==Introduction==

In this computational experiment, the Cope rearrangement of 1,5-hexadiene and two examples of Diels-Alder cycloaddition reaction were investigated. The structures of the transition state of these reactions are the focus of study. The geometry and energies of the transition structures were analysed. The Molecular orbitals of the reactants and transition states for the Diels-Alder reaction were studied in order to understand their interactions and the geometry of the transition structure. All molecules were modeled using GaussView 5.0 and all calculation was done by the Gaussian programme.

Gaussian applies the Born-Oppenheimer Approximation in which the nuclei is much heavier than the electrons and hence the motion of nuclei and electrons can be treated separately. A potential energy surface (PES) is created based on this approximation. It represents the potential energy as a function of the internuclear distance in a reaction. The use of different method and basis set for the calculation would affect the accuracy of the results compared to experimental data. These are discussed in the following sections.

==The Cope Rearrangement Tutorial==

===Optimization of 1,5-Hexadiene===

{| class="wikitable"
|+ Table 1 Optimized structure of 1,5-hexadiene
! !!Optimised "anti" 1,5-hexadiene molecule !! Optimised "gauche" 1,5-hexadiene molecule
|-
|Jmol
||
<jmol><jmolApplet>
<color>black</color>
<size>280</size>
<uploadedFileContents>15 HEXADIENE ANTI OPT6 HF.mol</uploadedFileContents>
</jmolApplet></jmol>
||
<jmol><jmolApplet>
<color>black</color>
<size>285</size>
<uploadedFileContents>15 HEXADIENE GAUCHE C1 OPT7 HF.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
|Method/Basic set
|HF/3-21G
|HF/3-21G
|-
|Energy (a.u.)
| -231.6853962
| -231.6926612
|-
|Point group
|C2h
|C1
|}

[[File:Newnabd projection for 15 hexadiene.jpg|frame|400px|Figure 1. Newman projection of conformers of 1,5-hexadiene. a) "Anti" structure, C2h. b) "Gauche" Structure, C1]]

The free rotations about the C-C single bonds give rise to many possible conformations in 1,5-hexadiene. Table 1 shows two optimizied 1,5-hexadiene molecules. One is antiperiplanar and the other is gauche. By comparing to [[Mod:phys3#Appendix 1|Appendix 1]], they are anti 3 and gauche 3. It was predicted that the 1,5-hexadiene with gauche linkage at the centre of the molecule would have a higher energy then the "anti" structure. The vinyl groups at the end of hexadiene are closer together in the gauche structure than in the "anti" structure. The gauche structure has a dihedral angle of 60 degree at the centre and repulsive steric interaction was expected to result in an increase in energy.

However, it was shown that the "anti" 1,5-hexadiene has a higher energy compared to the gauche conformer which has a slightly lower energy. By comparing to the table shown in [[Mod:phys3#Appendix 1|Appendix 1]], it was also found that the C1 gauche conformer is the lowest energy conformation of 1,5-hexadiene.

A possible explanation to this is that the gauche structure is stabilised by an attractive interaction between the protons on one vinyl group and the π-orbital on the other. A vinyl proton is covalently bonded to a carbon atom and weakly interacting with the π-orbital of the double bond. This is known as the CH/π interaction. In the antiperiplanar structure (Fig. 1a), such interaction is not possible as the vinyl groups are far apart. In the gauche structure (Fig. 1b), the vinyl groups are close to each other and therefore it is stabilised by this interaction.

1,5-hexadiene with an "anti" linkage, Ci conformation

{| class="wikitable"
|+ Table 2 Optimized structure of "anti" 1,5-hexadiene, Ci
! Method/basis set !! HF/3-21G !! DFT/6-31G*
|-
|Jmol
||
<jmol><jmolApplet>
<color>black</color>
<size>280</size>
<uploadedFileContents>15 HEXADIENE ANTI OPT15 HF 2.mol</uploadedFileContents>
</jmolApplet></jmol>
||
<jmol><jmolApplet>
<color>black</color>
<size>280</size>
<uploadedFileContents>15 HEXADIENE CI ANTI OPT16 DFT 631D.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
|Labelled molecule
|[[File:1,5 hexadiene HF Anti Ci Optimization.JPG|300px]]
||[[File:1,5 hexadiene DFT Anti Ci Optimization.JPG|300px]]
|-
|Energy (a.u.)
| -231.695353
| -234.559704
|}

The CI "anti 2" 1,5-hexadiene optimized at the HF/3-21G level of theory has an energy of -231.6925353 au. This value is the same as the one given in [[Mod:phys3#Appendix 1|Appendix 1]]. This is subsequently re-optimised at B3LYP/6-31G* level and yield a lower energy form than the one at HF/3-21G level. The structure from the HF/3-21G calculation closely resembles that from B3LYP/6-31G* calculation. Table 3 summarizes the dihedral angles and the bond lengths of both structures. The centre dihedral angle and all carbon-carbon bond lengths are similar in both 1,5-hexadiene. There is only a 4 degrees difference in the terminal dihedral angle between them. Overall, the change in geometry is minimal.

{| class="wikitable"
|+ Table 3 Geometry data "anti" 1,5-hexaidene optimized at HF/3-21G and DFT/6-31G* level; Ci
! Method !!colspan="3"| HF !! colspan="3"| DFT
|-
| Dihedral angle(C1-C4-C6-C9);(º) || colspan="3" align="center" |114.7 || colspan="3" align="center"| 118.8
|-
| Dihedral angle(C4-C6-C9-C12);(º) || colspan="3" align="center"|180.0 || colspan="3" align="center"| 180.0
|-
| || C1-C4 || C4-C6 || C6-C9 || C1-C4 || C4-C6 || C6-C9
|-
| align="center"| Bond length(Å) || 1.07 || 1.33 || 1.51 || 1.09 || 1.34 || 1.51
|}

===Frequency Analysis of "anti" 1,5-hexadiene, Ci conformation; DFT/6-31G===

[[Image:1,5 hexadiene DFT Anti Ci Freq spectrum.JPG|frame|centre|400px|Figure 2 Vibrational Spectrum of "anti" 1,5-hexadiene]]

Frequency analysis was carried out. It gives the second derivative of the potential energy surface. If all frequencies are positive, it means a minimum was resulted. The absence of imaginary (negative) frequencies shows that the structure is optimized to a minima. Table 4 shows the thermochemical analysis of the optimized structure.

{| class="wikitable"
|+ Table 4 Summary of energy
! !! Energy (in hatree)
|-
| Sum of electronic and zero point energies (E = Eelec + ZEP), at 0 K || align="center" |-234.469215
|-
| width="430" | Sum of electronic and thermal energies (E = E + Evib + Erot + Etrans), at 298.15 K and 1 atm|| align="center"| -234.461867
|-
| Sum of electronic and thermal enthalpies+ || align="center"| -234.460922
|-
| Sum of electronic and thermal free energies++ ||align="center"| -234.500800
|}

+ An additional correction for RT(H = E + RT)

++ Including entropic contribution to the free energy (G = H-TS)

===Optimization of "Chair" Transition Structures===

The Cope rearrangement have two different transition state: Chair and Boat.

Optimization and Frequency Analysis of Chair Transition Structure (Opt+Freq)

An allyl fragment (CH2CHCH2) was first optimized to TS(Berny) at HF/3-21G level. Two optimized fragments were arranged in the chair form and underwent optimization and frequency analysis. This optimized structure has an imaginary frequency at -818 cm-1. The negative second derivative of the potential energy surface corresponds to a maxima. This shows that the optimization of the chair transition state structure was successful. The imaginary frequency was also animated in table 5.

{| class="wikitable"
|+ Table 5 Results of optimizaed chair transition structure
! Jmol || colspan="3"|Animated vibration at -818 cm-1 || Vibrational spectrum
|-
|
<jmol><jmolApplet>
<title>(3-21G) optimized chair transition state</title>
<color>black</color>
<size>280</size>
<uploadedFileContents>TRANSITION ALLYL FRAG HF OPTFREQ5.mol</uploadedFileContents>
</jmolApplet></jmol>
|
|[[File:Imaginary vibration chair transition state animation.gif|50 x 50 px]]
|
|[[File:Trasition state OPT FREQ IR spectrum.JPG|400 px]]
|}

Alternatively the frozen coordinate method was used to optimize the transition structure. This was done by fixing the distance between the terminal carbons from both allyl fragments to 2.2 Å and then optimized to a minimum (HF/3-21G). A transition state optimization to TS(Berny) was carried out subsequently at HF/3-21G level. This allows the bond forming/breaking distances between the two fragments to be optimized as well. The table below summarizes the geometry data of the transition structures that were optimized differently.

Both optimized transition structures with either frozen or optimized bond forming/breaking distances, show similar C-C bond length and C-C-C angle within one allyl fragment. These are also similar in values compared to the structure from "Opt+Freq" calculation. The main difference lies in the distance between C1-C6 and C3-C4. When the bond forming/breaking distances were optimized, these values are more similar to that in the structure from "Opt+Freq" calculation. This shows that freezing the coordinate would give a less accurate optimization of structure.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Table 6 Geometry data of optimized chair transition structure
! !! Width="120"| Bond length (C1-C2)(Å) !! Width="120"|Angle (C1-C2-C3)(º) !! Width="120"| Distance between (C1-C6) (Å)!! Width="120"|Distance between (C3-C4) (Å)!! rowspan="4" |[[File:Chair Transition State Freeze HF Optimization with number.JPG|250 px]]
|-
| align="center"| Opt + Freq|| height="50" align="center"| 1.39 || align="center"|120.5 || align="center"|2.02 || align="center"|2.02
|-
| align="center"| Opt(Freeze Coordinate) || align="center"| 1.39 || align="center"| 121.8 || align="center"| 2.16 || align="center"| 2.20
|-
| align="center"| Opt(Derivative) || align="center"| 1.39|| align="center"| 120.5|| align="center"| 2.02 || align="center"| 2.02
|}

===Optimization of "Boat" Transition Structures===

{|style="margin: 0 auto;"
|[[File:Failed boat transition state.PNG|thumb|200 px|Figure 3. First attempt of QST2 calculation]]
|[[File:Boat QST2 rearrangement.JPG|thumb|350px|Figure 4. Rearrangement of butadiene]]
|[[File:Cope rearrangement scheme 2.JPG|thumb|200px|Figure 5. Cope Rearrangement]]
|}
The optimized Ci "anti" 1,5-hexadienes were optimized to a transition state and frequency analysis were carried out using the QST2 method. QST2 requires reactant and product as the input and all atoms must be labelled in the same way in both structure. The first calculation was done without any modification to the structure orientation. The job was failed and resulted in the transition structure shown in figure 3. The 1,5-hexadiene molecules were re-orientated so that they had the same arrangement as what shown in figure 4. The modified molecules had a dihedral angle of 0 degree at the centre and 100 degrees for the inside C-C-C angle. The QST2 calculation of the modified structure was successful and the following results (table 7) were obtained. The distance between the two fragments is 2.14 Å. The boat transition structure was optimized and it has an imaginary frequency at -840 cm-1.

The optimization was also carried out using the QST3 calculation. This requires 3 inputs in the following order: the reactant, product, and guess transition state structures. Similar to QST2, the atoms must be labelled in the same order. The energy and geometry of the optimized structure of QST3 calculation resembles that of QST2. It also has an imaginary frequency at -840 cm-1.

{| class="wikitable"
|+ Table 7 Results of optimized boat transition structure
! Jmol || colspan="3"|Vibration at -840 cm-1|| Vibrational spectrum
|-
|
<jmol><jmolApplet>
<title>(3-21G) optimised boat transition state</title>
<color>black</color>
<size>280</size>
<uploadedFileContents>BOAT TRANSITION HF OPTFREQ13 QST2.mol</uploadedFileContents>
</jmolApplet></jmol>
|
|[[File:Imaginary vibration Boat transition state QST2 animation.gif|50 x 50 px ]]
|
|[[File:Boat Transition State QST2 IR spectrum.JPG|400 px]]
|}

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

{| class="wikitable"
|+ Table 8 Results of IRC calculation
! First Calculation || Total Energy along IRC || RMS Gradient Norm along IRC
|-
|[[File:IRC Point 50 Chair Transition Forward Direction Always calculate force constant.gif|50 x 50 px|frame|centre|No. of points along IRC: 50]]
|[[File:IRC Point 50 Chair Transition Total Energy along IRC.JPG|400 px]]
|[[File:IRC Point 50 Chair Transition RMS Gradient Norm along IRC.JPG| 350 px]]
|}

It is difficult to predict which conformers of 1,5-hexadiene will form from the chair and boat transition structures. Intrinsic Reaction Coordinate (IRC) method was used to find out the structure that has the lowest energy. It allows the lowest energy reaction path from the transition state towards the reactants and products to be followed. Only the forward direction of the reaction coordinate was considered here. The number of data points along the IRC was set to 50 and the force constant was set to "calculate always" in the first attempt. Forty-four intermediates were obtained. A second attempt of IRC calculation with 100 points was carried out to ensure the minimum energy geometry was reached. There was no change to the energy graph and the gradient was closed to zero at the end of calculation. These prove that a minimum geometry has reached. A gauche conformer with an energy of -231.691608 a.u. (gradient: 0.00015154 a.u.) was found to be the minimum geometry from this calculation. This is gauche 2 in [[Mod:phys3#Appendix 1|Appendix 1]].

===Optimization of Chair and Boat Transition structures using B3LYP/6-31G*===

The HF/3-21G optimized chair and boat structure were re-optimized using B3LYP/6-31G* method. The following tables present a comparison for the geometries and different energies values. The chair transition structures optimized at HF/3-21G and B3LYP/6-31G* have very similar geometry compared to each other. The same applies to the boat transition structure. However, the energies are lower for the transition structures optimized at B3LYP/6-31G* level.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Table 9 Geometry data of chair and boat transition structure
! || height="40" colspan="3"|HF/3-21G || colspan="3"|B3LYP/6-31G* || rowspan="2" |[[File:Chair Transition State Freeze HF Optimization with number.JPG|170 px]]
|-
! !! Width="120" height="40"| Bond length (C1-C2)(Å) !! Width="120"|Angle (C1-C2-C3)(º) !! Width="120"| Distance between (C1-C6)/(C3-C4) (Å)!! Width="120"| Bond length (C1-C2)(Å) !! Width="120"|Angle (C1-C2-C3)(º) !! Width="120"| Distance between (C1-C6)/(C3-C4) (Å)
|-
| align="center"| Chair TS (Top)|| height="50" align="center"| 1.38 || align="center"|122.0 || align="center"|2.20 || align="center"|1.39 ||align="center"| 122.0 || align="center"|2.20 || rowspan="2" |[[File:Boat Transtion numbering.JPG|170 px]]
|-
| align="center" height="60"| Boat TS (bottom)|| align="center"| 1.41 || align="center"| 121.2 || align="center"| 2.14 || align="center"| 1.39 || align="center"| 121.1 || align="center"| 2.14
|}

{| class="wikitable"
|+ Table 10 Summary of Energy (in hatree)
! || colspan="3"|HF/3-21G || colspan="3"|B3LYP/6-31G*
|-
| || align="center" | Electronic energy || width="125" align="center" | Sum of electronic and zero-point energies (0 K)|| width="130" align="center" | Sum of electronic and thermal energies (298.15 K)|| align="center" | Electronic energy || width="125" align="center" | Sum of electronic and zero-point energies (0 K)|| width="130" align="center" | Sum of electronic and thermal energies (298.15 K)
|-
|align="center" | Chair TS|| align="center" | -231.619332 || align="center" | -231.466702 || align="center" | -231.461343 || align="center" | -234.553938 || align="center" | -234.413269 || align="center" | -234.406982
|-
|align="center" | Boat TS|| align="center" | -231.602802 || align="center" | -231.450928 || align="center" | -231.445299 ||align="center" | -234.542868 || align="center" | -234.401492 || align="center" | -234.395284
|-
|align="center" | Reactant (Ci; Anti)|| align="center" | -231.692535 || align="center" | -231.539539 || align="center" | -231.532565 || align="center" | -234.611712|| align="center" | -234.469215 || align="center" | -234.461867
|}

===Calculation of Activation Energies for Both Transition Structures===
{| class="wikitable"
|+ Table 11 Summary of Activation Energy (in kcal/mol)
! || colspan="2"|HF/3-21G || colspan="2"|B3LYP/6-31G* || Experimental value from [[Mod:phys3#Appendix 1|Appendix 1]]
|-
| align="center" | Temperature || width="50" align="center" | 0 K || align="center" | 298.15 K || width="50" align="center" | 0 K || align="center" | 298.15 K || align="center" | 0 K
|-
|align="center" | ∆E (Chair)|| width="50" align="center" | 45.70 || align="center" | 44.69 || width="50" align="center" | 35.12 || align="center" | 34.44 || align="center" | 33.5 ± 0.5
|-
|align="center" | ∆E (Boat)|| width="50" align="center" | 55.78 || align="center" | 54.93 || width="50" align="center" | 42.50 ||align="center" | 41.91 || align="center" | 44.7 ± 2.0
|}

The boat transition structure was found to have a higher activation energy than the chair. This can be due to the unfavourable repulsive interaction between the protons in the structure. The activation energies at 0 K of both transition structures optimized at B3LYP/6-31G* level are more similar to the experimental values . This can be explained by the choice of method and basis set. Electronic structure methods such as Hartree-Fock (HF) or Density functional theory (DFT) all approximate the exact solution in some ways. Generally, the lower the energy structure after a geometry optimization, the more suited the method is to describe the ground state.

The HF approximation describe non-interacting electrons under the influence of a mean electron field potential.It also accounts for the Pauli exclusion principle. DFT takes into account the electron correlation, but not the Pauli exclusion principle. The fact that electrons interaction is considered in the calculation gives a better approximation to strongly correlated problems. Different basis sets uses different number of functions to describe each atomic orbital and hence would affect the accuracy of calculation. The 6-31G* is a larger basis set compared to 3-21G in which more gaussian functions are used to describe each atomic orbital. 6-31G* also takes into account the distortion (polarisation) of the orbitals when molecules are formed. This in turn enables the basis set to describe the wavefunction more accurately.

==The Diels Alder Cycloaddition==

===Optimization of cis-butadiene and Molecular Orbitals Analysis===

[[File:Diels Alder reaction scheme.JPG|thumb|centre|Figure 6 Diels Alder reaction of ethene and cis-butadiene|450 px]]

Ethene and cis-butadiene were optimized to a minimum using the AM1 semi-empirical method. Their corresponding HOMO and LUMO were plotted as shown in table 12 and 13. The plane of symmetry bisect the C=C bond in ethene and centre C-C in butadiene.

{| class="wikitable"
|+ Table 12 Ethene MO
! HOMO, symmetric with respect to the plane || LUMO, antisymmetric with respect to the plane
|-
|[[File:Ethene HF HOMO.JPG|250 px]]
|[[File:Ethene HF LUMO.JPG|250 px]]
|}

{| class="wikitable"
|+ Table 13 Cis-butadiene
! Jmol || width="200" | HOMO, antisymmetry with respect to the planne || LUMO, symmetric with respect to the plane
|-
|
<jmol><jmolApplet>
<title>(AM1)optimised cis butadiene</title>
<color>black</color>
<size>250</size>
<uploadedFileContents>CIS BUTADIENE SEMI EMPIRICAL AM1 OPT.mol</uploadedFileContents>
</jmolApplet></jmol>
|[[File:Cis butadiene MO HOMO Transparent.JPG|250 px]]
|[[File:Cis butadiene MO LUMO Transparent.JPG|250 px]]
|}

{| class="wikitable"
|+ Table 14 Diels Alder Transition State
! Jmol || Vibration at -956 cm-1 || Vibration at 147 cm-1
|-
|
<jmol><jmolApplet>
<title>(AM1)optimised Diels Alder transition state</title>
<color>black</color>
<size>250</size>
<uploadedFileContents>DIELS ALDER TS HF OPTFREQ27test AM1.mol</uploadedFileContents>
</jmolApplet></jmol>
|[[File:Diels Alder TS HF OPTFREQ AM1 vibrational movie.gif|50 x 50 px ]]
|[[File:Diels Alder TS HF OPTFREQ AM1 vibrational lowest positive movie.gif|50 x 50 px ]]
|}

The imaginary frequency at -956 cm-1 corresponds to the bond forming/breaking of the Diels-Alder reaction. The animated vibration motion (table 14) shows that the bonds are formed synchronously. The lowest positive frequency does not show the same movement, and bonds do not seem to be forming or breaking.

{| class="wikitable"
|+ Table 15 Diels Alder Transition State MO
!HOMO, Antisymmetry with respect to the plane || LUMO, symmetric with respect to the plane
|-
|[[File:Diels Alder TS AM1 OPTFREQ HOMO 2 with line.jpg|260 px]]
|[[File:Diels Alder TS AM1 OPTFREQ LUMO 2 with line.jpg|250 px]]
|}

The Woodward–Hoffmann rules apply to cycloaddition reaction. It explains the stereochemical outcome of pericyclic reactions by considering the symmetry of the ‘frontier orbitals’ that contribute to the formation and breaking of bonds. A cycloaddition reaction involves the concerted formation of two or more σ bonds between the termini of two or more conjugated π systems. In Diels Alder reactions, two π systems are involved. The highest occupied molecular orbital (HOMO) of the dienophile overlaps with the lowest unoccupied molecular orbital (LUMO) of the diene. The diene contributes 4π electrons, and the dienophile contributes 2π electrons. This gives a total count of 6 electrons and hence the reaction is called [4πs + 2πs] cycloaddition. The reaction is thermally allowed and proceed suprafacially (new bonds form on the same face at both ends) via Hückel topology since it has 4n+2 (n=1) electrons in the system. Similarly, the Dewar and Zimmerman rules states that favourable pericyclic reactions will proceed via an aromatic transition state. If the reaction has a 4n+2 suprafacial topology, it is a Hückel system and reaction is allowed.

The reaction is favored by electron-donating groups such as COR, COOR and CN on the dienophile as this will lower the energy of LUMO. An electron-rich diene is also favoured. These would decrease the energy gap between the HOMO and LUMO. Since butadiene and ethene are discussed here, the effect of substituents is ignored.For an allowed reaction, the orbitals that overlap must have the same symmetry. The antisymmetric HOMO of butadiene interests with the antisymmetric LUMO of ethene to give rise to the antisymmetric HOMO of the transition state. Similarly, the same applied to the LUMO of the transition state. The symmetric HOMO of ethene overlaps with the symmetric LUMO of butadiene to form the symmteric HOMO of the transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Table 16 Geometry data of optimized Diels Alder transition structure
! height="60" Width="150"| Bond length (C1-C2),(C3-C4)(Å) !! Width="150"| Bond length (C2-C3)(Å) !! Width="150"| Distance between (C4-C5),(C1-C6) (Å) !! Width="150"| Distance between (C5-C6) (Å)!! rowspan="4" |[[File:Diels Alder TS numbering.JPG|200 px]]
|-
| align="center"| 1.38 || align="center"|1.39 || align="center"|2.12 || align="center"|1.38
|-
! colspan="2" | Typical sp 3 bond length (Å): 1.54
! colspan="2" | Typical sp 2 bond length (Å): 1.34
|-
! colspan="4" | van der Waals radius of C atom (Å): 1.70
|}

The distances of the bond forming/breaking in the transition structure are 2.12 Å. These distances are much greater than the sp2 and sp3 hybridised C-C bond. They are shorter than twice the van der Waals radius for carbon (3.40 Å) and are not close enough to experience repulsive interaction towards each other. Hence bond formation is favourable.

{| class="wikitable"
|+ Table 17 Results of IRC (No. of points: 60)
! Energy of product(a.u.): 0.0746648 !! Final gradient: 0.0005776
|-
|[[File:Diels alder TS AM1 OPT IRC 60 total energy graph.JPG|500 px]]
|[[File:Diels alder TS AM1 OPT IRC 60 gradient graph.JPG| 450 px]]
|}

An IRC calculation was carried out. This time, both direction was run. The energy graph (table 17) shows an expected reaction coordinate. A minimum geometry in the forward direction has an energy of 0.0746648 a.u. and a gradient close to zero. Increasing the number of data points to 70 made no difference to the results. This proves that a minimum geometry has reached.

===Cyclohexadiene-1,3-diene Reaction with Maleic Anhydride===

The AM1 semi-empirical method was applied for all calculation in this session. Maleic anhydride and cyclohexa-1,3-diene were optimized to a minimum. There HOMO and LUMO were plotted in table 18.

{| class="wikitable"
|+ Table 18 HOMO and LUMO of Maleic Anhydride and Cyclohexa-1,3-diene
! colspan="2" | Maleic Anhydride !! colspan="2" |Cyclohexa-1,3-diene
|-
! HOMO !! LUMO !! HOMO !! LUMO
|-
|[[File:Maleic anhydride HOMO.JPG|250 px]]
|[[File:Maleic anhydride LUMO.JPG|250 px]]
|[[File:Cyclohexadiene HOMO.JPG|250 px]]
|[[File:Cyclohexadiene LUMO.JPG|250 px]]
|}

The reactants were rearranged into a guess structure that resembles the exo transition state. They were optimized to TS(Berny) by freezing the coordinates and subsequently unfrozen. Frequency analysis was carried out. The imaginary frequency at -812 cm-1 confirms that a transition state was optimized.

{| class="wikitable"
|+ Table 19 Exo Transition State
! Jmol || Vibration at -812 cm-1 || HOMO
|-
|
<jmol><jmolApplet>
<title>optimised exo transition state </title>
<color>black</color>
<size>260</size>
<uploadedFileContents>EXO TRANSITION STATE OPT AM1.mol</uploadedFileContents>
</jmolApplet></jmol>
|[[File:Exo AM1 OPTFREQ vibration movie.gif|50 x 50 px]]
|[[File:Exo AM1 OPTFREQ HOMO.JPG|260 px]]
|}

IRC calculation was carried out for the exo transition structure and yielded the following results:

{| class="wikitable"
|+ Table 20 Results of IRC of Exo Transition Structure, No. of data points: 40
! Energy of the product (a.u.): -0.160168 !! Gradient: 0.0001171
|-
|[[File:Exo Transition state OPT AM1 IRC40 Total energy graph.JPG|320 px]]
|[[File:Exo Transition state OPT AM1 IRC40 Gradient graph.JPG| 320 px]]
|}

The reactants were rearranged into the endo transition state. They were optimized to TS(Berny) by freezing the coordinates and subsequently unfrozen. Frequency analysis was carried out. The imaginary frequency at -806 cm-1 confirms that a transition state was obtained.

{| class="wikitable"
|+ Table 21 Endo Transition State
! Jmol || Vibration at -806 cm-1 || HOMO
|-
|
<jmol><jmolApplet>
<title>(AM1) optimised exo transition state </title>
<color>black</color>
<size>260</size>
<uploadedFileContents>ENDO TRANSITION AM1 OPT.mol</uploadedFileContents>
</jmolApplet></jmol>
|[[File:Endo AM1 OPTFREQ3 vibration movie.gif|50 x 50 px ]]
|[[File:Endo AM1 OPT HOMO.JPG|260 px]]
|}

IRC calculation was carried out for the endo transition structure and yielded the following results (table 22). The gradient did not reach a value close to zero upon first calculation (middle). Another IRC calculation was run from the last point of the first calculation and yields results shown on the left hand side and right hand side. This shows that the minimum geometry was found.

{| class="wikitable"
|+ Table 22 Results of IRC of Endo Transition Structure, No. of data points:20
! Energy of product (a.u.): -0.159874 !! Gradient along IRC of first calculation !! Gradient:0.00002890
|-
|[[File:Endo Transition state OPT AM1 IRC20 Total energy graph.JPG|320 px]]
|[[File:Endo Transition state OPT AM1 IRC20 gradient graph.JPG| 320 px]]
|[[File:Endo AM1 IRC20 forward.JPG| 320 px]]
|}

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Table 23 Geometry data and energy of Exo and Endo transition structure
! !! Width="150" height="40"| Bonding forming distance (C2-C8),(C5-C7)(Å) !! Width="150"| Orientation (C3-C9),(C4-C11)(Å) !! width="150"|Maleic anhydride C=O bond length (Å)!! Width="150"|Maleic anhydride C-C bond length (C7-C8)/(C8-C9)(Å) !!Width="150"|Cyclohexadiene C-C bond length(Å) !! Width="150"|Cyclohexadiene C=C bond length(Å) !! Width="150"| Energy (a.u.) || rowspan="2" |[[File:Exo Transition State numbering.JPG|170 px]]
|-
| align="center"| Exo TS(Top)|| height="120" align="center"| 2.17|| align="center"|2.95 || align="center"|1.22 || align="center"|1.41/1.49 || align="center"|1.39/1.49 || align="center"|1.39 || align="center"|-0.0504198
|-
| align="center" height="60"| Endo TS (bottom)|| align="center"| 2.16 || align="center"| 2.89 || align="center"| 1.22 || align="center"|1.41/1.49 || align="center"|1.39/1.49 ||align="center"|1.39 || align="center"| -0.0515048|| rowspan="2" |[[File:Endo Transition state numbering.JPG|170 px]]
|}

====Analysis====

[[File:Diels Alder 2 reaction scheme.JPG|thumb|centre|400 px| Figure 7 Reaction scheme of Diels-Alder reaction of cyclohexadiene-1,3-diene with Maleic Anhydride]]

In cycloaddition, two new bonds are formed at the same time. Two filled p orbitals and two empty p orbitals need to be arranged at the right place and with the right symmetry in order to interact. In this Diels-Alder reaction, the LUMO of electron poor anhydride interacts with the HOMO of the diene. A node is present at the middle of HOMO of the diene and same in LUMO of dienophile. By Woodward–Hoffmann rules, it is an allowed interaction. The interaction of LUMO of diene and HOMO of anhydride also have the correct symmetry but due to the larger energy gap between them, it is less favourable. The HOMO of the diene and the LUMO of dienophile are closer in energy and gives a better overlap.

[[File:Second orbital effect.JPG|thumb|centre|400 px| Figure 8 Second orbital overlap effect of Diels-Alder reaction of cyclohexadiene-1,3-diene with Maleic Anhydride. a)Through space interaction between C=O and the back of diene. b)Primary and secondary orbital overlaps in endo structure. c) Overlap of orbitals in exo structure]]

Second orbital overlap effect was proposed by Woodward and Hoffmann. It is the positive overlap of inactive orbitals in the frontier molecular orbitals of a pericyclic reaction. In the endo transition structure (Figure 8b), it has the primary orbital overlap in which the p-orbitals of the anhydride LUMO interacts with the diene HOMO. However, the p-orbital on both side of the C=O also interacts with the p-orbitals at the back of the diene. These interactions are descriped as secondary as there are no change in the bonds. They interact strongly in the endo transition state (Figure 8a) but such interaction is not possible in the exo transition state (Figure 8c). The secondary overlap gives a stabilizing effect in the endo structure irrespective of the energies of the HOMO and LUMO.

Table 23 shows a comparison of structure and energy of the two transition structure. In general, both structures resemble each other. The main difference lies in the through space distance (Orientation distance) between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the C atoms of -CH2-CH2- in exo and -CH=CH- in endo . This distance is closer in endo. The endo structure also has a lower energy than the exo. These provides evidence that the endo structure is stabilized by the secondary overlap. Another way of analysing the presence of secondary overlaps, is to look at the MOs of the transition structure.

{| class="wikitable"
|+ Table 24 MOs of Endo and Exo Transition Structure
! !! HOMO - 4 !! HOMO !! LUMO !! LUMO + 1 !! LUMO + 2
|-
|Endo TS
|[[File:Endo HOMO-4.JPG|240 px]]
|[[File:Endo AM1 OPT HOMO.JPG| 240 px]]
|[[File:Endo AM1 OPTFREQ3 LUMO.JPG|240 px]]
|[[File:Endo LUMO+1.JPG| 240 px]]
|[[File:Endo LUMO+2.JPG| 240 px]]
|-
|Exo TS
|[[File:Exo HOMO-4.JPG|240 px]]
|[[File:Exo AM1 OPTFREQ HOMO.JPG| 240 px]]
|[[File:Exo AM1 OPTFREQ LUMO.JPG|240 px]]
|[[File:Exo LUMO+1.JPG| 240 px]]
|[[File:Exo LUMO+2.JPG| 240 px]]
|}

Table 24 shows a comparison of several MOs from the endo and exo transition state. The secondary orbitals overlap was not observed in the HOMO nor LUMO of the endo structure. The interaction was present in the HOMO-4, LUMO+1 and LUMO+2 instead. This is possibly due to the fact that orbital mixing was not taken into account in the calculation. A high level of theory such as HK or DFT might give a result closer to expectation. In the exo transition structure, no secondary orbitals overlap was observed which correlates with the discussion above.

Despite having endo form as the lower energy transition structure, it was shown that it leads to a higher energy product. Table 20 and table 22 shows the result of the IRC calculation. The energy of the endo product has an energy of -0.159874 a.u. where and the energy of the exo product has an energy of -0.160168 a.u.. This shows that the endo product is less stable. The structure experience steric repulsive interaction between the alkene of the six membered ring and the carbonyl groups of the dienophile. In an irreversible Diels-Alder reactions, therefore it would be the kinetic product of the reaction. The kinetic product is formed faster. If the reaction is under kinetic control, the energies of the transition states would dictate the outcome of the reaction. By Hammmond's postulate, the starting material, intermediate or product closest in energy to the transition state of the interest will be similar in structure.

Rep:Mod:WLL12Physicalcomplab

2015-01-30T01:20:31Z

Wll12: /* Cyclohexadiene-1,3-diene Reaction with Maleic Anhydride */

==Introduction==

In this computational experiment, the Cope rearrangement of 1,5-hexadiene and two examples of Diels-Alder cycloaddition reaction were investigated. The structures of the transition state of these reactions are the focus of study. The geometry and energies of the transition structures were analysed. The Molecular orbitals of the reactants and transition states for the Diels-Alder reaction were studied in order to understand their interactions and the geometry of the transition structure. All molecules were modeled using GaussView 5.0 and all calculation was done by the Gaussian programme.

Gaussian applies the Born-Oppenheimer Approximation in which the nuclei is much heavier than the electrons and hence the motion of nuclei and electrons can be treated separately. A potential energy surface (PES) is created based on this approximation. It represents the potential energy as a function of the internuclear distance in a reaction. The use of different method and basis set for the calculation would affect the accuracy of the results compared to experimental data. These are discussed in the following sections.

==The Cope Rearrangement Tutorial==

===Optimization of 1,5-Hexadiene===

{| class="wikitable"
|+ Table 1 Optimized structure of 1,5-hexadiene
! !!Optimised "anti" 1,5-hexadiene molecule !! Optimised "gauche" 1,5-hexadiene molecule
|-
|Jmol
||
<jmol><jmolApplet>
<color>black</color>
<size>280</size>
<uploadedFileContents>15 HEXADIENE ANTI OPT6 HF.mol</uploadedFileContents>
</jmolApplet></jmol>
||
<jmol><jmolApplet>
<color>black</color>
<size>285</size>
<uploadedFileContents>15 HEXADIENE GAUCHE C1 OPT7 HF.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
|Method/Basic set
|HF/3-21G
|HF/3-21G
|-
|Energy (a.u.)
| -231.6853962
| -231.6926612
|-
|Point group
|C2h
|C1
|}

[[File:Newnabd projection for 15 hexadiene.jpg|frame|400px|Figure 1. Newman projection of conformers of 1,5-hexadiene. a) "Anti" structure, C2h. b) "Gauche" Structure, C1]]

The free rotations about the C-C single bonds give rise to many possible conformations in 1,5-hexadiene. Table 1 shows two optimizied 1,5-hexadiene molecules. One is antiperiplanar and the other is gauche. By comparing to [[Mod:phys3#Appendix 1|Appendix 1]], they are anti 3 and gauche 3. It was predicted that the 1,5-hexadiene with gauche linkage at the centre of the molecule would have a higher energy then the "anti" structure. The vinyl groups at the end of hexadiene are closer together in the gauche structure than in the "anti" structure. The gauche structure has a dihedral angle of 60 degree at the centre and repulsive steric interaction was expected to result in an increase in energy.

However, it was shown that the "anti" 1,5-hexadiene has a higher energy compared to the gauche conformer which has a slightly lower energy. By comparing to the table shown in [[Mod:phys3#Appendix 1|Appendix 1]], it was also found that the C1 gauche conformer is the lowest energy conformation of 1,5-hexadiene.

A possible explanation to this is that the gauche structure is stabilised by an attractive interaction between the protons on one vinyl group and the π-orbital on the other. A vinyl proton is covalently bonded to a carbon atom and weakly interacting with the π-orbital of the double bond. This is known as the CH/π interaction. In the antiperiplanar structure (Fig. 1a), such interaction is not possible as the vinyl groups are far apart. In the gauche structure (Fig. 1b), the vinyl groups are close to each other and therefore it is stabilised by this interaction.

1,5-hexadiene with an "anti" linkage, Ci conformation

{| class="wikitable"
|+ Table 2 Optimized structure of "anti" 1,5-hexadiene, Ci
! Method/basis set !! HF/3-21G !! DFT/6-31G*
|-
|Jmol
||
<jmol><jmolApplet>
<color>black</color>
<size>280</size>
<uploadedFileContents>15 HEXADIENE ANTI OPT15 HF 2.mol</uploadedFileContents>
</jmolApplet></jmol>
||
<jmol><jmolApplet>
<color>black</color>
<size>280</size>
<uploadedFileContents>15 HEXADIENE CI ANTI OPT16 DFT 631D.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
|Labelled molecule
|[[File:1,5 hexadiene HF Anti Ci Optimization.JPG|300px]]
||[[File:1,5 hexadiene DFT Anti Ci Optimization.JPG|300px]]
|-
|Energy (a.u.)
| -231.695353
| -234.559704
|}

The CI "anti 2" 1,5-hexadiene optimized at the HF/3-21G level of theory has an energy of -231.6925353 au. This value is the same as the one given in [[Mod:phys3#Appendix 1|Appendix 1]]. This is subsequently re-optimised at B3LYP/6-31G* level and yield a lower energy form than the one at HF/3-21G level. The structure from the HF/3-21G calculation closely resembles that from B3LYP/6-31G* calculation. Table 3 summarizes the dihedral angles and the bond lengths of both structures. The centre dihedral angle and all carbon-carbon bond lengths are similar in both 1,5-hexadiene. There is only a 4 degrees difference in the terminal dihedral angle between them. Overall, the change in geometry is minimal.

{| class="wikitable"
|+ Table 3 Geometry data "anti" 1,5-hexaidene optimized at HF/3-21G and DFT/6-31G* level; Ci
! Method !!colspan="3"| HF !! colspan="3"| DFT
|-
| Dihedral angle(C1-C4-C6-C9);(º) || colspan="3" align="center" |114.7 || colspan="3" align="center"| 118.8
|-
| Dihedral angle(C4-C6-C9-C12);(º) || colspan="3" align="center"|180.0 || colspan="3" align="center"| 180.0
|-
| || C1-C4 || C4-C6 || C6-C9 || C1-C4 || C4-C6 || C6-C9
|-
| align="center"| Bond length(Å) || 1.07 || 1.33 || 1.51 || 1.09 || 1.34 || 1.51
|}

===Frequency Analysis of "anti" 1,5-hexadiene, Ci conformation; DFT/6-31G===

[[Image:1,5 hexadiene DFT Anti Ci Freq spectrum.JPG|frame|centre|400px|Figure 2 Vibrational Spectrum of "anti" 1,5-hexadiene]]

Frequency analysis was carried out. It gives the second derivative of the potential energy surface. If all frequencies are positive, it means a minimum was resulted. The absence of imaginary (negative) frequencies shows that the structure is optimized to a minima. Table 4 shows the thermochemical analysis of the optimized structure.

{| class="wikitable"
|+ Table 4 Summary of energy
! !! Energy (in hatree)
|-
| Sum of electronic and zero point energies (E = Eelec + ZEP), at 0 K || align="center" |-234.469215
|-
| width="430" | Sum of electronic and thermal energies (E = E + Evib + Erot + Etrans), at 298.15 K and 1 atm|| align="center"| -234.461867
|-
| Sum of electronic and thermal enthalpies+ || align="center"| -234.460922
|-
| Sum of electronic and thermal free energies++ ||align="center"| -234.500800
|}

+ An additional correction for RT(H = E + RT)

++ Including entropic contribution to the free energy (G = H-TS)

===Optimization of "Chair" Transition Structures===

The Cope rearrangement have two different transition state: Chair and Boat.

Optimization and Frequency Analysis of Chair Transition Structure (Opt+Freq)

An allyl fragment (CH2CHCH2) was first optimized to TS(Berny) at HF/3-21G level. Two optimized fragments were arranged in the chair form and underwent optimization and frequency analysis. This optimized structure has an imaginary frequency at -818 cm-1. The negative second derivative of the potential energy surface corresponds to a maxima. This shows that the optimization of the chair transition state structure was successful. The imaginary frequency was also animated in table 5.

{| class="wikitable"
|+ Table 5 Results of optimizaed chair transition structure
! Jmol || colspan="3"|Animated vibration at -818 cm-1 || Vibrational spectrum
|-
|
<jmol><jmolApplet>
<title>(3-21G) optimized chair transition state</title>
<color>black</color>
<size>280</size>
<uploadedFileContents>TRANSITION ALLYL FRAG HF OPTFREQ5.mol</uploadedFileContents>
</jmolApplet></jmol>
|
|[[File:Imaginary vibration chair transition state animation.gif|50 x 50 px]]
|
|[[File:Trasition state OPT FREQ IR spectrum.JPG|400 px]]
|}

Alternatively the frozen coordinate method was used to optimize the transition structure. This was done by fixing the distance between the terminal carbons from both allyl fragments to 2.2 Å and then optimized to a minimum (HF/3-21G). A transition state optimization to TS(Berny) was carried out subsequently at HF/3-21G level. This allows the bond forming/breaking distances between the two fragments to be optimized as well. The table below summarizes the geometry data of the transition structures that were optimized differently.

Both optimized transition structures with either frozen or optimized bond forming/breaking distances, show similar C-C bond length and C-C-C angle within one allyl fragment. These are also similar in values compared to the structure from "Opt+Freq" calculation. The main difference lies in the distance between C1-C6 and C3-C4. When the bond forming/breaking distances were optimized, these values are more similar to that in the structure from "Opt+Freq" calculation. This shows that freezing the coordinate would give a less accurate optimization of structure.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Table 6 Geometry data of optimized chair transition structure
! !! Width="120"| Bond length (C1-C2)(Å) !! Width="120"|Angle (C1-C2-C3)(º) !! Width="120"| Distance between (C1-C6) (Å)!! Width="120"|Distance between (C3-C4) (Å)!! rowspan="4" |[[File:Chair Transition State Freeze HF Optimization with number.JPG|250 px]]
|-
| align="center"| Opt + Freq|| height="50" align="center"| 1.39 || align="center"|120.5 || align="center"|2.02 || align="center"|2.02
|-
| align="center"| Opt(Freeze Coordinate) || align="center"| 1.39 || align="center"| 121.8 || align="center"| 2.16 || align="center"| 2.20
|-
| align="center"| Opt(Derivative) || align="center"| 1.39|| align="center"| 120.5|| align="center"| 2.02 || align="center"| 2.02
|}

===Optimization of "Boat" Transition Structures===

{|style="margin: 0 auto;"
|[[File:Failed boat transition state.PNG|thumb|200 px|Figure 3. First attempt of QST2 calculation]]
|[[File:Boat QST2 rearrangement.JPG|thumb|350px|Figure 4. Rearrangement of butadiene]]
|[[File:Cope rearrangement scheme 2.JPG|thumb|200px|Figure 5. Cope Rearrangement]]
|}
The optimized Ci "anti" 1,5-hexadienes were optimized to a transition state and frequency analysis were carried out using the QST2 method. QST2 requires reactant and product as the input and all atoms must be labelled in the same way in both structure. The first calculation was done without any modification to the structure orientation. The job was failed and resulted in the transition structure shown in figure 3. The 1,5-hexadiene molecules were re-orientated so that they had the same arrangement as what shown in figure 4. The modified molecules had a dihedral angle of 0 degree at the centre and 100 degrees for the inside C-C-C angle. The QST2 calculation of the modified structure was successful and the following results (table 7) were obtained. The distance between the two fragments is 2.14 Å. The boat transition structure was optimized and it has an imaginary frequency at -840 cm-1.

The optimization was also carried out using the QST3 calculation. This requires 3 inputs in the following order: the reactant, product, and guess transition state structures. Similar to QST2, the atoms must be labelled in the same order. The energy and geometry of the optimized structure of QST3 calculation resembles that of QST2. It also has an imaginary frequency at -840 cm-1.

{| class="wikitable"
|+ Table 7 Results of optimized boat transition structure
! Jmol || colspan="3"|Vibration at -840 cm-1|| Vibrational spectrum
|-
|
<jmol><jmolApplet>
<title>(3-21G) optimised boat transition state</title>
<color>black</color>
<size>280</size>
<uploadedFileContents>BOAT TRANSITION HF OPTFREQ13 QST2.mol</uploadedFileContents>
</jmolApplet></jmol>
|
|[[File:Imaginary vibration Boat transition state QST2 animation.gif|50 x 50 px ]]
|
|[[File:Boat Transition State QST2 IR spectrum.JPG|400 px]]
|}

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

{| class="wikitable"
|+ Table 8 Results of IRC calculation
! First Calculation || Total Energy along IRC || RMS Gradient Norm along IRC
|-
|[[File:IRC Point 50 Chair Transition Forward Direction Always calculate force constant.gif|50 x 50 px|frame|centre|No. of points along IRC: 50]]
|[[File:IRC Point 50 Chair Transition Total Energy along IRC.JPG|400 px]]
|[[File:IRC Point 50 Chair Transition RMS Gradient Norm along IRC.JPG| 350 px]]
|}

It is difficult to predict which conformers of 1,5-hexadiene will form from the chair and boat transition structures. Intrinsic Reaction Coordinate (IRC) method was used to find out the structure that has the lowest energy. It allows the lowest energy reaction path from the transition state towards the reactants and products to be followed. Only the forward direction of the reaction coordinate was considered here. The number of data points along the IRC was set to 50 and the force constant was set to "calculate always" in the first attempt. Forty-four intermediates were obtained. A second attempt of IRC calculation with 100 points was carried out to ensure the minimum energy geometry was reached. There was no change to the energy graph and the gradient was closed to zero at the end of calculation. These prove that a minimum geometry has reached. A gauche conformer with an energy of -231.691608 a.u. (gradient: 0.00015154 a.u.) was found to be the minimum geometry from this calculation. This is gauche 2 in [[Mod:phys3#Appendix 1|Appendix 1]].

===Optimization of Chair and Boat Transition structures using B3LYP/6-31G*===

The HF/3-21G optimized chair and boat structure were re-optimized using B3LYP/6-31G* method. The following tables present a comparison for the geometries and different energies values. The chair transition structures optimized at HF/3-21G and B3LYP/6-31G* have very similar geometry compared to each other. The same applies to the boat transition structure. However, the energies are lower for the transition structures optimized at B3LYP/6-31G* level.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Table 9 Geometry data of chair and boat transition structure
! || height="40" colspan="3"|HF/3-21G || colspan="3"|B3LYP/6-31G* || rowspan="2" |[[File:Chair Transition State Freeze HF Optimization with number.JPG|170 px]]
|-
! !! Width="120" height="40"| Bond length (C1-C2)(Å) !! Width="120"|Angle (C1-C2-C3)(º) !! Width="120"| Distance between (C1-C6)/(C3-C4) (Å)!! Width="120"| Bond length (C1-C2)(Å) !! Width="120"|Angle (C1-C2-C3)(º) !! Width="120"| Distance between (C1-C6)/(C3-C4) (Å)
|-
| align="center"| Chair TS (Top)|| height="50" align="center"| 1.38 || align="center"|122.0 || align="center"|2.20 || align="center"|1.39 ||align="center"| 122.0 || align="center"|2.20 || rowspan="2" |[[File:Boat Transtion numbering.JPG|170 px]]
|-
| align="center" height="60"| Boat TS (bottom)|| align="center"| 1.41 || align="center"| 121.2 || align="center"| 2.14 || align="center"| 1.39 || align="center"| 121.1 || align="center"| 2.14
|}

{| class="wikitable"
|+ Table 10 Summary of Energy (in hatree)
! || colspan="3"|HF/3-21G || colspan="3"|B3LYP/6-31G*
|-
| || align="center" | Electronic energy || width="125" align="center" | Sum of electronic and zero-point energies (0 K)|| width="130" align="center" | Sum of electronic and thermal energies (298.15 K)|| align="center" | Electronic energy || width="125" align="center" | Sum of electronic and zero-point energies (0 K)|| width="130" align="center" | Sum of electronic and thermal energies (298.15 K)
|-
|align="center" | Chair TS|| align="center" | -231.619332 || align="center" | -231.466702 || align="center" | -231.461343 || align="center" | -234.553938 || align="center" | -234.413269 || align="center" | -234.406982
|-
|align="center" | Boat TS|| align="center" | -231.602802 || align="center" | -231.450928 || align="center" | -231.445299 ||align="center" | -234.542868 || align="center" | -234.401492 || align="center" | -234.395284
|-
|align="center" | Reactant (Ci; Anti)|| align="center" | -231.692535 || align="center" | -231.539539 || align="center" | -231.532565 || align="center" | -234.611712|| align="center" | -234.469215 || align="center" | -234.461867
|}

===Calculation of Activation Energies for Both Transition Structures===
{| class="wikitable"
|+ Table 11 Summary of Activation Energy (in kcal/mol)
! || colspan="2"|HF/3-21G || colspan="2"|B3LYP/6-31G* || Experimental value from [[Mod:phys3#Appendix 1|Appendix 1]]
|-
| align="center" | Temperature || width="50" align="center" | 0 K || align="center" | 298.15 K || width="50" align="center" | 0 K || align="center" | 298.15 K || align="center" | 0 K
|-
|align="center" | ∆E (Chair)|| width="50" align="center" | 45.70 || align="center" | 44.69 || width="50" align="center" | 35.12 || align="center" | 34.44 || align="center" | 33.5 ± 0.5
|-
|align="center" | ∆E (Boat)|| width="50" align="center" | 55.78 || align="center" | 54.93 || width="50" align="center" | 42.50 ||align="center" | 41.91 || align="center" | 44.7 ± 2.0
|}

The boat transition structure was found to have a higher activation energy than the chair. This can be due to the unfavourable repulsive interaction between the protons in the structure. The activation energies at 0 K of both transition structures optimized at B3LYP/6-31G* level are more similar to the experimental values . This can be explained by the choice of method and basis set. Electronic structure methods such as Hartree-Fock (HF) or Density functional theory (DFT) all approximate the exact solution in some ways. Generally, the lower the energy structure after a geometry optimization, the more suited the method is to describe the ground state.

The HF approximation describe non-interacting electrons under the influence of a mean electron field potential.It also accounts for the Pauli exclusion principle. DFT takes into account the electron correlation, but not the Pauli exclusion principle. The fact that electrons interaction is considered in the calculation gives a better approximation to strongly correlated problems. Different basis sets uses different number of functions to describe each atomic orbital and hence would affect the accuracy of calculation. The 6-31G* is a larger basis set compared to 3-21G in which more gaussian functions are used to describe each atomic orbital. 6-31G* also takes into account the distortion (polarisation) of the orbitals when molecules are formed. This in turn enables the basis set to describe the wavefunction more accurately.

==The Diels Alder Cycloaddition==

===Optimization of cis-butadiene and Molecular Orbitals Analysis===

[[File:Diels Alder reaction scheme.JPG|thumb|centre|Figure 6 Diels Alder reaction of ethene and cis-butadiene|450 px]]

Ethene and cis-butadiene were optimized to a minimum using the AM1 semi-empirical method. Their corresponding HOMO and LUMO were plotted as shown in table 12 and 13. The plane of symmetry bisect the C=C bond in ethene and centre C-C in butadiene.

{| class="wikitable"
|+ Table 12 Ethene MO
! HOMO, symmetric with respect to the plane || LUMO, antisymmetric with respect to the plane
|-
|[[File:Ethene HF HOMO.JPG|250 px]]
|[[File:Ethene HF LUMO.JPG|250 px]]
|}

{| class="wikitable"
|+ Table 13 Cis-butadiene
! Jmol || width="200" | HOMO, antisymmetry with respect to the planne || LUMO, symmetric with respect to the plane
|-
|
<jmol><jmolApplet>
<title>(AM1)optimised cis butadiene</title>
<color>black</color>
<size>250</size>
<uploadedFileContents>CIS BUTADIENE SEMI EMPIRICAL AM1 OPT.mol</uploadedFileContents>
</jmolApplet></jmol>
|[[File:Cis butadiene MO HOMO Transparent.JPG|250 px]]
|[[File:Cis butadiene MO LUMO Transparent.JPG|250 px]]
|}

{| class="wikitable"
|+ Table 14 Diels Alder Transition State
! Jmol || Vibration at -956 cm-1 || Vibration at 147 cm-1
|-
|
<jmol><jmolApplet>
<title>(AM1)optimised Diels Alder transition state</title>
<color>black</color>
<size>250</size>
<uploadedFileContents>DIELS ALDER TS HF OPTFREQ27test AM1.mol</uploadedFileContents>
</jmolApplet></jmol>
|[[File:Diels Alder TS HF OPTFREQ AM1 vibrational movie.gif|50 x 50 px ]]
|[[File:Diels Alder TS HF OPTFREQ AM1 vibrational lowest positive movie.gif|50 x 50 px ]]
|}

The imaginary frequency at -956 cm-1 corresponds to the bond forming/breaking of the Diels-Alder reaction. The animated vibration motion (table 14) shows that the bonds are formed synchronously. The lowest positive frequency does not show the same movement, and bonds do not seem to be forming or breaking.

{| class="wikitable"
|+ Table 15 Diels Alder Transition State MO
!HOMO, Antisymmetry with respect to the plane || LUMO, symmetric with respect to the plane
|-
|[[File:Diels Alder TS AM1 OPTFREQ HOMO 2 with line.jpg|260 px]]
|[[File:Diels Alder TS AM1 OPTFREQ LUMO 2 with line.jpg|250 px]]
|}

The Woodward–Hoffmann rules apply to cycloaddition reaction. It explains the stereochemical outcome of pericyclic reactions by considering the symmetry of the ‘frontier orbitals’ that contribute to the formation and breaking of bonds. A cycloaddition reaction involves the concerted formation of two or more σ bonds between the termini of two or more conjugated π systems. In Diels Alder reactions, two π systems are involved. The highest occupied molecular orbital (HOMO) of the dienophile overlaps with the lowest unoccupied molecular orbital (LUMO) of the diene. The diene contributes 4π electrons, and the dienophile contributes 2π electrons. This gives a total count of 6 electrons and hence the reaction is called [4πs + 2πs] cycloaddition. The reaction is thermally allowed and proceed suprafacially (new bonds form on the same face at both ends) via Hückel topology since it has 4n+2 (n=1) electrons in the system. Similarly, the Dewar and Zimmerman rules states that favourable pericyclic reactions will proceed via an aromatic transition state. If the reaction has a 4n+2 suprafacial topology, it is a Hückel system and reaction is allowed.

The reaction is favored by electron-donating groups such as COR, COOR and CN on the dienophile as this will lower the energy of LUMO. An electron-rich diene is also favoured. These would decrease the energy gap between the HOMO and LUMO. Since butadiene and ethene are discussed here, the effect of substituents is ignored.For an allowed reaction, the orbitals that overlap must have the same symmetry. The antisymmetric HOMO of butadiene interests with the antisymmetric LUMO of ethene to give rise to the antisymmetric HOMO of the transition state. Similarly, the same applied to the LUMO of the transition state. The symmetric HOMO of ethene overlaps with the symmetric LUMO of butadiene to form the symmteric HOMO of the transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Table 16 Geometry data of optimized Diels Alder transition structure
! height="60" Width="150"| Bond length (C1-C2),(C3-C4)(Å) !! Width="150"| Bond length (C2-C3)(Å) !! Width="150"| Distance between (C4-C5),(C1-C6) (Å) !! Width="150"| Distance between (C5-C6) (Å)!! rowspan="4" |[[File:Diels Alder TS numbering.JPG|200 px]]
|-
| align="center"| 1.38 || align="center"|1.39 || align="center"|2.12 || align="center"|1.38
|-
! colspan="2" | Typical sp 3 bond length (Å): 1.54
! colspan="2" | Typical sp 2 bond length (Å): 1.34
|-
! colspan="4" | van der Waals radius of C atom (Å): 1.70
|}

The distances of the bond forming/breaking in the transition structure are 2.12 Å. These distances are much greater than the sp2 and sp3 hybridised C-C bond. They are shorter than twice the van der Waals radius for carbon (3.40 Å) and are not close enough to experience repulsive interaction towards each other. Hence bond formation is favourable.

{| class="wikitable"
|+ Table 17 Results of IRC (No. of points: 60)
! Energy of product(a.u.): 0.0746648 !! Final gradient: 0.0005776
|-
|[[File:Diels alder TS AM1 OPT IRC 60 total energy graph.JPG|500 px]]
|[[File:Diels alder TS AM1 OPT IRC 60 gradient graph.JPG| 450 px]]
|}

An IRC calculation was carried out. This time, both direction was run. The energy graph (table 17) shows an expected reaction coordinate. A minimum geometry in the forward direction has an energy of 0.0746648 a.u. and a gradient close to zero. Increasing the number of data points to 70 made no difference to the results. This proves that a minimum geometry has reached.

===Cyclohexadiene-1,3-diene Reaction with Maleic Anhydride===

The AM1 semi-empirical method was applied for all calculation in this session. Maleic anhydride and cyclohexa-1,3-diene were optimized to a minimum. There HOMO and LUMO were plotted in table 18.

{| class="wikitable"
|+ Table 18 HOMO and LUMO of Maleic Anhydride and Cyclohexa-1,3-diene
! colspan="2" | Maleic Anhydride !! colspan="2" |Cyclohexa-1,3-diene
|-
! HOMO !! LUMO !! HOMO !! LUMO
|-
|[[File:Maleic anhydride HOMO.JPG|250 px]]
|[[File:Maleic anhydride LUMO.JPG|250 px]]
|[[File:Cyclohexadiene HOMO.JPG|250 px]]
|[[File:Cyclohexadiene LUMO.JPG|250 px]]
|}

The reactants were rearranged into a guess structure that resembles the exo transition state. They were optimized to TS(Berny) by freezing the coordinates and subsequently unfrozen. Frequency analysis was carried out. The imaginary frequency at -812 cm-1 confirms that a transition state was optimized.

{| class="wikitable"
|+ Table 19 Exo Transition State
! Jmol || Vibration at -812 cm-1 || HOMO
|-
|
<jmol><jmolApplet>
<title>optimised exo transition state </title>
<color>black</color>
<size>260</size>
<uploadedFileContents>EXO TRANSITION STATE OPT AM1.mol</uploadedFileContents>
</jmolApplet></jmol>
|[[File:Exo AM1 OPTFREQ vibration movie.gif|50 x 50 px]]
|[[File:Exo AM1 OPTFREQ HOMO.JPG|260 px]]
|}

IRC calculation was carried out for the exo transition structure and yielded the following results:

{| class="wikitable"
|+ Table 20 Results of IRC of Exo Transition Structure, No. of data points: 40
! Energy of the product (a.u.): -0.160168 !! Gradient: 0.0001171
|-
|[[File:Exo Transition state OPT AM1 IRC40 Total energy graph.JPG|250 px]]
|[[File:Exo Transition state OPT AM1 IRC40 Gradient graph.JPG| 250 px]]
|}

The reactants were rearranged into the endo transition state. They were optimized to TS(Berny) by freezing the coordinates and subsequently unfrozen. Frequency analysis was carried out. The imaginary frequency at -806 cm-1 confirms that a transition state was obtained.

{| class="wikitable"
|+ Table 21 Endo Transition State
! Jmol || Vibration at -806 cm-1 || HOMO
|-
|
<jmol><jmolApplet>
<title>(AM1) optimised exo transition state </title>
<color>black</color>
<size>260</size>
<uploadedFileContents>ENDO TRANSITION AM1 OPT.mol</uploadedFileContents>
</jmolApplet></jmol>
|[[File:Endo AM1 OPTFREQ3 vibration movie.gif|50 x 50 px ]]
|[[File:Endo AM1 OPT HOMO.JPG|260 px]]
|}

IRC calculation was carried out for the endo transition structure and yielded the following results (table 22). The gradient did not reach a value close to zero upon first calculation (middle). Another IRC calculation was run from the last point of the first calculation and yields results shown on the left hand side and right hand side. This shows that the minimum geometry was found.

{| class="wikitable"
|+ Table 22 Results of IRC of Endo Transition Structure, No. of data points:20
! Energy of product (a.u.): -0.159874 !! Gradient along IRC of first calculation !! Gradient:0.00002890
|-
|[[File:Endo Transition state OPT AM1 IRC20 Total energy graph.JPG|250 px]]
|[[File:Endo Transition state OPT AM1 IRC20 gradient graph.JPG| 250 px]]
|[[File:Endo AM1 IRC20 forward.JPG| 250 px]]
|}

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Table 23 Geometry data and energy of Exo and Endo transition structure
! !! Width="150" height="40"| Bonding forming distance (C2-C8),(C5-C7)(Å) !! Width="150"| Orientation (C3-C9),(C4-C11)(Å) !! width="150"|Maleic anhydride C=O bond length (Å)!! Width="150"|Maleic anhydride C-C bond length (C7-C8)/(C8-C9)(Å) !!Width="150"|Cyclohexadiene C-C bond length(Å) !! Width="150"|Cyclohexadiene C=C bond length(Å) !! Width="150"| Energy (a.u.) || rowspan="2" |[[File:Exo Transition State numbering.JPG|170 px]]
|-
| align="center"| Exo TS(Top)|| height="120" align="center"| 2.17|| align="center"|2.95 || align="center"|1.22 || align="center"|1.41/1.49 || align="center"|1.39/1.49 || align="center"|1.39 || align="center"|-0.0504198
|-
| align="center" height="60"| Endo TS (bottom)|| align="center"| 2.16 || align="center"| 2.89 || align="center"| 1.22 || align="center"|1.41/1.49 || align="center"|1.39/1.49 ||align="center"|1.39 || align="center"| -0.0515048|| rowspan="2" |[[File:Endo Transition state numbering.JPG|170 px]]
|}

====Analysis====

[[File:Diels Alder 2 reaction scheme.JPG|thumb|centre|400 px| Figure 7 Reaction scheme of Diels-Alder reaction of cyclohexadiene-1,3-diene with Maleic Anhydride]]

In cycloaddition, two new bonds are formed at the same time. Two filled p orbitals and two empty p orbitals need to be arranged at the right place and with the right symmetry in order to interact. In this Diels-Alder reaction, the LUMO of electron poor anhydride interacts with the HOMO of the diene. A node is present at the middle of HOMO of the diene and same in LUMO of dienophile. By Woodward–Hoffmann rules, it is an allowed interaction. The interaction of LUMO of diene and HOMO of anhydride also have the correct symmetry but due to the larger energy gap between them, it is less favourable. The HOMO of the diene and the LUMO of dienophile are closer in energy and gives a better overlap.

[[File:Second orbital effect.JPG|thumb|centre|400 px| Figure 8 Second orbital overlap effect of Diels-Alder reaction of cyclohexadiene-1,3-diene with Maleic Anhydride. a)Through space interaction between C=O and the back of diene. b)Primary and secondary orbital overlaps in endo structure. c) Overlap of orbitals in exo structure]]

Second orbital overlap effect was proposed by Woodward and Hoffmann. It is the positive overlap of inactive orbitals in the frontier molecular orbitals of a pericyclic reaction. In the endo transition structure (Figure 8b), it has the primary orbital overlap in which the p-orbitals of the anhydride LUMO interacts with the diene HOMO. However, the p-orbital on both side of the C=O also interacts with the p-orbitals at the back of the diene. These interactions are descriped as secondary as there are no change in the bonds. They interact strongly in the endo transition state (Figure 8a) but such interaction is not possible in the exo transition state (Figure 8c). The secondary overlap gives a stabilizing effect in the endo structure irrespective of the energies of the HOMO and LUMO.

Table 23 shows a comparison of structure and energy of the two transition structure. In general, both structures resemble each other. The main difference lies in the through space distance (Orientation distance) between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the C atoms of -CH2-CH2- in exo and -CH=CH- in endo . This distance is closer in endo. The endo structure also has a lower energy than the exo. These provides evidence that the endo structure is stabilized by the secondary overlap. Another way of analysing the presence of secondary overlaps, is to look at the MOs of the transition structure.

{| class="wikitable"
|+ Table 24 MOs of Endo and Exo Transition Structure
! !! HOMO - 4 !! HOMO !! LUMO !! LUMO + 1 !! LUMO + 2
|-
|Endo TS
|[[File:Endo HOMO-4.JPG|240 px]]
|[[File:Endo AM1 OPT HOMO.JPG| 240 px]]
|[[File:Endo AM1 OPTFREQ3 LUMO.JPG|240 px]]
|[[File:Endo LUMO+1.JPG| 240 px]]
|[[File:Endo LUMO+2.JPG| 240 px]]
|-
|Exo TS
|[[File:Exo HOMO-4.JPG|240 px]]
|[[File:Exo AM1 OPTFREQ HOMO.JPG| 240 px]]
|[[File:Exo AM1 OPTFREQ LUMO.JPG|240 px]]
|[[File:Exo LUMO+1.JPG| 240 px]]
|[[File:Exo LUMO+2.JPG| 240 px]]
|}

Table 24 shows a comparison of several MOs from the endo and exo transition state. The secondary orbitals overlap was not observed in the HOMO nor LUMO of the endo structure. The interaction was present in the HOMO-4, LUMO+1 and LUMO+2 instead. This is possibly due to the fact that orbital mixing was not taken into account in the calculation. A high level of theory such as HK or DFT might give a result closer to expectation. In the exo transition structure, no secondary orbitals overlap was observed which correlates with the discussion above.

Despite having endo form as the lower energy transition structure, it was shown that it leads to a higher energy product. Table 20 and table 22 shows the result of the IRC calculation. The energy of the endo product has an energy of -0.159874 a.u. where and the energy of the exo product has an energy of -0.160168 a.u.. This shows that the endo product is less stable. The structure experience steric repulsive interaction between the alkene of the six membered ring and the carbonyl groups of the dienophile. In an irreversible Diels-Alder reactions, therefore it would be the kinetic product of the reaction. The kinetic product is formed faster. If the reaction is under kinetic control, the energies of the transition states would dictate the outcome of the reaction. By Hammmond's postulate, the starting material, intermediate or product closest in energy to the transition state of the interest will be similar in structure.

2015-01-30T01:08:18Z

Wll12: /* Optimization of 1,5-Hexadiene */

==Introduction==

In this computational experiment, the Cope rearrangement of 1,5-hexadiene and two examples of Diels-Alder cycloaddition reaction were investigated. The structures of the transition state of these reactions are the focus of study. The geometry and energies of the transition structures were analysed. The Molecular orbitals of the reactants and transition states for the Diels-Alder reaction were studied in order to understand their interactions and the geometry of the transition structure. All molecules were modeled using GaussView 5.0 and all calculation was done by the Gaussian programme.

Gaussian applies the Born-Oppenheimer Approximation in which the nuclei is much heavier than the electrons and hence the motion of nuclei and electrons can be treated separately. A potential energy surface (PES) is created based on this approximation. It represents the potential energy as a function of the internuclear distance in a reaction. The use of different method and basis set for the calculation would affect the accuracy of the results compared to experimental data. These are discussed in the following sections.

==The Cope Rearrangement Tutorial==

===Optimization of 1,5-Hexadiene===

{| class="wikitable"
|+ Table 1 Optimized structure of 1,5-hexadiene
! !!Optimised "anti" 1,5-hexadiene molecule !! Optimised "gauche" 1,5-hexadiene molecule
|-
|Jmol
||
<jmol><jmolApplet>
<color>black</color>
<size>280</size>
<uploadedFileContents>15 HEXADIENE ANTI OPT6 HF.mol</uploadedFileContents>
</jmolApplet></jmol>
||
<jmol><jmolApplet>
<color>black</color>
<size>285</size>
<uploadedFileContents>15 HEXADIENE GAUCHE C1 OPT7 HF.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
|Method/Basic set
|HF/3-21G
|HF/3-21G
|-
|Energy (a.u.)
| -231.6853962
| -231.6926612
|-
|Point group
|C2h
|C1
|}

[[File:Newnabd projection for 15 hexadiene.jpg|frame|400px|Figure 1. Newman projection of conformers of 1,5-hexadiene. a) "Anti" structure, C2h. b) "Gauche" Structure, C1]]

The free rotations about the C-C single bonds give rise to many possible conformations in 1,5-hexadiene. Table 1 shows two optimizied 1,5-hexadiene molecules. One is antiperiplanar and the other is gauche. By comparing to [[Mod:phys3#Appendix 1|Appendix 1]], they are anti 3 and gauche 3. It was predicted that the 1,5-hexadiene with gauche linkage at the centre of the molecule would have a higher energy then the "anti" structure. The vinyl groups at the end of hexadiene are closer together in the gauche structure than in the "anti" structure. The gauche structure has a dihedral angle of 60 degree at the centre and repulsive steric interaction was expected to result in an increase in energy.

However, it was shown that the "anti" 1,5-hexadiene has a higher energy compared to the gauche conformer which has a slightly lower energy. By comparing to the table shown in [[Mod:phys3#Appendix 1|Appendix 1]], it was also found that the C1 gauche conformer is the lowest energy conformation of 1,5-hexadiene.

A possible explanation to this is that the gauche structure is stabilised by an attractive interaction between the protons on one vinyl group and the π-orbital on the other. A vinyl proton is covalently bonded to a carbon atom and weakly interacting with the π-orbital of the double bond. This is known as the CH/π interaction. In the antiperiplanar structure (Fig. 1a), such interaction is not possible as the vinyl groups are far apart. In the gauche structure (Fig. 1b), the vinyl groups are close to each other and therefore it is stabilised by this interaction.

1,5-hexadiene with an "anti" linkage, Ci conformation

{| class="wikitable"
|+ Table 2 Optimized structure of "anti" 1,5-hexadiene, Ci
! Method/basis set !! HF/3-21G !! DFT/6-31G*
|-
|Jmol
||
<jmol><jmolApplet>
<color>black</color>
<size>280</size>
<uploadedFileContents>15 HEXADIENE ANTI OPT15 HF 2.mol</uploadedFileContents>
</jmolApplet></jmol>
||
<jmol><jmolApplet>
<color>black</color>
<size>280</size>
<uploadedFileContents>15 HEXADIENE CI ANTI OPT16 DFT 631D.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
|Labelled molecule
|[[File:1,5 hexadiene HF Anti Ci Optimization.JPG|300px]]
||[[File:1,5 hexadiene DFT Anti Ci Optimization.JPG|300px]]
|-
|Energy (a.u.)
| -231.695353
| -234.559704
|}

The CI "anti 2" 1,5-hexadiene optimized at the HF/3-21G level of theory has an energy of -231.6925353 au. This value is the same as the one given in [[Mod:phys3#Appendix 1|Appendix 1]]. This is subsequently re-optimised at B3LYP/6-31G* level and yield a lower energy form than the one at HF/3-21G level. The structure from the HF/3-21G calculation closely resembles that from B3LYP/6-31G* calculation. Table 3 summarizes the dihedral angles and the bond lengths of both structures. The centre dihedral angle and all carbon-carbon bond lengths are similar in both 1,5-hexadiene. There is only a 4 degrees difference in the terminal dihedral angle between them. Overall, the change in geometry is minimal.

{| class="wikitable"
|+ Table 3 Geometry data "anti" 1,5-hexaidene optimized at HF/3-21G and DFT/6-31G* level; Ci
! Method !!colspan="3"| HF !! colspan="3"| DFT
|-
| Dihedral angle(C1-C4-C6-C9);(º) || colspan="3" align="center" |114.7 || colspan="3" align="center"| 118.8
|-
| Dihedral angle(C4-C6-C9-C12);(º) || colspan="3" align="center"|180.0 || colspan="3" align="center"| 180.0
|-
| || C1-C4 || C4-C6 || C6-C9 || C1-C4 || C4-C6 || C6-C9
|-
| align="center"| Bond length(Å) || 1.07 || 1.33 || 1.51 || 1.09 || 1.34 || 1.51
|}

===Frequency Analysis of "anti" 1,5-hexadiene, Ci conformation; DFT/6-31G===

[[Image:1,5 hexadiene DFT Anti Ci Freq spectrum.JPG|frame|centre|400px|Figure 2 Vibrational Spectrum of "anti" 1,5-hexadiene]]

Frequency analysis was carried out. It gives the second derivative of the potential energy surface. If all frequencies are positive, it means a minimum was resulted. The absence of imaginary (negative) frequencies shows that the structure is optimized to a minima. Table 4 shows the thermochemical analysis of the optimized structure.

{| class="wikitable"
|+ Table 4 Summary of energy
! !! Energy (in hatree)
|-
| Sum of electronic and zero point energies (E = Eelec + ZEP), at 0 K || align="center" |-234.469215
|-
| width="430" | Sum of electronic and thermal energies (E = E + Evib + Erot + Etrans), at 298.15 K and 1 atm|| align="center"| -234.461867
|-
| Sum of electronic and thermal enthalpies+ || align="center"| -234.460922
|-
| Sum of electronic and thermal free energies++ ||align="center"| -234.500800
|}

+ An additional correction for RT(H = E + RT)

++ Including entropic contribution to the free energy (G = H-TS)

===Optimization of "Chair" Transition Structures===

The Cope rearrangement have two different transition state: Chair and Boat.

Optimization and Frequency Analysis of Chair Transition Structure (Opt+Freq)

An allyl fragment (CH2CHCH2) was first optimized to TS(Berny) at HF/3-21G level. Two optimized fragments were arranged in the chair form and underwent optimization and frequency analysis. This optimized structure has an imaginary frequency at -818 cm-1. The negative second derivative of the potential energy surface corresponds to a maxima. This shows that the optimization of the chair transition state structure was successful. The imaginary frequency was also animated in table 5.

{| class="wikitable"
|+ Table 5 Results of optimizaed chair transition structure
! Jmol || colspan="3"|Animated vibration at -818 cm-1 || Vibrational spectrum
|-
|
<jmol><jmolApplet>
<title>(3-21G) optimized chair transition state</title>
<color>black</color>
<size>280</size>
<uploadedFileContents>TRANSITION ALLYL FRAG HF OPTFREQ5.mol</uploadedFileContents>
</jmolApplet></jmol>
|
|[[File:Imaginary vibration chair transition state animation.gif|50 x 50 px]]
|
|[[File:Trasition state OPT FREQ IR spectrum.JPG|400 px]]
|}

Alternatively the frozen coordinate method was used to optimize the transition structure. This was done by fixing the distance between the terminal carbons from both allyl fragments to 2.2 Å and then optimized to a minimum (HF/3-21G). A transition state optimization to TS(Berny) was carried out subsequently at HF/3-21G level. This allows the bond forming/breaking distances between the two fragments to be optimized as well. The table below summarizes the geometry data of the transition structures that were optimized differently.

Both optimized transition structures with either frozen or optimized bond forming/breaking distances, show similar C-C bond length and C-C-C angle within one allyl fragment. These are also similar in values compared to the structure from "Opt+Freq" calculation. The main difference lies in the distance between C1-C6 and C3-C4. When the bond forming/breaking distances were optimized, these values are more similar to that in the structure from "Opt+Freq" calculation. This shows that freezing the coordinate would give a less accurate optimization of structure.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Table 6 Geometry data of optimized chair transition structure
! !! Width="120"| Bond length (C1-C2)(Å) !! Width="120"|Angle (C1-C2-C3)(º) !! Width="120"| Distance between (C1-C6) (Å)!! Width="120"|Distance between (C3-C4) (Å)!! rowspan="4" |[[File:Chair Transition State Freeze HF Optimization with number.JPG|250 px]]
|-
| align="center"| Opt + Freq|| height="50" align="center"| 1.39 || align="center"|120.5 || align="center"|2.02 || align="center"|2.02
|-
| align="center"| Opt(Freeze Coordinate) || align="center"| 1.39 || align="center"| 121.8 || align="center"| 2.16 || align="center"| 2.20
|-
| align="center"| Opt(Derivative) || align="center"| 1.39|| align="center"| 120.5|| align="center"| 2.02 || align="center"| 2.02
|}

===Optimization of "Boat" Transition Structures===

{|style="margin: 0 auto;"
|[[File:Failed boat transition state.PNG|thumb|200 px|Figure 3. First attempt of QST2 calculation]]
|[[File:Boat QST2 rearrangement.JPG|thumb|350px|Figure 4. Rearrangement of butadiene]]
|[[File:Cope rearrangement scheme 2.JPG|thumb|200px|Figure 5. Cope Rearrangement]]
|}
The optimized Ci "anti" 1,5-hexadienes were optimized to a transition state and frequency analysis were carried out using the QST2 method. QST2 requires reactant and product as the input and all atoms must be labelled in the same way in both structure. The first calculation was done without any modification to the structure orientation. The job was failed and resulted in the transition structure shown in figure 3. The 1,5-hexadiene molecules were re-orientated so that they had the same arrangement as what shown in figure 4. The modified molecules had a dihedral angle of 0 degree at the centre and 100 degrees for the inside C-C-C angle. The QST2 calculation of the modified structure was successful and the following results (table 7) were obtained. The distance between the two fragments is 2.14 Å. The boat transition structure was optimized and it has an imaginary frequency at -840 cm-1.

The optimization was also carried out using the QST3 calculation. This requires 3 inputs in the following order: the reactant, product, and guess transition state structures. Similar to QST2, the atoms must be labelled in the same order. The energy and geometry of the optimized structure of QST3 calculation resembles that of QST2. It also has an imaginary frequency at -840 cm-1.

{| class="wikitable"
|+ Table 7 Results of optimized boat transition structure
! Jmol || colspan="3"|Vibration at -840 cm-1|| Vibrational spectrum
|-
|
<jmol><jmolApplet>
<title>(3-21G) optimised boat transition state</title>
<color>black</color>
<size>280</size>
<uploadedFileContents>BOAT TRANSITION HF OPTFREQ13 QST2.mol</uploadedFileContents>
</jmolApplet></jmol>
|
|[[File:Imaginary vibration Boat transition state QST2 animation.gif|50 x 50 px ]]
|
|[[File:Boat Transition State QST2 IR spectrum.JPG|400 px]]
|}

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

{| class="wikitable"
|+ Table 8 Results of IRC calculation
! First Calculation || Total Energy along IRC || RMS Gradient Norm along IRC
|-
|[[File:IRC Point 50 Chair Transition Forward Direction Always calculate force constant.gif|50 x 50 px|frame|centre|No. of points along IRC: 50]]
|[[File:IRC Point 50 Chair Transition Total Energy along IRC.JPG|400 px]]
|[[File:IRC Point 50 Chair Transition RMS Gradient Norm along IRC.JPG| 350 px]]
|}

It is difficult to predict which conformers of 1,5-hexadiene will form from the chair and boat transition structures. Intrinsic Reaction Coordinate (IRC) method was used to find out the structure that has the lowest energy. It allows the lowest energy reaction path from the transition state towards the reactants and products to be followed. Only the forward direction of the reaction coordinate was considered here. The number of data points along the IRC was set to 50 and the force constant was set to "calculate always" in the first attempt. Forty-four intermediates were obtained. A second attempt of IRC calculation with 100 points was carried out to ensure the minimum energy geometry was reached. There was no change to the energy graph and the gradient was closed to zero at the end of calculation. These prove that a minimum geometry has reached. A gauche conformer with an energy of -231.691608 a.u. (gradient: 0.00015154 a.u.) was found to be the minimum geometry from this calculation. This is gauche 2 in Appendix 1.

===Optimization of Chair and Boat Transition structures using B3LYP/6-31G*===

The HF/3-21G optimized chair and boat structure were re-optimized using B3LYP/6-31G* method. The following tables present a comparison for the geometries and different energies values. The chair transition structures optimized at HF/3-21G and B3LYP/6-31G* have very similar geometry compared to each other. The same applies to the boat transition structure. However, the energies are lower for the transition structures optimized at B3LYP/6-31G* level.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Table 9 Geometry data of chair and boat transition structure
! || height="40" colspan="3"|HF/3-21G || colspan="3"|B3LYP/6-31G* || rowspan="2" |[[File:Chair Transition State Freeze HF Optimization with number.JPG|170 px]]
|-
! !! Width="120" height="40"| Bond length (C1-C2)(Å) !! Width="120"|Angle (C1-C2-C3)(º) !! Width="120"| Distance between (C1-C6)/(C3-C4) (Å)!! Width="120"| Bond length (C1-C2)(Å) !! Width="120"|Angle (C1-C2-C3)(º) !! Width="120"| Distance between (C1-C6)/(C3-C4) (Å)
|-
| align="center"| Chair TS (Top)|| height="50" align="center"| 1.38 || align="center"|122.0 || align="center"|2.20 || align="center"|1.39 ||align="center"| 122.0 || align="center"|2.20 || rowspan="2" |[[File:Boat Transtion numbering.JPG|170 px]]
|-
| align="center" height="60"| Boat TS (bottom)|| align="center"| 1.41 || align="center"| 121.2 || align="center"| 2.14 || align="center"| 1.39 || align="center"| 121.1 || align="center"| 2.14
|}

{| class="wikitable"
|+ Table 10 Summary of Energy (in hatree)
! || colspan="3"|HF/3-21G || colspan="3"|B3LYP/6-31G*
|-
| || align="center" | Electronic energy || width="125" align="center" | Sum of electronic and zero-point energies (0 K)|| width="130" align="center" | Sum of electronic and thermal energies (298.15 K)|| align="center" | Electronic energy || width="125" align="center" | Sum of electronic and zero-point energies (0 K)|| width="130" align="center" | Sum of electronic and thermal energies (298.15 K)
|-
|align="center" | Chair TS|| align="center" | -231.619332 || align="center" | -231.466702 || align="center" | -231.461343 || align="center" | -234.553938 || align="center" | -234.413269 || align="center" | -234.406982
|-
|align="center" | Boat TS|| align="center" | -231.602802 || align="center" | -231.450928 || align="center" | -231.445299 ||align="center" | -234.542868 || align="center" | -234.401492 || align="center" | -234.395284
|-
|align="center" | Reactant (Ci; Anti)|| align="center" | -231.692535 || align="center" | -231.539539 || align="center" | -231.532565 || align="center" | -234.611712|| align="center" | -234.469215 || align="center" | -234.461867
|}

===Calculation of Activation Energies for Both Transition Structures===
{| class="wikitable"
|+ Table 11 Summary of Activation Energy (in kcal/mol)
! || colspan="2"|HF/3-21G || colspan="2"|B3LYP/6-31G* || Experimental value
|-
| align="center" | Temperature || width="50" align="center" | 0 K || align="center" | 298.15 K || width="50" align="center" | 0 K || align="center" | 298.15 K || align="center" | 0 K
|-
|align="center" | ∆E (Chair)|| width="50" align="center" | 45.70 || align="center" | 44.69 || width="50" align="center" | 35.12 || align="center" | 34.44 || align="center" | 33.5 ± 0.5
|-
|align="center" | ∆E (Boat)|| width="50" align="center" | 55.78 || align="center" | 54.93 || width="50" align="center" | 42.50 ||align="center" | 41.91 || align="center" | 44.7 ± 2.0
|}

The boat transition structure was found to have a higher activation energy than the chair. This can be due to the unfavourable repulsive interaction between the protons in the structure. The activation energies at 0 K of both transition structures optimized at B3LYP/6-31G* level are more similar to the experimental values . This can be explained by the choice of method and basis set. Electronic structure methods such as Hartree-Fock (HF) or Density functional theory (DFT) all approximate the exact solution in some ways. Generally, the lower the energy structure after a geometry optimization, the more suited the method is to describe the ground state.

The HF approximation describe non-interacting electrons under the influence of a mean electron field potential.It also accounts for the Pauli exclusion principle. DFT takes into account the electron correlation, but not the Pauli exclusion principle. The fact that electrons interaction is considered in the calculation gives a better approximation to strongly correlated problems. Different basis sets uses different number of functions to describe each atomic orbital and hence would affect the accuracy of calculation. The 6-31G* is a larger basis set compared to 3-21G in which more gaussian functions are used to describe each atomic orbital. 6-31G* also takes into account the distortion (polarisation) of the orbitals when molecules are formed. This in turn enables the basis set to describe the wavefunction more accurately.

==The Diels Alder Cycloaddition==

===Optimization of cis-butadiene and Molecular Orbitals Analysis===

http://web.chem.ucsb.edu/~kalju/chem226/public/semiemp_intro.html

[[File:Diels Alder reaction scheme.JPG|thumb|centre|Figure 6 Diels Alder reaction of ethene and cis-butadiene|450 px]]

Ethene and cis-butadiene were optimized to a minimum using the AM1 semi-empirical method. Their corresponding HOMO and LUMO were plotted as shown in table 12 and 13. The plane of symmetry bisect the C=C bond in ethene and centre C-C in butadiene.

{| class="wikitable"
|+ Table 12 Ethene MO
! HOMO, symmetric with respect to the plane || LUMO, antisymmetric with respect to the plane
|-
|[[File:Ethene HF HOMO.JPG|250 px]]
|[[File:Ethene HF LUMO.JPG|250 px]]
|}

{| class="wikitable"
|+ Table 13 Cis-butadiene
! Jmol || width="200" | HOMO, antisymmetry with respect to the planne || LUMO, symmetric with respect to the plane
|-
|
<jmol><jmolApplet>
<title>(AM1)optimised cis butadiene</title>
<color>black</color>
<size>250</size>
<uploadedFileContents>CIS BUTADIENE SEMI EMPIRICAL AM1 OPT.mol</uploadedFileContents>
</jmolApplet></jmol>
|[[File:Cis butadiene MO HOMO Transparent.JPG|250 px]]
|[[File:Cis butadiene MO LUMO Transparent.JPG|250 px]]
|}

{| class="wikitable"
|+ Table 14 Diels Alder Transition State
! Jmol || Vibration at -956 cm-1 || Vibration at 147 cm-1
|-
|
<jmol><jmolApplet>
<title>(AM1)optimised Diels Alder transition state</title>
<color>black</color>
<size>250</size>
<uploadedFileContents>DIELS ALDER TS HF OPTFREQ27test AM1.mol</uploadedFileContents>
</jmolApplet></jmol>
|[[File:Diels Alder TS HF OPTFREQ AM1 vibrational movie.gif|50 x 50 px ]]
|[[File:Diels Alder TS HF OPTFREQ AM1 vibrational lowest positive movie.gif|50 x 50 px ]]
|}

The imaginary frequency at -956 cm-1 corresponds to the bond forming/breaking of the Diels-Alder reaction. The animated vibration motion (table 14) shows that the bonds are formed synchronously. The lowest positive frequency does not show the same movement, and bonds do not seem to be forming or breaking.

{| class="wikitable"
|+ Table 15 Diels Alder Transition State MO
!HOMO, Antisymmetry with respect to the plane || LUMO, symmetric with respect to the plane
|-
|[[File:Diels Alder TS AM1 OPTFREQ HOMO 2 with line.jpg|260 px]]
|[[File:Diels Alder TS AM1 OPTFREQ LUMO 2 with line.jpg|250 px]]
|}

The Woodward–Hoffmann rules apply to cycloaddition reaction. It explains the stereochemical outcome of pericyclic reactions by considering the symmetry of the ‘frontier orbitals’ that contribute to the formation and breaking of bonds. A cycloaddition reaction involves the concerted formation of two or more σ bonds between the termini of two or more conjugated π systems. In Diels Alder reactions, two π systems are involved. The highest occupied molecular orbital (HOMO) of the dienophile overlaps with the lowest unoccupied molecular orbital (LUMO) of the diene. The diene contributes 4π electrons, and the dienophile contributes 2π electrons. This gives a total count of 6 electrons and hence the reaction is called [4πs + 2πs] cycloaddition. The reaction is thermally allowed and proceed suprafacially (new bonds form on the same face at both ends) via Hückel topology since it has 4n+2 (n=1) electrons in the system. Similarly, the Dewar and Zimmerman rules states that favourable pericyclic reactions will proceed via an aromatic transition state. If the reaction has a 4n+2 suprafacial topology, it is a Hückel system and reaction is allowed.

The reaction is favored by electron-donating groups such as COR, COOR and CN on the dienophile as this will lower the energy of LUMO. An electron-rich diene is also favoured. These would decrease the energy gap between the HOMO and LUMO. Since butadiene and ethene are discussed here, the effect of substituents is ignored.For an allowed reaction, the orbitals that overlap must have the same symmetry. The antisymmetric HOMO of butadiene interests with the antisymmetric LUMO of ethene to give rise to the antisymmetric HOMO of the transition state. Similarly, the same applied to the LUMO of the transition state. The symmetric HOMO of ethene overlaps with the symmetric LUMO of butadiene to form the symmteric HOMO of the transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Table 16 Geometry data of optimized Diels Alder transition structure
! height="60" Width="150"| Bond length (C1-C2),(C3-C4)(Å) !! Width="150"| Bond length (C2-C3)(Å) !! Width="150"| Distance between (C4-C5),(C1-C6) (Å) !! Width="150"| Distance between (C5-C6) (Å)!! rowspan="4" |[[File:Diels Alder TS numbering.JPG|200 px]]
|-
| align="center"| 1.38 || align="center"|1.39 || align="center"|2.12 || align="center"|1.38
|-
! colspan="2" | Typical sp 3 bond length (Å): 1.54
! colspan="2" | Typical sp 2 bond length (Å): 1.34
|-
! colspan="4" | van der Waals radius of C atom (Å): 1.70
|}

The distances of the bond forming/breaking in the transition structure are 2.12 Å. These distances are much greater than the sp2 and sp3 hybridised C-C bond. They are shorter than twice the van der Waals radius for carbon (3.40 Å) and are not close enough to experience repulsive interaction towards each other. Hence bond formation is favourable.

{| class="wikitable"
|+ Table 17 Results of IRC (No. of points: 60)
! Energy of product(a.u.): 0.0746648 !! Final gradient: 0.0005776
|-
|[[File:Diels alder TS AM1 OPT IRC 60 total energy graph.JPG|500 px]]
|[[File:Diels alder TS AM1 OPT IRC 60 gradient graph.JPG| 450 px]]
|}

An IRC calculation was carried out. This time, both direction was run. The energy graph (table 17) shows an expected reaction coordinate. A minimum geometry in the forward direction has an energy of 0.0746648 a.u. and a gradient close to zero. Increasing the number of data points to 70 made no difference to the results. This proves that a minimum geometry has reached.

===Cyclohexadiene-1,3-diene Reaction with Maleic Anhydride===

The AM1 semi-empirical method was applied for all calculation in this session. Maleic anhydride and cyclohexa-1,3-diene were optimized to a minimum. There HOMO and LUMO were plotted in table 18.

{| class="wikitable"
|+ Table 18 HOMO and LUMO of Maleic Anhydride and Cyclohexa-1,3-diene
! colspan="2" | Maleic Anhydride !! colspan="2" |Cyclohexa-1,3-diene
|-
! HOMO !! LUMO !! HOMO !! LUMO
|-
|[[File:Maleic anhydride HOMO.JPG|250 px]]
|[[File:Maleic anhydride LUMO.JPG|250 px]]
|[[File:Cyclohexadiene HOMO.JPG|250 px]]
|[[File:Cyclohexadiene LUMO.JPG|250 px]]
|}

The reactants were rearranged into a guess structure that resembles the exo transition state. They were optimized to TS(Berny) by freezing the coordinates and subsequently unfrozen. Frequency analysis was carried out. The imaginary frequency at -812 cm-1 confirms that a transition state was optimized.

{| class="wikitable"
|+ Table 19 Exo Transition State
! Jmol || Vibration at -812 cm-1 || HOMO
|-
|
<jmol><jmolApplet>
<title>optimised exo transition state </title>
<color>black</color>
<size>260</size>
<uploadedFileContents>EXO TRANSITION STATE OPT AM1.mol</uploadedFileContents>
</jmolApplet></jmol>
|[[File:Exo AM1 OPTFREQ vibration movie.gif|50 x 50 px]]
|[[File:Exo AM1 OPTFREQ HOMO.JPG|260 px]]
|}

IRC calculation was carried out for the exo transition structure and yielded the following results:

{| class="wikitable"
|+ Table 20 Results of IRC of Exo Transition Structure, No. of data points: 40
! Energy of the product (a.u.): -0.160168 !! Gradient: 0.0001171
|-
|[[File:Exo Transition state OPT AM1 IRC40 Total energy graph.JPG|250 px]]
|[[File:Exo Transition state OPT AM1 IRC40 Gradient graph.JPG| 250 px]]
|}

The reactants were rearranged into the endo transition state. They were optimized to TS(Berny) by freezing the coordinates and subsequently unfrozen. Frequency analysis was carried out. The imaginary frequency at -806 cm-1 confirms that a transition state was obtained.

{| class="wikitable"
|+ Table 21 Endo Transition State
! Jmol || Vibration at -806 cm-1 || HOMO
|-
|
<jmol><jmolApplet>
<title>(AM1) optimised exo transition state </title>
<color>black</color>
<size>260</size>
<uploadedFileContents>ENDO TRANSITION AM1 OPT.mol</uploadedFileContents>
</jmolApplet></jmol>
|[[File:Endo AM1 OPTFREQ3 vibration movie.gif|50 x 50 px ]]
|[[File:Endo AM1 OPT HOMO.JPG|260 px]]
|}

IRC calculation was carried out for the endo transition structure and yielded the following results. Another IRC calculation was run from the last point of the first calculation and yields the same results. This shows that the minimum geometry was found.

{| class="wikitable"
|+ Table 22 Results of IRC of Endo Transition Structure, No. of data points:20
! Energy of product (a.u.): -0.159874 !! Gradient:0.00002890
|-
|[[File:Endo Transition state OPT AM1 IRC20 Total energy graph.JPG|250 px]]
|[[File:Endo Transition state OPT AM1 IRC20 gradient graph.JPG| 250 px]]
|}

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ Table 23 Geometry data and energy of Exo and Endo transition structure
! !! Width="150" height="40"| Bonding forming distance (C2-C8),(C5-C7)(Å) !! Width="150"| Orientation (C3-C9),(C4-C11)(Å) !! width="150"|Maleic anhydride C=O bond length (Å)!! Width="150"|Maleic anhydride C-C bond length (C7-C8)/(C8-C9)(Å) !!Width="150"|Cyclohexadiene C-C bond length(Å) !! Width="150"|Cyclohexadiene C=C bond length(Å) !! Width="150"| Energy (a.u.) || rowspan="2" |[[File:Exo Transition State numbering.JPG|170 px]]
|-
| align="center"| Exo TS(Top)|| height="120" align="center"| 2.17|| align="center"|2.95 || align="center"|1.22 || align="center"|1.41/1.49 || align="center"|1.39/1.49 || align="center"|1.39 || align="center"|-0.0504198
|-
| align="center" height="60"| Endo TS (bottom)|| align="center"| 2.16 || align="center"| 2.89 || align="center"| 1.22 || align="center"|1.41/1.49 || align="center"|1.39/1.49 ||align="center"|1.39 || align="center"| -0.0515048|| rowspan="2" |[[File:Endo Transition state numbering.JPG|170 px]]
|}

====Analysis====

[[File:Diels Alder 2 reaction scheme.JPG|thumb|centre|400 px| Figure 7 Reaction scheme of Diels-Alder reaction of cyclohexadiene-1,3-diene with Maleic Anhydride]]

In cycloaddition, two new bonds are formed at the same time. Two filled p orbitals and two empty p orbitals need to be arranged at the right place and with the right symmetry in order to interact. In this Diels-Alder reaction, the LUMO of electron poor anhydride interacts with the HOMO of the diene. A node is present at the middle of HOMO of the diene and same in LUMO of dienophile. By Woodward–Hoffmann rules, it is an allowed interaction. The interaction of LUMO of diene and HOMO of anhydride also have the correct symmetry but due to the larger energy gap between them, it is less favourable. The HOMO of the diene and the LUMO of dienophile are closer in energy and gives a better overlap.

[[File:Second orbital effect.JPG|thumb|centre|400 px| Figure 8 Second orbital overlap effect of Diels-Alder reaction of cyclohexadiene-1,3-diene with Maleic Anhydride. a)Through space interaction between C=O and the back of diene. b)Primary and secondary orbital overlaps in endo structure. c) Overlap of orbitals in exo structure]]

Second orbital overlap effect was proposed by Woodward and Hoffmann. It is the positive overlap of inactive orbitals in the frontier molecular orbitals of a pericyclic reaction. In the endo transition structure (Figure 8b), it has the primary orbital overlap in which the p-orbitals of the anhydride LUMO interacts with the diene HOMO. However, the p-orbital on both side of the C=O also interacts with the p-orbitals at the back of the diene. These interactions are descriped as secondary as there are no change in the bonds. They interact strongly in the endo transition state (Figure 8a) but such interaction is not possible in the exo transition state (Figure 8c). The secondary overlap gives a stabilizing effect in the endo structure irrespective of the energies of the HOMO and LUMO.

Table 23 shows a comparison of structure and energy of the two transition structure. In general, both structures resemble each other. The main difference lies in the through space distance (Orientation distance) between the -(C=O)-O-(C=O)- fragment of maleic anhydride and the C atoms of -CH2-CH2- in exo and -CH=CH- in endo . This distance is closer in endo. The endo structure also has a lower energy than the exo. These provides evidence that the endo structure is stabilized by the secondary overlap. Another way of analysing the presence of secondary overlaps, is to look at the MOs of the transition structure.

{| class="wikitable"
|+ Table 24 MOs of Endo and Exo Transition Structure
! !! HOMO - 4 !! HOMO !! LUMO !! LUMO + 1 !! LUMO + 2
|-
|Endo TS
|[[File:Endo HOMO-4.JPG|240 px]]
|[[File:Endo AM1 OPT HOMO.JPG| 240 px]]
|[[File:Endo AM1 OPTFREQ3 LUMO.JPG|240 px]]
|[[File:Endo LUMO+1.JPG| 240 px]]
|[[File:Endo LUMO+2.JPG| 240 px]]
|-
|Exo TS
|[[File:Exo HOMO-4.JPG|240 px]]
|[[File:Exo AM1 OPTFREQ HOMO.JPG| 240 px]]
|[[File:Exo AM1 OPTFREQ LUMO.JPG|240 px]]
|[[File:Exo LUMO+1.JPG| 240 px]]
|[[File:Exo LUMO+2.JPG| 240 px]]
|}

Table 24 shows a comparison of several MOs from the endo and exo transition state. The secondary orbitals overlap was not observed in the HOMO nor LUMO of the endo structure. The interaction was present in the HOMO-4, LUMO+1 and LUMO+2 instead. This is possibly due to the fact that orbital mixing was not taken into account in the calculation. A high level of theory such as HK or DFT might give a result closer to expectation. In the exo transition structure, no secondary orbitals overlap was observed which correlates with the discussion above.

Despite having endo form as the lower energy transition structure, it was shown that it leads to a higher energy product. Table 20 and table 22 shows the result of the IRC calculation. The energy of the endo product has an energy of -0.159874 a.u. where and the energy of the exo product has an energy of -0.160168 a.u.. This shows that the endo product is less stable. The structure experience steric repulsive interaction between the alkene of the six membered ring and the carbonyl groups of the dienophile. In an irreversible Diels-Alder reactions, therefore it would be the kinetic product of the reaction. The kinetic product is formed faster. If the reaction is under kinetic control, the energies of the transition states would dictate the outcome of the reaction. By Hammmond's postulate, the starting material, intermediate or product closest in energy to the transition state of the interest will be similar in structure.