electron exchange is accounted for by the use of the single Slater determinant for the wavefunction, but electron-electron correlation is not considered. Therefore, HF approximations tend to overestimate the total energy of the system, which may also result in shorter bond length predictions.

Density Functional Theory (DFT) contrasts from ''ab initio'' HF approximations by expressing the total energy in terms of a functional of electron charge density instead of the many electron wavefunctions. The total energy functional consists of a Kohn-Sham kinetic energy, electron nuclear attractive energy, electron electron repulsive energy and the exchange correlation energy functional. The Kohn Sham equation is a pseudo Schrodinger equation which resembles the HF equation except that it uses a classical method for electron electron repulsions. Both electron electron correlation and exchange are accounted for by the exchange correlation energy functional. Hence the accuracy of DFT depends critically on the exchange correlation energy functional. B3LYP is a hybrid functional which involves Becke's three parameter functional and a fraction of the non local electron electron Hartree Fock exchange energy functional and the Lee Young Parr correlation functional.

The basis set represents the mathematical approximations for the shape of the molecular orbitals. Gaussian functions are usually used for the expansion of molecular orbitals. In this study split-valence sets are used, where each valence shell orbital is calculated as the sum of two Slater-type orbitals (STO) multiplied by a proportionality coefficient, ''d''. The Slater Orbitals are a function of radius (r) and zeta (ζ), which determines the diffuseness of the orbital. The inner-shell electrons are defined by a single Slater Orbital. These basis sets are expressed as K-LMG where K, L and M are integers. The basis sets used are K=3/6, L=2/3, and M=1 ie., 3-21G or 6-31G*. In the 3-21G basis set, the inner shell orbital is the sum of three Gaussian functions. The valence orbital, split into the first and second STO, is defined with two and one gaussian functions respectively. The same methodology applies for 6-31G*, where the asterisk accounts for the polarization effect for 'p' orbitals by adding 'd' character.

===Background===
[[File:Etz13 Cope.png|thumb|frame|right|Figure 1. Cope rearrangement|400px]]
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement. A sigmatropic rearrangement refers to a concerted pericyclic reaction where a σ bond is formed and another σ bond is broken. [3,3] refers to the number of carbons between the formed and broken σ bonds. Pericyclic reactions must follow Woodward-Hoffman rules, which state that the sum of the number of (4q + 2)s and (4r)a components must be odd to be thermally allowed. 'S' and 'a' are acronyms for suprafacial and antarafacial respectively. However if it is an even number it is photochemically allowed.

The Cope rearrangement can take place through both chair and boat transition states as they both follow the Woodward-Hoffman rules. For example, the chair transition structure has 3 components, π2a, π2s, σ2a, which add up to 1. The reaction has an equilibrium constant K = 1 as the reactant and product is the same.

Previously, a stepwise mechanism involving a diradical intermediate was proposed. However the concerted pericyclic mechanism is now widely accepted.

===Optimization the Reactants and Products===
====Optimization of 1,5-hexadiene Conformers====

Multiple conformers of 1,5-hexadiene were prepared with GaussView 5.0 and Gaussian 09W. Initially the structures were drawn and cleaned with GaussView. These structures were then optimized using the HF/3-21G level of theory and are presented in the table below, along with the corresponding point group and energy. The point group was then obtained through the symmetrize function. The conformer names correspond to the structures provided in the Appendix of the script, [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable"
!Conformer
!Structure
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_ANTI4.LOG ''Anti4'']
|[[File:Etz13 anti1.png|330px]]
|[[File:Etz13 anti1data.png|250px]]
|style="text-align: center;"| -231.69097
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE4.LOG ''Gauche4'']
|[[File:Etz13 gauche4.png|330px]]
|[[File:Etz13 gauche4data.png|250px]]
|style="text-align: center;"| -231.69153
|style="text-align: center;"|C2
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_GAUCHE3.LOG ''Gauche3'']
|[[File:Etz13 gauche3.png|330px]]
|[[File:Etz13 gauche3data.png|250px]]
|style="text-align: center;"| -231.69266
|style="text-align: center;"|C1
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HFANTI2_TRY2.LOG ''Anti2'']
|[[File:Etz13 hfanti2.png|330px]]
|[[File:Etz13 hfanti2data.png|250px]]
|style="text-align: center;"| -231.69254
|style="text-align: center;"|Ci
|}

[[File:Etz13 geom hexadiene.png|right|thumb|Figure 2. Labelled conformer of 1,5-cyclohexadiene|400px]]
By definition, the 4 central carbon atoms (C2-C3-C4-C5) of the ''anti'' conformers have a dihedral angle of 180°, whereas the ''gauche'' conformers have that of 60°. The factors that differentiate the specific ''anti'' / ''gauche'' conformers from each other are the dihedral angles of C1-C2-C3-C4 and C3-C4-C5-C6. The computed conformer energies are in agreement with the values provided in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. All the conformers adopt A1,3 eclipsed conformations at both of their double bonds. This conformation is preferred over the A1,2 eclipsed conformation electronically due to a good alignment of the σC-H/C and the π*C=C orbitals (θ ~ 0°). This results in 2 hyperconjugation interactions.

As shown, the ''anti2'' has a lower energy than ''anti4'' conformer and thus is more stable. This is because there is a destabilizing A1,3 strain at one of the double bonds of the ''anti2'' conformer. Generally the ''gauche'' conformer is expected to be more unstable than the ''anti'' conformer due to unfavorable steric (gauche) interactions between the two alkenyl groups. However ''gauche3'' is shown to be the most stable. This arises due to an in-phase overlapping π clouds in the HOMO and CH/π hydrogen bonds.<ref>M. Nishio, CH/π hydrogen bonds in crystals, CrystEngComm, 2004, 6(27), 130-158</ref> For the same reason, the ''gauche3'' is more stable than the ''gauche4'' conformer. A Jmol is provided [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_gauche3_conformer_of_1.2C5-hexadiene here].

[[File:Etz13 Pi system.png|thumb|frame|none|Figure 3. Overlap of π electron clouds in HOMO |400px]]

====Comparison of different levels of theory====
The ''anti2'' conformer was further optimized using B3LYP/6-31G(d) as shown below.

{| class="wikitable"
!Conformer
!Structure
!B3LYP/6-31G(d) Calculation
!Energy / hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_B3_ANTI2_TRY3.LOG ''Anti2'']
|[[File:Etz13 b3anti2.png|330px]]
|[[File:Etz13 b3anti2data.png|250px]]
|style="text-align: center;"| -231.61171
|style="text-align: center;"|Ci
|}

As different basis sets and methods are used, the difference in energies cannot be compared. Hence comparisons must be made with regards to the overall geometry. Comparison of the model predictions and experimentally measured bond lengths and angles clearly show that the DFT calculations are more accurate. The molecular orbital calculation using HF/3-21G gives both shorter bond lengths and smaller dihedral angles owing to the neglect of electron-electron interaction energy. The exception is the C3-C4 bond length, which is longer due to the underestimation of the dihedral angle.

{| class="wikitable"
!colspan="4"|Bond Lengths
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / Å
!style="text-align: center;"|HF/3-21G / Å
!style="text-align: center;"|Experimental<ref>G. Schultz and J. Hagittai, Conformational Investigation of Gaseous 1,5-hexadiene by Electron Diffraction and Molecular Mechanics, Journal of Molecular Structure, 346 (1995), 63-69</ref> / Å
|-
|C1-C2
|1.33350
|1.31615
|1.340 ± 0.003
|-
|C2-C3
|1.50420
|1.50893
|1.508 ± 0.012
|-
|C3-C4
|1.54816
|1.55280
|1.538 ± 0.027
|}

{| class="wikitable"
!colspan="4"| Dihedral Angle
|-
!style="text-align: center;"|Atoms
!style="text-align: center;"|B3LYP/6-31G(d) / °
!style="text-align: center;"|HF/3-21G / °
!style="text-align: center;"|Experimental / °
|-
|C1-C2-C3-C4
|118.58635
|114.66373
|120
|-
|C2-C3-C4-C5
|180.00000
|179.99970
|168
|}

The frequency is proportional to the square root of the eigenvalue from the diagonal matrix of the Hessian (Harmonic oscillator equation), where the Hessian is calculated from the second derivatives of energy. If the second derivative is negative, this indicates an energy maximum. Hence, if the eigenvalue is negative, the frequency is imaginary and is represented as a negative number with Gaussian. As a result the presence of negative frequencies indicate that the structure has not been optimized to its minima. A frequency calculation of the ''anti2'' conformer of 1,5-hexadiene at the B3LYP/6-31G level of theory revealed that all the vibrational modes have a positive frequency as shown below, and so a minima has been achieved.

====Thermochemistry====
{| class="wikitable"
!
!Calculation at 298 K / Hartrees
!Calculation at 0 K / Hartrees
|-
|Sum of electronic and zero-point energies
| -234.469215
| -234.469203
|-
|Sum of electronic and thermal energies
| -234.461867
| -234.469204
|-
|Sum of electronic and thermal enthalpies
| -234.460923
| -234.469203
|-
|Sum of electronic and thermal free energies
| -234.500799
| -234.469203
|}

===Optimization of Chair and Boat Transition Structures===

====Optimization of Chair TS with Guess Method====

The Guess method involves creating a "guess" structure which is energetically similar to the transition state and then finding the energy maxima using the Berny Algorithm. The Berny algorithm is an eigenvector following algorithm, where the eigenvectors are obtained through calculating the second derivative of the potential energy (from the Hessian) with respect to the nuclear displacement aka. the force constant.

This method is heavily dependent on the initial "guess" structure which needs to be as close to the transition structure as possible. Hence the allyl fragments are first optimised at the HF/3-21G level of theory.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_HFALLYL.LOG Allyl Fragment Structure]
!HF/3-21G Calculation
!Energy / hartrees
!Point Group
|-
|[[File:Etz13 hfallyl.png|330px]]
|[[File:Etz13 hfallyldata.png|250px]]
|style="text-align: center;"| -115.82304
|style="text-align: center;"|C2V
|}

A "guess" chair transition state is then made, comprimising of two allyl fragments with the terminal carbons 2.2 Å apart. This structure was then optimized to the transition state using the Berny algorithm, and calculating the force constants once. The keyword opt=noeigen is used to prevent Gaussian from crashing when an imaginary frequency is calculated. To verify that a transition state is found, the vibrational spectrum must yield one imaginary frequency, meaning that the energy is a maximum in one direction and a minimum in the other orthogonal directions. As shown below a single imaginary frequency of -817.91cm-1 was found, indicating the transition state.

{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.91cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 hf chair ts.png|x250px]]
|[[File:Etz13 hf chair ts data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 45; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932
|}

====Optimization of Chair TS with the Frozen Method====

Utilizing the redundant coordinate editor, specific potential energy surfaces can be scanned with Gaussian. Firstly, by freezing the terminal carbons of the allyl fragments to be 2.2 Å apart, the lowest energy configuration of the allyl fragments can be found at the HF/3-21G level of theory. The TS is then computed with the Berny algorithm but limiting the direction of movement by keeping the allyl fragments fixed but changing the distance between them. This is achieved through setting the terminal carbons of the allyl fragments as a "derivative".

As shown below, a single negative frequency is found. As described before, this indicates the presence of the transition state.
{| class="wikitable"
!Chair TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-817.89cm-1)
!Energy / hartrees
|-
|[[File:Etz13 chair TS redundant.png|250px]]
|[[File:Etz13 chair TS redundant data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 19; vibration 1;</script>
<uploadedFileContents>ETZ13 CHAIR TS FREEZE2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.61932

|}

====Comparison of Guess and Frozen Coordinates method====

The energies of the Guess and Frozen Coordinates methods are the same to 6 decimal points (-231.61932229 and -231.61932233 Hartrees, respectively). The discrepancy of further decimal points arises from error originating from the step size of the algorithm. Similarly, the imaginary frequencies are equal to 1 decimal point (-817.91 and -817.89cm-1. Furthermore, the bond lengths are the same to 3 decimal points as shown below. Hence both methods are perfectly viable for calculating the transition structure.

{| class="wikitable"
!
!Bond forming bond length / Å
!Bond breaking bond length / Å
|-
|TS from Guess method
|2.02039
|1.38928
|-
|TS from Redundant coordinates method
|2.02071
|1.38930
|}

====Optimization to Boat TS utilizing QST2====

The QST2 method is another method to find the transition state, where the stuctures of the reactant and product are specified. Clearly this method is more advantageous as a 'guess' transition structure is not required (Note that QST3 uses a guess transition structure). QST2 utilizes the Synchronous Transit-Guided Quasi-Newton (SQTN) Method. Firstly it uses a linear synchronous transit approach, where it assumes that the position of the atom in the transition state lies halfway between the position of the reactant and that of the product. Once at the geometry, it uses an eigenvector following algorithm to find the energy maxima.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 reactant initial.png|x250px]]
|[[File:Etz13 qst2 product initial.png|x250px]]
|}

[[File:Etz13 qst2 initial.png|thumb|frame|right|Figure 4. Failed QST2 Transition Structure|300px]]

The first attempt to search for the transition state using QST2 was made at the HF/3-21G level of theory. The labelled ''anti2'' conformers used are shown above. However as shown in '''Figure 4.''', the transition state produced is clearly wrong. This is believed to be due to significant differences in the geometries between the initial and final product with the transition structure. As a result the midpoints of the two geometries does not accurately reflect the transition state structure so the eigenvector following algorithm finds the wrong maxima. As a result QST2 can fail when the geometries of the reactants and products are far from the transition state.

The ''anti2'' conformer was then twisted to form a pseudo boat geometry reflecting the boat-like transition state, as shown below.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
!Reactant
!Product
|-
|[[File:Etz13 qst2 product.png|x250px]]
|[[File:Etz13 qst2 reactant.png|x250px]]
|}

Using the same methodology, the transition state was then calculated. A single vibrational frequency of -839.96cm-1 and a corresponding energy, -231.60280197 Hartrees is found. The transition state energy is slightly higher than that calculated for the chair-like transition state previously and so the boat-like transition state is less stable.

{| class="wikitable"
!Boat TS Structure
!HF/3-21G Calculation
!Imaginary Frequency (-839.96cm-1)
!Energy / hartrees
|-
|[[File:Etz13 qst2 TS.png|250px]]
|[[File:Etz13 qst2 TS data.png|250px]]
|
<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 41; vibration 1;</script>
<uploadedFileContents>Etz13 QST2BOAT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -231.60280
|}

====Intrinsic Reaction Coordinate of Chair TS====

[[File:Etz13 irc 1.png|thumb|frame|right|Figure 5. Total Energy along IRC of the chair transition structure of 1,5-hexadiene|500px]]
[[File:Etz13 irc 2.png|thumb|frame|right|Figure 6. RMS Gradient Norm along IRC of the chair transition structure of 1,5-hexadiene|500px]]
The Intrinsic Reaction Coordinate (IRC) utilizes the Hessian-based Predictor integrator algorithm (HPC) to calculate the forward and backward reaction paths from the transition state structure. In addition, it provides geometries of the structures along the reaction path, defined by the number of points, N. An IRC was run for the chair transition state for 1,5-hexadiene at the HF/3-21G level of theory. The force constants were calculated at each step to ensure an optimum reaction path.

As the reactant and product are the same, the following calculation only calculated the IRC in one direction with N = 50. As shown in '''Figure 5.''' the reaction terminated when N = 44 with an energy of -231.69157870 Hartrees. Furthermore, '''Figure 6.''' shows that this corresponds to the region where the RMS Gradient Norm approaches zero showing that the energy no longer changes. An optimization for the structure was then run at the HF/3-21G level of theory to ensure the minimum energy structure was found. This was revealed to be the ''gauche2'' conformer shown in the appendix of the script.

{| class="wikitable"
![https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:ETZ13_HF_CHAIR_IRC_2_CHECK.LOG Chair TS Structure]
!HF/3-21G Calculation
!Energy / hartrees
|-
|[[File:Etz13 Irc check.png|250px]]
|[[File:Etz13 Irc check data.png|250px]]
|style="text-align: center;"| -231.691667
|}

====Optimization of chair and boat TS with B3LYP/6-31G(d)====

The chair and boat transition structures of 1,5-hexadiene were then reoptimized with the B3LYP/6-31G(d) level of theory. As before this was achieved with the Berny algorithm. The boat and chair transition structures had an imaginary frequency of -530.36 and -565.54cm-1 respectively, indicating the transition structure has been found.

{| class="wikitable"
!Boat Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-530.36cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 boat.png|250px]]
|[[File:B3 boat TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>ETZ13 B3 BOAT TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.543093
|}
{| class="wikitable"
!Chair Transition Structure
!B3LYP/6-31G(d) Calculation
!Imaginary Frequency (-565.54cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 b3 chair.png|250px]]
|[[File:B3 chair TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 13; vibration 1;rotate x -20; </script>
<uploadedFileContents>B3 CHAIR TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -234.556983
|}

====Activation Energies====

The thermochemical data obtained from the frequency calculations of the ''anti2'' conformer, boat TS using HF/3-21G and B3LYP/6-31G levels of theory are presented below.
{| class="wikitable"
!
!colspan="3" | HF/3-21G
!colspan="3" | B3LYP/6-31G(d)
|-
!
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
!width="125" | Electronic Energy / Hartrees
!width="125" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="125" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Chair TS
| -231.619322
| -231.466699
| -231.461340
| -234.556983
| -234.414929
| -234.409009
|-
!Boat TS
| -231.602802
| -231.450928
| -231.445299
| -234.543093
| -234.402342
| -234.396008
|-
!Reactant (''anti2'')
| -231.692535
| -231.539540
| -231.532566
| -234.611710
| -234.469203
| -234.461856
|}

The activation energy could then be calculated and converted to kcal mol-1. The difference in levels of theory HF/3-21G calculations significantly overestimate the activation energies whereas B3LYP/6-31G(d) slightly underestimate the activation energies. This discrepancy is attributed to the higher electron repulsion in the HF meanfield approximation method.

{| class="wikitable"
!
!width="125" | HF/3-21G at 0K
!width="125" | HF/3-21G at 298.15K
!width="125" | B3LYP/6-31G(d) at 0K
!width="125" | B3LYP/6-31G(d) at 298.15K
!width="125" | Expt. at 0K
|-
|ΔE (Chair) / kJ mol-1
| 45.71
| 44.69
| 34.06
| 33.16
| 33.5 ± 0.5
|-
|ΔE (Boat) / kJ mol-1
| 55.60
| 54.76
| 41.96
| 41.32
| 44.7 ± 2.0
|}

==The Diels Alder Cycloaddition==

===Background===

====Diels-Alder====

[[File:Etz13 Da mech.png|Figure 7. Cis-butadiene and ethylene|thumb|right|300px]]
[[File:Etz13 Maleic mech.png|Figure 8. Maleic Anhydride and cyclohexa-1,3-diene exo and endo selectivity|thumb|right|300px]]
The Diels-Alder reaction is a [4+2] cycloaddition. It is a pericyclic reaction involving the concerted formation of two σ bonds to join together two conjugated π systems. The [4+2] refers to the number of π electrons in each component, where 4 represents the diene and 2 the dienophile. The diene is required to adopt an s-''cis'' conformation. Electron withdrawing groups (EWG)/conjugation on the dienophile favour the reaction as it lowers the energy of the LUMO. Similarly, electron releasing groups (ERG) on the diene favour the reaction as it increases the energy of the HOMO. This allows for a better interaction between the HOMO of the diene and the LUMO of the dienophile

The reaction is regioselective, where an ERG group on the diene prefers to be ortho or para to the EWG on the dienophile. This can be deduced from a frontier molecular orbital analysis. It is also stereoselective and follows the ''Alder Rule'', which states that the ''endo'' (kinetic) product is preferably formed in irreversible reactions. The ''endo'' product is defined where the dienophile substituent faces the diene. This is due to secondary orbital effects between the π systems as well as other effects to be mentioned later. Furthermore it is thermally allowed as a Woodward Hoffman analysis of its components, π4s and π2s yields a sum of 1 (odd).

In the following sections the Diels-Alder reaction of ''cis''-butadiene with ethylene, and cyclohexa-1,3-diene with maleic anhydride will be examined. The strongly electron withdrawing character of maleic anhydride make the Diels Alder reaction highly favourable.

Previous proposals included a stepwise mechanism but has been since revealed to have a higher activation barrier.<ref>E. Goldstein, B. Beno, K. N. Houk, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanisms of the Diels-Alder Reaction of Butadiene and Ethylene, J. Am. Chem. Soc. 1996, 118, 6036-6043</ref>

====Computational Background====

In the following calculations, the Semi-empirical/Austin Model 1 (AM1) method is used. Semi-empirical methods use Hartree Fock for its framework but include approximations and parameters based on empirical (experimental) data. It uses a minimum basis set (STO-3G) for the occupied orbitals of the valence electrons and considers the core electrons only for their nuclear shielding effect. Furthermore, the AM1 is an extension of the NDDO (Neglect of Differential Diatomic Orbital Overlap) integral approximation by introducing radial Gaussian functions to modify the core-core repulsions. The advantage of the semi-empirical method is its computational efficiency and suitability for large systems. Its disadvantage lies in its limited accuracy which is dependent on experimental data.

===Diels Alder Cycloaddition of ''cis''-butadiene and ethylene===
====Optimization of ''cis''-butadiene and ethylene====
''Cis''-butadiene and ethylene were drawn and cleaned with GaussView. The structures were then optimized at the Semi-Empirical/AM1 level of theory. A further frequency analysis revealed that there were no negative frequencies showing that they were the minimum energy structures.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CIS_BUTADIENE_ORBITALS.LOG ''Cis''-butadiene]
|[[File:Etz13 Cis butadiene.png|330px]]
|[[File:Etz13 Cis butadiene data.png|250px]]
|style="text-align: center;"| 0.04879718
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_ETHENE_ORBITALS_2.LOG Ethylene]
|[[File:Etz13 Ethene.png|x220px]]
|[[File:Etz13 Ethene data.png|250px]]
|style="text-align: center;"| 0.02619027
|style="text-align: center;"| D2h
|}

With respect to the plane perpendicular to the central C-C bond, the HOMO of ''cis''-butadiene is antisymmetric, and the LUMO is symmetric. With respect to the plane perpendicular to the C=C bond, the HOMO for ethylene is symmetric, and the LUMO is antisymmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Cis-butadiene
|[[File:Etz13 Cis butadiene HOMO.png|x250px]]
|[[File:Etz13 Cis butadiene LUMO.png|x250px]]
|-
|style="text-align: center;"|Ethylene
|[[File:Etz13 Ethene HOMO.png|center|x250px]]
|[[File:Etz13 Ethene LUMO.png|center|x250px]]
|}

====Optimization of cis-butadiene and ethylene TS====

Despite the ethylene not having an electronegative substituent, it is assumed that this reaction can still proceed as the difference in energy between the HOMO of the diene and the LUMO of the dienophile is only 0.40483 Hartrees. The transition structure of this reaction was found using the Berny Algorithm using the Semi-Empirical/AM1 level of theory. The guess structure consisted of the optimized diene and dienophile with the terminal carbons 2 Å apart, which provided a satisfactory initial Hessian.

A lone imaginary frequency of -956.24 cm-1 was obtained indicating the presence of the transition state, showing the concerted formation of two σ bonds as the bonds are stretched and compressed. The next lowest frequency is 147.24 cm-1. As previously mentioned, this shows that there is only one energy maxima, corresponding to the imaginary frequency vibration and the rest of the vibrations result in an energy minima. As shown below, the vibration with the frequency 147.24cm-1 shows a rotation of the dienophile relative to the diene (or vice versa), and does not involve any bond compression or extension which could be regarded as the formation or breakage of bonds.
{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-956.24cm-1)
!Energy / Hartrees
!Lowest Positive Frequency (147.24cm-1)
|-
|[[File:Etz13 Butadiene ethene DA.png|250px]]
|[[File:Etz13 Butadiene ethene DA data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 29; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 30; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 BUTADIENE ETHENE DA.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

|}

Relative to the plane perpendicular to the central C-C bond of the diene and C=C bond of ethene, the HOMO is antisymmetric and the LUMO is symmetric. The HOMO consists of the addition of the HOMO of ''cis''-butadiene (antisymmetric) and the LUMO of ethylene (antisymmetric). The LUMO consists of the addition of the LUMO of ''cis''-butadiene (symmetric) and the HOMO of ethylene (symmetric). Links to Jmol: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#HOMO_of_cis-butadiene_ethylene_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals#LUMO_of_cis-butadiene_ethylene_TS LUMO]

{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 butadiene ethene HOMO.png|x300px]]
|[[File:Etz13 butadiene ethene LUMO.png|x300px]]
|}

[[File:Etz13 geometry butadiene ethene.png|thumb|frame|Figure 9. Labelled Transition State diagram|450px]]

Through the analysis of the bond lengths of the transition state, one can examine the partially forming and breaking of bonds. For reference, the van der Waals radii of carbon is 1.70 Å, Csp3-Csp3 bond lengths are 1.54 Å and C=C double bond lengths are 1.34 Å. In addition, only half of the bond lengths are examined as the transition state has an internal mirror plane (meso).<ref>Fox, Marye Anne; Whitesell, James K. Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. (1995)</ref>

Twice of the van der Waals radius indicates the minimum distance for two atoms to be non-bonding. As shown, the distance between the two terminal atoms, C1 and C3, are within this distance and hence shows a bonding interaction. This indicates the formation of the new σ bond. Simultaneously, the C1-2, C3-4 bond length increases so that the atoms gain sp3 character (as the single bond forms) to facilitate for the new σ bond formations between C1 and C3. This also results in the C4-C5 bond compression, forming a double bond and the carbons gaining sp2 character. Note that this all happens in a concerted fashion.

{| class="wikitable"
!Atoms
!Bond Length / Å
|-
|C1-C2
|1.38291
|-
|C3-C4
|1.38186
|-
|C4-C5
|1.39748
|-
|C1-C3
|2.11927
|}

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

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

Maleic anhydride and cyclohexa-1,3-diene are drawn and cleaned in GaussView and optimised in Gaussian with the Semi-empirical/AM1 level of theory. Further frequency analysis showed no negative frequencies in the IR spectrum indicating that the minimum energy structure has been generated.

{| class="wikitable"
!Molecule
!Structure
!Semi-Empirical/AM1 Calculation
!Energy / Hartrees
!Point Group
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_MALEIC_ANHYDRIDE_ORBITALS.LOG Maleic Anhydride]
|[[File:Etz13 Maleic anhydride.png|330px]]
|[[File:Etz13 Maleic anhydride data.png|250px]]
|style="text-align: center;"| -0.12182418
|style="text-align: center;"|C2v
|-
|style="text-align: center;"|[https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:Etz13_CYCLOHEXADIENE_ORBITALS.LOG Cyclohexa-1,3-diene]
|[[File:Etz13 Cyclohexadiene.png|x220px]]
|[[File:Etz13 Cyclohexadiene data.png|250px]]
|style="text-align: center;"| 0.02771129
|style="text-align: center;"| C2
|}

For maleic anhydride, with respect to the plane perpendicular to the C=C bond, the HOMO is symmetric and the LUMO is antisymmetric. For cyclohexa-1,3-diene, with respect to the plane perpendicular to the sp2-sp2 single bond, the HOMO is antisymmetric and the LUMO is symmetric.

{| class="wikitable"
!Molecule
!HOMO
!LUMO
|-
|style="text-align: center;"|Maleic Anhydride
|[[File:Etz13 Maleic anhydride HOMO.png|x250px]]
|[[File:Etz13 Maleic anhydride LUMO.png|x250px]]
|-
|style="text-align: center;"|Cyclohexa-1,3-diene
|[[File:Etz13 Cyclohexadiene HOMO.png|x250px]]
|[[File:Etz13 Cyclohexadiene LUMO.png|x250px]]
|}

====Optimization of Exo Transition State====

The ''exo'' transition state was optimized using the Berny Algorithm at the Semi-empirical/AM1 level of theory. In order to provide a suitable guess structure so that the initial Hessian's eigenvectors can lead to the transition state, the ''exo'' product was firstly optimised. This was achieved through drawing and cleaning the ''exo'' product, and optimising it at Semi-empirical/AM1 level of theory. The bond lengths between the maleic anhydride and the cyclohexa-1,3-diene fragments were then elongated to 2 Å and the double bond on the cyclohexa-1,3-diene fragment was elongated to 1.45 Å. These chosen lengths were roughly based on the values from the ethene ''cis''-butadiene transition structures. This method was successful as a single imaginary frequency of -812 cm-1 was found, indicating an energy maxima for the vibration. Furthermore the vibration shows the elongation and compression of the bonds to form the ''exo'' product structure.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-812.16cm-1)
!Energy / Hartrees
|-
|[[File:Etz13 Exo maleic TS.png|250px]]
|[[File:Etz13 Exo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 25; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 EXO DA TRY3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| 0.11165474
|}

With respect to the plane perpendicular to the C=C double bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Exo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Exo maleic TS HOMO.png|x300px]]
|[[File:Etz13 Exo maleic TS LUMO.png|x300px]]
|}

====Optimization of Endo Transition State====

The ''endo'' transition state was optimised in a similar fashion as the ''exo'' transition state, where the only factor that changes is that the ''endo'' product is used. This method succeeds as a single imaginary frequency of -806 cm-1 is generated. This vibrational mode shows the bond elongation and compression to generate the ''endo'' product.

{| class="wikitable"
!Structure
!Semi-Empirical/AM1 Calculation
!Imaginary Frequency (-806.43cm-1)
!Energy / Hartrees
|-
|[[File:Endo maleic TS.png|250px]]
|[[File:Etz13 Endo maleic TS data.png|250px]]
|<jmol>
<jmolApplet>
<color>white</color>
<size>250</size>
<script>frame 27; vibration 1;rotate x -20; </script>
<uploadedFileContents>Etz13 ENDO DA TRY3 TS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|style="text-align: center;"| -0.16017081
|}

With respect to the plane perpendicular to the C=C bond, the HOMO and LUMO are both antisymmetric. Jmol hyperlinks: [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#HOMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS HOMO], [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:ETZ13TSOrbitals2#LUMO_of_maleic_anhydride_and_cyclohexa-1.2C3-diene_Endo_TS LUMO]
{| class="wikitable"
!HOMO
!LUMO
|-
|[[File:Etz13 Endo maleic TS HOMO.png|x300px]]
|[[File:Endo maleic TS LUMO.png|x300px]]
|}

====Comparison of the activation energies of the endo and exo TS====

Although the ''endo'' TS is more sterically crowded than that of the ''exo'' TS, the ''endo'' TS has a lower energy and so results in the kinetic product. This preference has been coined the ''endo rule''. In general this has been attributed to secondary orbital interactions, greater dispersion forces and a greater electrostatic attraction. The secondary orbital interaction for the ''endo'' TS originates from the overlap of the π orbitals of the cyclohexa-1,3-diene with the C=O π orbitals from maleic anhydride.

Fernández ''et al.'' further elucidated this selectivity using the activation strain model, and proposed that the difference arises from the energy required to deform the reactants.<ref>I. Fernandez, F. M. Bickelhaupt, Origin of the "endo rule" in Diels-Alder reactions, Journal of Computational Chemistry, 2014, 35, 371-376</ref> For the exo transition state, the oxygen on maleic anhydride and the Csp3-Csp3 on cyclohexa-1,3-diene are repelled due to Pauli repulsion of the O lone pairs and C-H sp3 orbitals, causing a significant deformation. Hence the strain energy required for the deformation is higher for the ''exo'' TS than ''endo'' TS.

An analysis of the nodal properties of the HOMO between the -(C=O)-O-(C=O)= fragment and the remainder of the system reveals the orbitals ***

{| class="wikitable floatright"
!
!Semi-empirical/AM1 at 0K
!Semi-empirical/AM1 at 298.15K
|-
|ΔE Exo TS / kcal mol-1
| 28.69
| 28.46
|-
|ΔE Endo TS / kcal mol-1
|27.82
|27.70
|}
{| class="wikitable"
!
!colspan="3" | Semi-Empirical/AM1
|-
!
!width="150" | Electronic Energy / Hartrees
!width="150" | Sum of electronic and zero-point energies at 0K / Hartrees
!width="150" | Sum of electronic and thermal energies at 298.15K / Hartrees
|-
!Exo TS
|style="text-align: center;"| -0.050420
|style="text-align: center;"| 0.134879
|style="text-align: center;"| 0.144881
|-
!Endo TS
|style="text-align: center;"| -0.051505
|style="text-align: center;"| 0.133494
|style="text-align: center;"| 0.143683
|-
!Maleic Anhydride
|style="text-align: center;"| -0.121824
|style="text-align: center;"| -0.063346
|style="text-align: center;"| -0.058192
|-
!Cyclohexa-1,3-diene
|style="text-align: center;"| 0.027711
|style="text-align: center;"| 0.152502
|style="text-align: center;"| 0.157726
|}

====Comparison of bond lengths of exo and endo TS====
{| class="wikitable floatright"
!Bond
!Exo TS / Å
!Endo TS / Å
|-
|C1-C2
|1.397
|1.397
|-
|C2-C3
|1.394
|1.393
|-
|C3-C4
|1.490
|1.491
|-
|C4-C5
|1.522
|1.523
|-
|C6-C7
|2.170
|2.162
|-
|C7-C8
|1.410
|1.408
|-
|C8-C9
|1.488
|1.489
|-
|C9-O10
|1.410
|1.409
|}

Steric congestion can be observed in the exo transition structure as the C4-C9 bond distance (2.95 Å) is smaller than the combined van der Waals radii (3.40 Å). As a result, the ''exo'' has a larger interfragment distance C6-C7 than that of the ''endo'' structure. Although they both are within the combined van der Waals radii which indicate a bonding interaction of in phase orbitals, the ''endo'''s distance is smaller indicating a greater orbital overlap and stabilization.

The remaining bonds are equal in length and indicate the extension and compression of bonds as the reactant transitions to the product and vice versa.

{| class="wikitable"
!Labelled Exo TS Diagram
!Labelled Endo TS Diagram
|-
|[[File:Etz13 Exo maleic TS geom.png|x300px]]
|[[File:Etz13 Endo maleic TS geom.png|x300px]]
|}

==Conclusion==
The computational study of the Cope rearrangement of 1,5-hexadiene has further confirmed that the reaction proceeds ''via'' a concerted pericyclic mechanism instead of a previously debunked theory of the diradical intermediate through an analysis of the transition state. Furthermore, the calculations have revealed that although both the chair and boat transition states are thermally allowed, the chair transition state is energetically favoured. Furthermore the limitations of the transition state determination algorithms were discussed; the Berny algorithm needs a suitable 'guess' structure, and the Frozen Coordinate method is a better alternative. The QST2 method requires the reactant and products to be structurally similar to the transition state. Finally, the calculations have also highlighted the limits of HF theory, showing that the predicted energies are higher than experimental values (36% and 24% for the chair and boat respectively).

For the study of the Diels Alder cycloaddition of ''cis''-butadiene with ethylene, further understanding of the orbital interactions was developed. The HOMO and LUMO interactions of the reactants were examined and were related to the formed products. The Berny algorithms limitations were demonstrated again.

Finally for the study of the Diels Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene, the ''endo'' rule was examined. The interactions destabilizing the ''exo'' transition state included the strain energy to deform the reactants and the steric congestion of the transition state, which limited the inphase orbital bonding interactions. The interactions stabilizing the ''endo'' transition state included secondary orbital effects. The advantages of the Berny Algorithm were shown with this method as it provided a fast and facile way to locate the transition state.

==References==