ChemWiki - User contributions [en]

Rep:Mod:PLZ

2016-03-11T09:50:19Z

Oc113:

==Introduction==

This experiment aims to determine the energy of the transition state for the Cope Rearrangement and the Diels-Alder reaction, two concerted cyclic reactions. Transition states for these reactions will be determined using a variety of methods, using the Gaussian program which provides numerical solutions to the Schrodinger equation.

==The Cope Rearrangement==

The Cope Rearrangement is a concerted sigmatropic rearrangement of 1,5 dienes <ref name="Cope def"/>. There are both boat and chair transition states and the energies of these will be investigated.

===Optimising Hexadiene===

1,5 Hexadiene has several possible conformations, with varying energies. A selection of conformers were optimised using the 3-21G Hartree-Fock (HF) method in order to investigate the relative energies.

====3-21G Hartree-Fock Optimisations====

The various conformations were obtained by drawing a molecule of 1,5 hexadiene in an orientation close to the desired conformer, using the clean tool to adjust the geometry, and then performing the optimisation. The point group tool was then used to investigate the symmetry of the molecule. Energies and point groups for the four conformers investigated are summarised in the table below.

{| class="wikitable"
|'''Conformer'''
|'''Energy (hartrees)'''
|'''Geometry'''
|'''Point Group'''
|'''Log File'''
|-
|Anti 1
| -231.69260235
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZAnti1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|C2
|[https://wiki.ch.ic.ac.uk/wiki/images/5/52/PLZANTI1.LOG Log file]
|-
|Anti 2
| -231.69253528
||<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZAnti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|Ci
|[https://wiki.ch.ic.ac.uk/wiki/images/c/c5/PLZANTI2T.LOG Log file]
|-
|Gauche 3
| -231.69266121
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZGauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|C1
|[https://wiki.ch.ic.ac.uk/wiki/images/d/d6/PLZGAUCHE3T.LOG Log file]
|-
|Gauche 4
| -231.69153032
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZGauche4.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|C2
|[https://wiki.ch.ic.ac.uk/wiki/images/8/8e/PLZGAUCHE4.LOG Log file]
|}

As can be seen from the table that the gauche 3 conformer is the lowest energy conformation. Considering steric interactions alone, this is unexpected as the antiperiplanar conformations have less strain and therefore would be expected to have a lower energy. Examination of the molecular orbitals (MOs) of the gauche 3 conformer shows a favourable interaction between the π orbitals on the terminal carbon atoms, leading to a lower overall energy.

<jmol>
<jmolApplet>
<size>300</size><color>white</color>
<script>isosurface color orange purple "images/a/a8/PLZhexadieneMO.jvxl" translucent;</script>
<uploadedFileContents>PLZhexadieneMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>

====6-31G* Density Functional Theory Optimisations====

The anti 2 conformer were then optimised using Density Functional Theory (DFT) with a 6-31G* basis set. As this differs from the 3-21G basis set used with the HF optimisation, the energies obtained cannot be meaningfully compared. As the 6-31G* basis set is more sophisticated than 3-21G (adding polarisation functions to atoms beyond H), it would be expected that the energies obtained are more accurate. Geometries of the two different optimisations have been examined and displayed in the table below in order to provide a level of comparison between the basis sets. The DFT method models the system as a series of functions of spatial electron density <ref name="DFT"/>, compared to a series of single electron wavefunctions in HF <ref name="HF"/>. [https://wiki.ch.ic.ac.uk/wiki/images/f/f9/ANTI2TB3LYP_631G_D.LOG Log file]

{| class="wikitable"
|rowspan="2" |'''Calculation Type'''
|colspan="3" |'''Bond Length (Å)'''
|'''Dihedral angle'''
|-
|C1-C2
|C2-C3
|C3-C4
|C1-C2-C3-C4
|-
|HF 3-21G
|1.316
|1.509
|1.553
| -144.66
|-
|DFT B3LYP 6-31G*
|1.334
|1.504
|1.548
| -118.58
|}

The data show that there is not a particularly large difference in the geometries, illustrating the role of less accurate basis sets in providing a good approximation of molecules in a shorter space of time.

=====Frequency Calculations=====

A frequency calculation was performed on the anti 2 conformer, this allowed the IR spectrum to be produced and thermochemistry data to be determined. This is summarised in the table below.
{| class="wikitable"
|'''Quantity'''
|'''Energy (Hartrees)'''
|-
|Zero-point Energy
|0.142497
|-
|Sum of electronic and zero-point Energies
| -234.469215
|-
|Sum of electronic and thermal Energies
| -234.461866
|-
|Sum of electronic and thermal Enthalpies
| -234.460922
|-
|Sum of electronic and thermal Free Energies
| -234.500800
|}

The Zero-point energy is the absolute minimum energy for the system. This is combined with the electronic energy (at 0K), then the thermal energy (the energy of translations, vibrations and rotations at 298.15K), an enthalpic adjustment is then made, and finally a free energy adjustment is made.

===Optimising a Transition State===

A transition state was optimised using three different methods, a IRC was run and activation energies were calculated.

Transition states do not occur at energy minimums and so optimisations can be difficult as they can proceed towards nonsensical geometries. In order to reduce this risk, optimisations can be started from a "best guess" geometry that is close to the true transition state, a frozen co-ordinate method, where sections of the molecule are optimised individually, or the QST2 method, which specifies the products and reactants and attempts to find an intermediate transition state. The Cope Rearrangement has both a chair and a boat transition state, the chair transition state was determined using the "best guess" and frozen co-ordinate methods, and the boat transition state was determined using the QST2 method.

Numerical results from these optimisations (such as energy and imaginary vibration frequencies are summarised at the end.

===="Best Guess" Type Method====

Transition states can be determined by calculating a Hessian matrix that contains one negative eigenvalue, which is then continuously updated as the optimisation proceeds towards the transition state. In this optimisation, a CH<sub>2</sub>CHCH<sub>2</sub> allyl fragment was drawn and optimised using a HF method with a 3-21G basis set. Two of these were then positioned roughly 2.2 Å apart, and orientated so that they formed a transition state-like structure. [https://wiki.ch.ic.ac.uk/wiki/images/5/53/PLSCOPEGUESS.LOG Log file]

<jmol>
<jmolApplet>
<title>Chair TS Vibration</title>
<color>white</color>
<size>250</size>
<script>frame 1.25;vibration 1;</script>
<uploadedFileContents>PLSCOPEGUESS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Co-ordinate Method====

This approach seeks to eliminate some of the degrees of freedom of the system by freezing the reaction co-ordinate and optimising the rest of the molecule in order to produce a structure closer to the transition state. The reaction co-ordinate is then unfrozen and a transition state is found by differentiating down the reaction co-ordinate. This optimisation using the same starting orientation as the "best guess" method, with the addition of the frozen co-ordinates. [https://wiki.ch.ic.ac.uk/wiki/images/1/17/PLZCOPEFROZE.LOG Log file]

<jmol>
<jmolApplet>
<title>Chair TS Vibration</title>
<color>white</color>
<size>250</size>
<script>frame 1.21;vibration 1;</script>
<uploadedFileContents>PLZCOPEFROZE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====QST2 Method====

This method used adjusted versions of the previously optimised anti 2 conformer of 1,5 hexadiene. Atom labeling was used to specify the location of each atom in the reactant and product, and allow a transition state to be formed. The geometries were also altered in order to produce structures closer to the transition state, as the original linear structures were not close enough to produce a transition state from optimisation. [https://wiki.ch.ic.ac.uk/wiki/images/3/3a/PLZCOPEQT.LOG Log file]

<jmol>
<jmolApplet>
<title>Boat TS Vibration</title>
<color>white</color>
<size>250</size>
<script>frame 1.43;vibration 1;</script>
<uploadedFileContents>PLZCOPEQT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====IRC====

While transition states have been determined, it is not possible to tell which conformers of 1,5 hexadiene they link. Therefore an IRC was run to profile the reaction. This was performed on the chair transition state produced by the frozen co-ordinate method. The IRC calculation follows the steepest gradient down the potential energy surface in order to locate the resultant geometry. The results of the IRC calculation are summarised below, and the resultant conformer is gauche 4 (based on the final energies reached in the IRC path). [https://wiki.ch.ic.ac.uk/wiki/images/c/cf/PLZCOPEIRC.LOG Log file]

{|
|[[File:PLZIRCen.png|none|thumb|500px|Fig14. IRC plot for the chair TS]]
|[[File:PLZIRCgrad.png|none|thumb|500px|Fig15. RMS plot for the chair TS]]
|}

====Activation Energies and Other Numerical Data====

The chair and boat transition states were repeated at the DFT B3LYP 6-31G* level of theory, and the results are summarised here. [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/PLZCOPECHAIRDFT.LOG Chair log file] [https://wiki.ch.ic.ac.uk/wiki/images/1/1c/PLZCOPEBOATDFT.LOG Boat log file]

{| class="wikitable"
|
|colspan="3"|'''HF 3-21G'''
|colspan="3"|'''DFT BLYP 6-31G*'''
|-
|
|Electronic Energy (Hartrees)
|Sum of Electronic and Zero-Point Energies (Hartrees)
|Sum of Electronic and Thermal Energies at 298.15K (Hartrees)
|Electronic Energy (Hartrees)
|Sum of Electronic and Zero-Point Energies (Hartrees)
|Sum of Electronic and Thermal Energies at 298.15K (Hartrees)
|-
|Chair TS
| -231.619322
| -231.466698
| -231.461339
| -234.610701
| -234.468248
| -234.460958
|-
|Boat TS
| -231.602802
| -231.450930
| -231.445302
| -234.611329
| -234.468693
| -234.461464
|-
|Anti 2 Conformer
| -231.692535
| -231.539540
| -231.532566
| -234.611711
| -234.469215
| -234.461866
|}

A summary of activation energies is provided below (kcal/mol). These appear to suggest that the HF method produces vastly more difference in energies than the DFT method.
{| class="wikitable"
|
|colspan="2"|'''HF 3-21G'''
|colspan="2"|'''DFT BLYP 6-31G*'''
|-
|
|0K
|298.15K
|0K
|298.15K
|-
|ΔE Chair
|45.709010578
|44.695583543
|0.606801203
|0.569778172
|-
|ΔE Boat
|55.60357249
|54.758945376
|0.327559698
|0.252258618
|}

==Diels-Alder Cycloaddition==

The Diels-Alder reaction is a reaction between a conjugated diene and a dienophile, with the concerted formation of two sigma bonds and the breaking of two pi bonds with a cyclic transition state.

Two examples of this reaction were examined, the reaction of ethene and cis-butadiene, and the reaction of malaeic anhydride and cyclohexa-1,3-diene.

The calculations performed all used the semi-empirical AM1 method (semi-empirical refers to the use of some empirical data to make approximations on the HF method, allowing calculations to be run at a greater speed but with reduced accuracy). The AM1 method is an adjusted version of the Neglect of Diatomic Differential Overlap method, which ignores certain integrals such as two electron replusion integrals in order to save time on calculations.

===The Reaction of Ethene and cis-Butadiene===

Ethene was drawn and optimised, and the HOMO and LUMO are displayed below. It can be seen that the HOMO is symmetric and the LUMO is antisymmetric. [https://wiki.ch.ic.ac.uk/wiki/images/4/4c/PLZETHENE.LOG Log file]
{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/c/cf/PLZetheneHOMO.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZetheneHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/b/b1/PLZetheneLUMO.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZetheneLUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}
This process was repeated with cis-butadiene with a dihedral angle of 0<sup>o</sup>, the antisymmetric HOMO and symmetric LUMO are displayed below. [https://wiki.ch.ic.ac.uk/wiki/images/2/20/PLZBUTADIENE.LOG Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/7/72/PLZbutadieneHOMO.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZbutadieneHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/a/a1/Cisbutdiene_opt_mo12.cub.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZbutadieneHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}
The transition state of the reaction was then calculated using the frozen co-ordinate method. An imaginary frequency was found and the vibration corresponding to the transition state is shown below. The synchronised motion in the vibration is indicative of a concerted reaction. The first positive vibration has also been shown to illustrate that it is not involved in the formation of the transition state. [https://wiki.ch.ic.ac.uk/wiki/images/0/07/PLZda1.LOG Log file]

<jmol>
<jmolApplet>
<title>Imaginary Vibration</title><color>white</color><size>300</size>
<script>frame 1.7;vibration 1</script>
<uploadedFileContents>PLZda1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
<jmol>
<jmolApplet>
<title>Lowest Positive Vibration</title><color>white</color><size>300</size>
<script>frame 1.8;vibration 1</script>
<uploadedFileContents>PLZda1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
MO data has been calculated for the transition state and is displayed below.

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/0/01/PLZda1HOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZda1HOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/4/4a/PLZda1LUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZda1LUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}

As can be seen from these 6 MOs, the HOMO of the transition state is comprised of the HOMO from cis-butadiene and the LUMO from ethene, and is also antisymmetric. Likewise the LUMO of the transition state is comprised of the LUMO from cis-butadiene and the HOMO from ethene and is also symmetric.

The transition structure is shown below with a forming bond length of 2.21 Å. Typical bond lengths for sp<sup>2</sup> and sp<sup>3</sup> hybridised carbon carbon bond are 1.34 Å and 1.54 Å respectively <ref name="Clen"/>. The Wan der Waals radius of a carbon atom is 1.70 Å <ref name="Cwaal"/>. The bond lengths in the transition state are between two Wan der Waals radii and a normal carbon-carbon bond, suggesting a bond is in the process of forming.

===The Reaction of Maleic Anhydride and Cyclohexa-1,5-diene===

Maleic Anydride was first drawn and optimised, and MOs were calculated. [https://wiki.ch.ic.ac.uk/wiki/images/e/ef/PLZMAL.LOG Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/7/70/PLZMALHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZMALHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/0/08/PLZMALLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZMALLUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>

|}

The same was done for Cyclohexa-1,5-diene. [https://wiki.ch.ic.ac.uk/wiki/images/7/7b/PLZCYCLO.LOG Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/d/d2/PLZCYHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZCY.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/d/da/PLZCYLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZCY.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}

The endo and exo transition states were determined separately, through the frozen co-ordinate method.

====Endo State====

The energy of the transition state is -0.05150480, the MOs and imaginary vibration are displayed below, together with the bond distances. [https://wiki.ch.ic.ac.uk/wiki/images/3/36/PLZENDO.log Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/0/0b/PLZENDOHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZENDOHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/3/3e/PLZENDOLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZENDOLUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Vibration</title><color>white</color><size>300</size>
<script>frame 1.7;vibration 1</script>
<uploadedFileContents>PLZENDO.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Bond Distances</title><color>white</color><size>500</size>
<script>;measure 7 17; measure 1 16;measure RANGE 1.0 2 ALL (carbon) (carbon); measure 15 4; measure 18 6</script>
<uploadedFileContents>PLZENDO.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|}

====Exo State====

The energy of the transition state is -0.05041985, the MOs and imaginary vibration are displayed below together with bond distances. [https://wiki.ch.ic.ac.uk/wiki/images/0/0a/PLZEXO.LOG Log file]
{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/e/e4/PLZEXOHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZEXOHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/9/9c/PLZEXOLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZEXOHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Vibration</title><color>white</color><size>300</size>
<script>frame 1.9;vibration 1</script>
<uploadedFileContents>PLZEXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Bond Distances</title><color>white</color><size>500</size>
<script>;measure 7 17; measure 1 16;measure RANGE 1.0 2 ALL (carbon) (carbon); measure 15 13; measure 18 11</script>
<uploadedFileContents>PLZEXO.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|}

====Comparison of Transition States====
The exo state can be seen to have a slightly higher energy. Examining the C-C bond distances shows that all existing bond distances are becoming equal, indicating a concerted reaction. Equally, there are two through space carbon-carbon distances that are less than two van der waals radii, indicating that bonds are in the process of being formed. There does not appear to be any significant difference in bond lengths between the exo and endo states, despite the different geometry. It might be theorised that there is a steric interaction between the O=C-O-C=O fragment of the maleic anhydride and the hydrogen on the diene. As can be seen above, the conformer withsp<sup>3</sup> hydrogens are significantly closer to the maleic anhydride, which could slightly raise the energy of the exo state.

Looking for secondary orbital effects, it does not appear that the C=O π* orbitals are close enough to the diene π orbitals to interact favourably. They are noticeably closer on the endo state however, and it is possible there is a weak interaction beyond the drawn boundary surface (as they do not represent finite boundaries).

==Conclusions==

The experiment determined that the chair transition state appears to have a marginally lower energy than the boat transition state. Additionally it was found that the endo product for the diels alder reaction between cyclo-1-5-hexadiene and maleic anhydride is favoured, due to steric interactions.

While difficult to document, the experiment also provided an insight into using appropriate basis for computations, in order to balance speed with the accuracy of the data required, particularly with longer calculations such as IRC paths on larger molecules.

==References==
<references>
<ref name="Cope def"> S. J. Rhoads, N. R. Raulins. <i>Organic Reactions</i>. 1975(22): 1–252. DOI:10.1002/0471264180.or022.01</ref>
<ref name="DFT"> R. G. Parr, Electron Distributions and the Chemical Bond, 1982, 95–100.</ref>
<ref name="HF"> 2 J. C. Slater, Physical Review, 1951, 81, 385–390. </ref>
<ref name="Clen">E. Lippert. The Strengths of Chemical Bonds. Butterworths Publications Ltd., London 1958. </ref>
<ref name="Cwaal"> A. Bondi. van der Waals Volumes and Radii. J Phys Chem. 1964;68(3):441–51. </ref>
</references>

Rep:Mod:PLZ

2016-03-11T09:40:56Z

Oc113:

==The Cope Rearrangement==

The Cope Rearrangement is a concerted sigmatropic rearrangement of 1,5 dienes <ref name="Cope def"/>. There are both boat and chair transition states and the energies of these will be investigated.

===Optimising Hexadiene===

1,5 Hexadiene has several possible conformations, with varying energies. A selection of conformers were optimised using the 3-21G Hartree-Fock (HF) method in order to investigate the relative energies.

====3-21G Hartree-Fock Optimisations====

The various conformations were obtained by drawing a molecule of 1,5 hexadiene in an orientation close to the desired conformer, using the clean tool to adjust the geometry, and then performing the optimisation. The point group tool was then used to investigate the symmetry of the molecule. Energies and point groups for the four conformers investigated are summarised in the table below.

{| class="wikitable"
|'''Conformer'''
|'''Energy (hartrees)'''
|'''Geometry'''
|'''Point Group'''
|'''Log File'''
|-
|Anti 1
| -231.69260235
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZAnti1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|C2
|[https://wiki.ch.ic.ac.uk/wiki/images/5/52/PLZANTI1.LOG Log file]
|-
|Anti 2
| -231.69253528
||<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZAnti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|Ci
|[https://wiki.ch.ic.ac.uk/wiki/images/c/c5/PLZANTI2T.LOG Log file]
|-
|Gauche 3
| -231.69266121
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZGauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|C1
|[https://wiki.ch.ic.ac.uk/wiki/images/d/d6/PLZGAUCHE3T.LOG Log file]
|-
|Gauche 4
| -231.69153032
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZGauche4.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|C2
|[https://wiki.ch.ic.ac.uk/wiki/images/8/8e/PLZGAUCHE4.LOG Log file]
|}

As can be seen from the table that the gauche 3 conformer is the lowest energy conformation. Considering steric interactions alone, this is unexpected as the antiperiplanar conformations have less strain and therefore would be expected to have a lower energy. Examination of the molecular orbitals (MOs) of the gauche 3 conformer shows a favourable interaction between the π orbitals on the terminal carbon atoms, leading to a lower overall energy.

<jmol>
<jmolApplet>
<size>300</size><color>white</color>
<script>isosurface color orange purple "images/a/a8/PLZhexadieneMO.jvxl" translucent;</script>
<uploadedFileContents>PLZhexadieneMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>

====6-31G* Density Functional Theory Optimisations====

The anti 2 conformer were then optimised using Density Functional Theory (DFT) with a 6-31G* basis set. As this differs from the 3-21G basis set used with the HF optimisation, the energies obtained cannot be meaningfully compared. As the 6-31G* basis set is more sophisticated than 3-21G (adding polarisation functions to atoms beyond H), it would be expected that the energies obtained are more accurate. Geometries of the two different optimisations have been examined and displayed in the table below in order to provide a level of comparison between the basis sets. The DFT method models the system as a series of functions of spatial electron density <ref name="DFT"/>, compared to a series of single electron wavefunctions in HF <ref name="HF"/>. [https://wiki.ch.ic.ac.uk/wiki/images/f/f9/ANTI2TB3LYP_631G_D.LOG Log file]

{| class="wikitable"
|rowspan="2" |'''Calculation Type'''
|colspan="3" |'''Bond Length (Å)'''
|'''Dihedral angle'''
|-
|C1-C2
|C2-C3
|C3-C4
|C1-C2-C3-C4
|-
|HF 3-21G
|1.316
|1.509
|1.553
| -144.66
|-
|DFT B3LYP 6-31G*
|1.334
|1.504
|1.548
| -118.58
|}

The data show that there is not a particularly large difference in the geometries, illustrating the role of less accurate basis sets in providing a good approximation of molecules in a shorter space of time.

=====Frequency Calculations=====

A frequency calculation was performed on the anti 2 conformer, this allowed the IR spectrum to be produced and thermochemistry data to be determined. This is summarised in the table below.
{| class="wikitable"
|'''Quantity'''
|'''Energy (Hartrees)'''
|-
|Zero-point Energy
|0.142497
|-
|Sum of electronic and zero-point Energies
| -234.469215
|-
|Sum of electronic and thermal Energies
| -234.461866
|-
|Sum of electronic and thermal Enthalpies
| -234.460922
|-
|Sum of electronic and thermal Free Energies
| -234.500800
|}

The Zero-point energy is the absolute minimum energy for the system. This is combined with the electronic energy (at 0K), then the thermal energy (the energy of translations, vibrations and rotations at 298.15K), an enthalpic adjustment is then made, and finally a free energy adjustment is made.

===Optimising a Transition State===

A transition state was optimised using three different methods, a IRC was run and activation energies were calculated.

Transition states do not occur at energy minimums and so optimisations can be difficult as they can proceed towards nonsensical geometries. In order to reduce this risk, optimisations can be started from a "best guess" geometry that is close to the true transition state, a frozen co-ordinate method, where sections of the molecule are optimised individually, or the QST2 method, which specifies the products and reactants and attempts to find an intermediate transition state. The Cope Rearrangement has both a chair and a boat transition state, the chair transition state was determined using the "best guess" and frozen co-ordinate methods, and the boat transition state was determined using the QST2 method.

Numerical results from these optimisations (such as energy and imaginary vibration frequencies are summarised at the end.

===="Best Guess" Type Method====

Transition states can be determined by calculating a Hessian matrix that contains one negative eigenvalue, which is then continuously updated as the optimisation proceeds towards the transition state. In this optimisation, a CH<sub>2</sub>CHCH<sub>2</sub> allyl fragment was drawn and optimised using a HF method with a 3-21G basis set. Two of these were then positioned roughly 2.2 Å apart, and orientated so that they formed a transition state-like structure. [https://wiki.ch.ic.ac.uk/wiki/images/5/53/PLSCOPEGUESS.LOG Log file]

<jmol>
<jmolApplet>
<title>Chair TS Vibration</title>
<color>white</color>
<size>250</size>
<script>frame 1.25;vibration 1;</script>
<uploadedFileContents>PLSCOPEGUESS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Co-ordinate Method====

This approach seeks to eliminate some of the degrees of freedom of the system by freezing the reaction co-ordinate and optimising the rest of the molecule in order to produce a structure closer to the transition state. The reaction co-ordinate is then unfrozen and a transition state is found by differentiating down the reaction co-ordinate. This optimisation using the same starting orientation as the "best guess" method, with the addition of the frozen co-ordinates. [https://wiki.ch.ic.ac.uk/wiki/images/1/17/PLZCOPEFROZE.LOG Log file]

<jmol>
<jmolApplet>
<title>Chair TS Vibration</title>
<color>white</color>
<size>250</size>
<script>frame 1.21;vibration 1;</script>
<uploadedFileContents>PLZCOPEFROZE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====QST2 Method====

This method used adjusted versions of the previously optimised anti 2 conformer of 1,5 hexadiene. Atom labeling was used to specify the location of each atom in the reactant and product, and allow a transition state to be formed. The geometries were also altered in order to produce structures closer to the transition state, as the original linear structures were not close enough to produce a transition state from optimisation. [https://wiki.ch.ic.ac.uk/wiki/images/3/3a/PLZCOPEQT.LOG Log file]

<jmol>
<jmolApplet>
<title>Boat TS Vibration</title>
<color>white</color>
<size>250</size>
<script>frame 1.43;vibration 1;</script>
<uploadedFileContents>PLZCOPEQT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====IRC====

While transition states have been determined, it is not possible to tell which conformers of 1,5 hexadiene they link. Therefore an IRC was run to profile the reaction. This was performed on the chair transition state produced by the frozen co-ordinate method. The IRC calculation follows the steepest gradient down the potential energy surface in order to locate the resultant geometry. The results of the IRC calculation are summarised below, and the resultant conformer is gauche 4 (based on the final energies reached in the IRC path). [https://wiki.ch.ic.ac.uk/wiki/images/c/cf/PLZCOPEIRC.LOG Log file]

{|
|[[File:PLZIRCen.png|none|thumb|500px|Fig14. IRC plot for the chair TS]]
|[[File:PLZIRCgrad.png|none|thumb|500px|Fig15. RMS plot for the chair TS]]
|}

====Activation Energies and Other Numerical Data====

The chair and boat transition states were repeated at the DFT B3LYP 6-31G* level of theory, and the results are summarised here. [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/PLZCOPECHAIRDFT.LOG Chair log file] [https://wiki.ch.ic.ac.uk/wiki/images/1/1c/PLZCOPEBOATDFT.LOG Boat log file]

{| class="wikitable"
|
|colspan="3"|'''HF 3-21G'''
|colspan="3"|'''DFT BLYP 6-31G*'''
|-
|
|Electronic Energy (Hartrees)
|Sum of Electronic and Zero-Point Energies (Hartrees)
|Sum of Electronic and Thermal Energies at 298.15K (Hartrees)
|Electronic Energy (Hartrees)
|Sum of Electronic and Zero-Point Energies (Hartrees)
|Sum of Electronic and Thermal Energies at 298.15K (Hartrees)
|-
|Chair TS
| -231.619322
| -231.466698
| -231.461339
| -234.610701
| -234.468248
| -234.460958
|-
|Boat TS
| -231.602802
| -231.450930
| -231.445302
| -234.611329
| -234.468693
| -234.461464
|-
|Anti 2 Conformer
| -231.692535
| -231.539540
| -231.532566
| -234.611711
| -234.469215
| -234.461866
|}

A summary of activation energies is provided below (kcal/mol). These appear to suggest that the HF method produces vastly more difference in energies than the DFT method.
{| class="wikitable"
|
|colspan="2"|'''HF 3-21G'''
|colspan="2"|'''DFT BLYP 6-31G*'''
|-
|
|0K
|298.15K
|0K
|298.15K
|-
|ΔE Chair
|45.709010578
|44.695583543
|0.606801203
|0.569778172
|-
|ΔE Boat
|55.60357249
|54.758945376
|0.327559698
|0.252258618
|}

==Diels-Alder Cycloaddition==

The Diels-Alder reaction is a reaction between a conjugated diene and a dienophile, with the concerted formation of two sigma bonds and the breaking of two pi bonds with a cyclic transition state.

Two examples of this reaction were examined, the reaction of ethene and cis-butadiene, and the reaction of malaeic anhydride and cyclohexa-1,3-diene.

The calculations performed all used the semi-empirical AM1 method (semi-empirical refers to the use of some empirical data to make approximations on the HF method, allowing calculations to be run at a greater speed but with reduced accuracy). The AM1 method is an adjusted version of the Neglect of Diatomic Differential Overlap method, which ignores certain integrals such as two electron replusion integrals in order to save time on calculations.

===The Reaction of Ethene and cis-Butadiene===

Ethene was drawn and optimised, and the HOMO and LUMO are displayed below. It can be seen that the HOMO is symmetric and the LUMO is antisymmetric. [https://wiki.ch.ic.ac.uk/wiki/images/4/4c/PLZETHENE.LOG Log file]
{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/c/cf/PLZetheneHOMO.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZetheneHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/b/b1/PLZetheneLUMO.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZetheneLUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}
This process was repeated with cis-butadiene with a dihedral angle of 0<sup>o</sup>, the antisymmetric HOMO and symmetric LUMO are displayed below. [https://wiki.ch.ic.ac.uk/wiki/images/2/20/PLZBUTADIENE.LOG Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/7/72/PLZbutadieneHOMO.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZbutadieneHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/a/a1/Cisbutdiene_opt_mo12.cub.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZbutadieneHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}
The transition state of the reaction was then calculated using the frozen co-ordinate method. An imaginary frequency was found and the vibration corresponding to the transition state is shown below. The synchronised motion in the vibration is indicative of a concerted reaction. The first positive vibration has also been shown to illustrate that it is not involved in the formation of the transition state. [https://wiki.ch.ic.ac.uk/wiki/images/0/07/PLZda1.LOG Log file]

<jmol>
<jmolApplet>
<title>Imaginary Vibration</title><color>white</color><size>300</size>
<script>frame 1.7;vibration 1</script>
<uploadedFileContents>PLZda1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
<jmol>
<jmolApplet>
<title>Lowest Positive Vibration</title><color>white</color><size>300</size>
<script>frame 1.8;vibration 1</script>
<uploadedFileContents>PLZda1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
MO data has been calculated for the transition state and is displayed below.

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/0/01/PLZda1HOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZda1HOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/4/4a/PLZda1LUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZda1LUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}

As can be seen from these 6 MOs, the HOMO of the transition state is comprised of the HOMO from cis-butadiene and the LUMO from ethene, and is also antisymmetric. Likewise the LUMO of the transition state is comprised of the LUMO from cis-butadiene and the HOMO from ethene and is also symmetric.

The transition structure is shown below with a forming bond length of 2.21 Å. Typical bond lengths for sp<sup>2</sup> and sp<sup>3</sup> hybridised carbon carbon bond are 1.34 Å and 1.54 Å respectively <ref name="Clen"/>. The Wan der Waals radius of a carbon atom is 1.70 Å <ref name="Cwaal"/>. The bond lengths in the transition state are between two Wan der Waals radii and a normal carbon-carbon bond, suggesting a bond is in the process of forming.

===The Reaction of Maleic Anhydride and Cyclohexa-1,5-diene===

Maleic Anydride was first drawn and optimised, and MOs were calculated. [https://wiki.ch.ic.ac.uk/wiki/images/e/ef/PLZMAL.LOG Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/7/70/PLZMALHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZMALHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/0/08/PLZMALLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZMALLUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>

|}

The same was done for Cyclohexa-1,5-diene. [https://wiki.ch.ic.ac.uk/wiki/images/7/7b/PLZCYCLO.LOG Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/d/d2/PLZCYHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZCY.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/d/da/PLZCYLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZCY.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}

The endo and exo transition states were determined separately, through the frozen co-ordinate method.

====Endo State====

The energy of the transition state is -0.05150480, the MOs and imaginary vibration are displayed below, together with the bond distances. [https://wiki.ch.ic.ac.uk/wiki/images/3/36/PLZENDO.log Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/0/0b/PLZENDOHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZENDOHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/3/3e/PLZENDOLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZENDOLUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Vibration</title><color>white</color><size>300</size>
<script>frame 1.7;vibration 1</script>
<uploadedFileContents>PLZENDO.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Bond Distances</title><color>white</color><size>500</size>
<script>;measure 7 17; measure 1 16;measure RANGE 1.0 2 ALL (carbon) (carbon); measure 15 4; measure 18 6</script>
<uploadedFileContents>PLZENDO.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|}

====Exo State====

The energy of the transition state is -0.05041985, the MOs and imaginary vibration are displayed below together with bond distances. [https://wiki.ch.ic.ac.uk/wiki/images/0/0a/PLZEXO.LOG Log file]
{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/e/e4/PLZEXOHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZEXOHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/9/9c/PLZEXOLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZEXOHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Vibration</title><color>white</color><size>300</size>
<script>frame 1.9;vibration 1</script>
<uploadedFileContents>PLZEXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Bond Distances</title><color>white</color><size>500</size>
<script>;measure 7 17; measure 1 16;measure RANGE 1.0 2 ALL (carbon) (carbon); measure 15 13; measure 18 11</script>
<uploadedFileContents>PLZEXO.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|}

====Comparison of Transition States====
The exo state can be seen to have a slightly higher energy. Examining the C-C bond distances shows that all existing bond distances are becoming equal, indicating a concerted reaction. Equally, there are two through space carbon-carbon distances that are less than two van der waals radii, indicating that bonds are in the process of being formed. There does not appear to be any significant difference in bond lengths between the exo and endo states, despite the different geometry. It might be theorised that there is a steric interaction between the O=C-O-C=O fragment of the maleic anhydride and the hydrogen on the diene. As can be seen above, the conformer withsp<sup>3</sup> hydrogens are significantly closer to the maleic anhydride, which could slightly raise the energy of the exo state.

Looking for secondary orbital effects, it does not appear that the C=O π* orbitals are close enough to the diene π orbitals to interact favourably. They are noticeably closer on the endo state however, and it is possible there is a weak interaction beyond the drawn boundary surface (as they do not represent finite boundaries).

==References==
<references>
<ref name="Cope def"> S. J. Rhoads, N. R. Raulins. <i>Organic Reactions</i>. 1975(22): 1–252. DOI:10.1002/0471264180.or022.01</ref>
<ref name="DFT"> R. G. Parr, Electron Distributions and the Chemical Bond, 1982, 95–100.</ref>
<ref name="HF"> 2 J. C. Slater, Physical Review, 1951, 81, 385–390. </ref>
<ref name="Clen">E. Lippert. The Strengths of Chemical Bonds. Butterworths Publications Ltd., London 1958. </ref>
<ref name="Cwaal"> A. Bondi. van der Waals Volumes and Radii. J Phys Chem. 1964;68(3):441–51. </ref>
</references>

Rep:Mod:PLZ

2016-03-11T09:37:40Z

Oc113: /* The Reaction of Maleic Anhydride and Cyclohexa-1,5-diene */

==The Cope Rearrangement==

The Cope Rearrangment is a concerted sigmatropic rerrangement of 1,5 dienes <ref name="Cope def"/>. There are both boat and chair transition states and the energies of these will be investigated.

===Optimising Hexadiene===

1,5 Hexadiene has several possible conformations, with varying energies. A selection of conformers were optimised using the 3-21G Hartree-Frock (HF) method in order to investigate the relative energies.

====3-21G Hartree-Frock Optimisations====

The various conformations were obtained by drawing a molecule of 1,5 hexadiene in an orientation close to the desired conformer, using the clean tool to adjust the geometry, and then performing the optimisation. The point group tool was then used to investigate the symmetry of the molecule. Energies and point groups for the four conformers investigated are summarised in the table below.

{| class="wikitable"
|'''Conformer'''
|'''Energy (hartrees)'''
|'''Geometry'''
|'''Point Group'''
|'''Log File'''
|-
|Anti 1
| -231.69260235
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZAnti1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|C2
|[https://wiki.ch.ic.ac.uk/wiki/images/5/52/PLZANTI1.LOG Log file]
|-
|Anti 2
| -231.69253528
||<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZAnti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|Ci
|[https://wiki.ch.ic.ac.uk/wiki/images/c/c5/PLZANTI2T.LOG Log file]
|-
|Gauche 3
| -231.69266121
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZGauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|C1
|[https://wiki.ch.ic.ac.uk/wiki/images/d/d6/PLZGAUCHE3T.LOG Log file]
|-
|Gauche 4
| -231.69153032
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZGauche4.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|C2
|[https://wiki.ch.ic.ac.uk/wiki/images/8/8e/PLZGAUCHE4.LOG Log file]
|}

As can be seen from the table that the gauche 3 conformer is the lowest energy conformation. Considering steric interactions alone, this is unexpected as the antiperiplanar conformations have less strain and therefore would be expected to have a lower energy. Examiniation of the molecular orbitals (MOs) of the gauche 3 conformer shows a favourable interaction between the π orbitals on the terminal carbon atoms, leading to a lower overall energy.

<jmol>
<jmolApplet>
<size>300</size><color>white</color>
<script>isosurface color orange purple "images/a/a8/PLZhexadieneMO.jvxl" translucent;</script>
<uploadedFileContents>PLZhexadieneMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>

====6-31G* Density Functional Theory Optimisations====

The anti 2 conforner were then optimised using Denisty Functional Theory (DFT) with a 6-31G* basis set. As this differs from the 3-21G basis set used with the HF optimisation, the energies obtained cannot be meaninfully compared. As the 6-31G* basis set is more sophisticated than 3-21G (adding polarisation functions to atoms beyond H), it would be expected that the energies obtained are more accurate. Geometries of the two different optimisations have been examined and displayed in the table below in order to provide a level of comparison between the basis sets. The DFT method models the system as a series of functions of spatial electron density <ref name="DFT"/>, compared to a series of single electron wavefunctions in HF <ref name="HF"/>. [https://wiki.ch.ic.ac.uk/wiki/images/f/f9/ANTI2TB3LYP_631G_D.LOG Log file]

{| class="wikitable"
|rowspan="2" |'''Calculation Type'''
|colspan="3" |'''Bond Length (Å)'''
|'''Dihedral angle'''
|-
|C1-C2
|C2-C3
|C3-C4
|C1-C2-C3-C4
|-
|HF 3-21G
|1.316
|1.509
|1.553
| -144.66
|-
|DFT B3LYP 6-31G*
|1.334
|1.504
|1.548
| -118.58
|}

The data show that there is not a particularly large difference in the geometries, illustrating the role of less accurate basis sets in providing a good approximation of molecules in a shorter space of time.

=====Frequency Calculations=====

A frequency calculation was performed on the anti 2 conformer, this allowed the IR spectrum to be produced and thermochemistry data to be determined. This is summarised in the table below.
{| class="wikitable"
|'''Quantity'''
|'''Energy (Hartrees)'''
|-
|Zero-point Energy
|0.142497
|-
|Sum of electronic and zero-point Energies
| -234.469215
|-
|Sum of electronic and thermal Energies
| -234.461866
|-
|Sum of electronic and thermal Enthalpies
| -234.460922
|-
|Sum of electronic and thermal Free Energies
| -234.500800
|}

The Zero-point energy is the absolute minimum energy for the system. This is combined with the electronic energy (at 0K), then the thermal energy (the energy of translations, vibrations and rotations at 298.15K), an enthalpic adjustment is then made, and finally a free energy adjustment is made.

===Optimising a Transition State===

A transition state was optimised using three different methods, a IRC was run and activation energies were calculated.

Transition states do not occur at energy minimums and so optimisations can be difficult as they can proceed towards nonsensical geometries. In order to reduce this risk, optimisations can be started from a "best guess" geometry that is close to the true transition state, a frozen co-ordinate method, where sections of the molecule are optimised individually, or the QST2 method, which specifies the products and reactants and attempts to find an intermediate transition state. The Cope Rearrangement has both a chair and a boat transition state, the chair transition state was determined using the "best guess" and frozen co-ordinate methods, and the boat transition state was determined using the QST2 method.

Numerical results from these optimisations (such as energy and imaginary vibration frequencies are summarised at the end.

===="Best Guess" Type Method====

Transition states can be determined by calculating a Hessian matrix that contains one negative eigenvalue, which is then continuously updated as the optimisation proceeds towards the transition state. In this optimisation, a CH<sub>2</sub>CHCH<sub>2</sub> allyl fragment was drawn and optimised using a HF method with a 3-21G basis set. Two of these were then positioned roughly 2.2 Å apart, and orientated so that they formed a transition state-like structure. [https://wiki.ch.ic.ac.uk/wiki/images/5/53/PLSCOPEGUESS.LOG Log file]

<jmol>
<jmolApplet>
<title>Chair TS Vibration</title>
<color>white</color>
<size>250</size>
<script>frame 1.25;vibration 1;</script>
<uploadedFileContents>PLSCOPEGUESS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Co-ordinate Method====

This approach seeks to eliminate some of the degrees of freedom of the system by freezing the reaction co-ordinate and optimising the rest of the molecule in order to produce a structure closer to the transition state. The reaction co-ordinate is then unfrozen and a transition state is found by differentiating down the reaction co-ordinate. This optimisation using the same starting orientation as the "best guess" method, with the addition of the frozen co-ordinates. [https://wiki.ch.ic.ac.uk/wiki/images/1/17/PLZCOPEFROZE.LOG Log file]

<jmol>
<jmolApplet>
<title>Chair TS Vibration</title>
<color>white</color>
<size>250</size>
<script>frame 1.21;vibration 1;</script>
<uploadedFileContents>PLZCOPEFROZE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====QST2 Method====

This method used adjusted versions of the previously optimised anti 2 conformer of 1,5 hexadiene. Atom labeling was used to specify the location of each atom in the reactant and product, and allow a transition state to be formed. The geometries were also altered in order to produce structures closer to the transition state, as the original linear structures were not close enough to produce a transition state from optimisation. [https://wiki.ch.ic.ac.uk/wiki/images/3/3a/PLZCOPEQT.LOG Log file]

<jmol>
<jmolApplet>
<title>Boat TS Vibration</title>
<color>white</color>
<size>250</size>
<script>frame 1.43;vibration 1;</script>
<uploadedFileContents>PLZCOPEQT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====IRC====

While transition states have been determined, it is not possible to tell which conformers of 1,5 hexadiene they link. Therefore an IRC was run to profile the reaction. This was performed on the chair transition state produced by the frozen co-ordinate method. The IRC calculation follows the steepest gradient down the potential energy surface in order to locate the resultant geometry. The results of the IRC calculation are summarised below, and the resultant conformer is gauche 4 (based on the final energies reached in the IRC path). [https://wiki.ch.ic.ac.uk/wiki/images/c/cf/PLZCOPEIRC.LOG Log file]

{|
|[[File:PLZIRCen.png|none|thumb|500px|Fig14. IRC plot for the chair TS]]
|[[File:PLZIRCgrad.png|none|thumb|500px|Fig15. RMS plot for the chair TS]]
|}

====Activation Energies and Other Numerical Data====

The chair and boat transition states were repeated at the DFT B3LYP 6-31G* level of theory, and the results are summarised here. [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/PLZCOPECHAIRDFT.LOG Chair log file] [https://wiki.ch.ic.ac.uk/wiki/images/1/1c/PLZCOPEBOATDFT.LOG Boat log file]

{| class="wikitable"
|
|colspan="3"|'''HF 3-21G'''
|colspan="3"|'''DFT BLYP 6-31G*'''
|-
|
|Electronic Energy (Hartrees)
|Sum of Electronic and Zero-Point Energies (Hartrees)
|Sum of Electronic and Thermal Energies at 298.15K (Hartrees)
|Electronic Energy (Hartrees)
|Sum of Electronic and Zero-Point Energies (Hartrees)
|Sum of Electronic and Thermal Energies at 298.15K (Hartrees)
|-
|Chair TS
| -231.619322
| -231.466698
| -231.461339
| -234.610701
| -234.468248
| -234.460958
|-
|Boat TS
| -231.602802
| -231.450930
| -231.445302
| -234.611329
| -234.468693
| -234.461464
|-
|Anti 2 Conformer
| -231.692535
| -231.539540
| -231.532566
| -234.611711
| -234.469215
| -234.461866
|}

A summary of activation energies is provided below (kcal/mol). These appear to suggest that the HF method produces vastly more difference in energies than the DFT method.
{| class="wikitable"
|
|colspan="2"|'''HF 3-21G'''
|colspan="2"|'''DFT BLYP 6-31G*'''
|-
|
|0K
|298.15K
|0K
|298.15K
|-
|ΔE Chair
|45.709010578
|44.695583543
|0.606801203
|0.569778172
|-
|ΔE Boat
|55.60357249
|54.758945376
|0.327559698
|0.252258618
|}

==Diels-Alder Cycloaddition==

The Diels-Alder reaction is a reaction between a conjugated diene and a dienophile, with the concerted formation of two sigma bonds and the breaking of two pi bonds with a cyclic transition state.

Two examples of this reaction were examined, the reaction of ethene and cis-butadiene, and the reaction of malaeic anhydride and cyclohexa-1,3-diene.

The calculations performed all used the semi-empirical AM1 method (semi-empirical refers to the use of some empirical data to make approximations on the HF method, allowing calculations to be run at a greater speed but with reduced accuracy). The AM1 method is an adjusted version of the Neglect of Diatomic Differential Overlap method, which ignores certain integrals such as two electron replusion integrals in order to save time on calculations.

===The Reaction of Ethene and cis-Butadiene===

Ethene was drawn and optimised, and the HOMO and LUMO are displayed below. It can be seen that the HOMO is symmetric and the LUMO is antisymmetric. [https://wiki.ch.ic.ac.uk/wiki/images/4/4c/PLZETHENE.LOG Log file]
{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/c/cf/PLZetheneHOMO.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZetheneHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/b/b1/PLZetheneLUMO.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZetheneLUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}
This process was repeated with cis-butadiene with a dihedral angle of 0<sup>o</sup>, the antisymmetric HOMO and symmetric LUMO are displayed below. [https://wiki.ch.ic.ac.uk/wiki/images/2/20/PLZBUTADIENE.LOG Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/7/72/PLZbutadieneHOMO.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZbutadieneHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/a/a1/Cisbutdiene_opt_mo12.cub.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZbutadieneHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}
The transition state of the reaction was then calculated using the frozen co-ordinate method. An imaginary frequency was found and the vibration corresponding to the transition state is shown below. The synchronised motion in the vibration is indicative of a concerted reaction. The first positive vibration has also been shown to illustrate that it is not involved in the formation of the transition state. [https://wiki.ch.ic.ac.uk/wiki/images/0/07/PLZda1.LOG Log file]

<jmol>
<jmolApplet>
<title>Imaginary Vibration</title><color>white</color><size>300</size>
<script>frame 1.7;vibration 1</script>
<uploadedFileContents>PLZda1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
<jmol>
<jmolApplet>
<title>Lowest Positive Vibration</title><color>white</color><size>300</size>
<script>frame 1.8;vibration 1</script>
<uploadedFileContents>PLZda1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
MO data has been calculated for the transition state and is displayed below.

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/0/01/PLZda1HOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZda1HOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/4/4a/PLZda1LUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZda1LUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}

As can be seen from these 6 MOs, the HOMO of the transition state is comprised of the HOMO from cis-butadiene and the LUMO from ethene, and is also antisymmetric. Likewise the LUMO of the transition state is comprised of the LUMO from cis-butadiene and the HOMO from ethene and is also symmetric.

The transition structure is shown below with a forming bond length of 2.21 Å. Typical bond lengths for sp<sup>2</sup> and sp<sup>3</sup> hybridised carbon carbon bond are 1.34 Å and 1.54 Å respectively <ref name="Clen"/>. The Wan der Waals radius of a carbon atom is 1.70 Å <ref name="Cwaal"/>. The bond lengths in the transition state are between two Wan der Waals radii and a normal carbon-carbon bond, suggesting a bond is in the process of forming.

===The Reaction of Maleic Anhydride and Cyclohexa-1,5-diene===

Maleic Anydride was first drawn and optimised, and MOs were calculated. [https://wiki.ch.ic.ac.uk/wiki/images/e/ef/PLZMAL.LOG Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/7/70/PLZMALHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZMALHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/0/08/PLZMALLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZMALLUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>

|}

The same was done for Cyclohexa-1,5-diene. [https://wiki.ch.ic.ac.uk/wiki/images/7/7b/PLZCYCLO.LOG Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/d/d2/PLZCYHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZCY.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/d/da/PLZCYLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZCY.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}

The endo and exo transition states were determined separately, through the frozen co-ordinate method.

====Endo State====

The energy of the transition state is -0.05150480, the MOs and imaginary vibration are displayed below, together with the bond distances. [https://wiki.ch.ic.ac.uk/wiki/images/3/36/PLZENDO.log Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/0/0b/PLZENDOHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZENDOHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/3/3e/PLZENDOLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZENDOLUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Vibration</title><color>white</color><size>300</size>
<script>frame 1.7;vibration 1</script>
<uploadedFileContents>PLZENDO.log</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Bond Distances</title><color>white</color><size>500</size>
<script>;measure 7 17; measure 1 16;measure RANGE 1.0 2 ALL (carbon) (carbon); measure 15 4; measure 18 6</script>
<uploadedFileContents>PLZENDO.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|}

====Exo State====

The energy of the transition state is -0.05041985, the MOs and imaginary vibration are displayed below together with bond distances. [https://wiki.ch.ic.ac.uk/wiki/images/0/0a/PLZEXO.LOG Log file]
{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/e/e4/PLZEXOHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZEXOHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/9/9c/PLZEXOLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZEXOHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Vibration</title><color>white</color><size>300</size>
<script>frame 1.9;vibration 1</script>
<uploadedFileContents>PLZEXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Bond Distances</title><color>white</color><size>500</size>
<script>;measure 7 17; measure 1 16;measure RANGE 1.0 2 ALL (carbon) (carbon); measure 15 13; measure 18 11</script>
<uploadedFileContents>PLZEXO.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|}

====Comparison of Transition States====
The exo state can be seen to have a slightly higher energy. Examining the C-C bond distances shows that all existing bond distances are becoming equal, indicating a concerted reaction. Equally, there are two through space carbon-carbon distances that are less than two van der waals radii, indicating that bonds are in the process of being formed. There does not appear to be any significant difference in bond lengths between the exo and endo states, despite the different geometry. It might be theorised that there is a steric interaction between the O=C-O-C=O fragment of the maleic anhydride and the hydrogen on the diene. As can be seen above, the conformer withsp<sup>3</sup> hydrogens are significantly closer to the maleic anhydride, which could slightly raise the energy of the exo state.

Looking for secondary orbital effects, it does not appear that the C=O π* orbitals are close enough to the diene π orbitals to interact favourably. They are noticeably closer on the endo state however, and it is possible there is a weak interaction beyond the drawn boundary surface (as they do not represent finite boundaries).

==References==
<references>
<ref name="Cope def"> S. J. Rhoads, N. R. Raulins. <i>Organic Reactions</i>. 1975(22): 1–252. DOI:10.1002/0471264180.or022.01</ref>
<ref name="DFT"> R. G. Parr, Electron Distributions and the Chemical Bond, 1982, 95–100.</ref>
<ref name="HF"> 2 J. C. Slater, Physical Review, 1951, 81, 385–390. </ref>
<ref name="Clen">E. Lippert. The Strengths of Chemical Bonds. Butterworths Publications Ltd., London 1958. </ref>
<ref name="Cwaal"> A. Bondi. van der Waals Volumes and Radii. J Phys Chem. 1964;68(3):441–51. </ref>
</references>

File:PLZENDO.mol

2016-03-11T09:16:28Z

Oc113:

File:PLZEXO.mol

2016-03-11T09:01:03Z

Oc113:

Rep:Mod:PLZ

2016-03-11T08:43:11Z

Oc113: /* Activation Energies and Other Numerical Data */

==The Cope Rearrangement==

The Cope Rearrangment is a concerted sigmatropic rerrangement of 1,5 dienes <ref name="Cope def"/>. There are both boat and chair transition states and the energies of these will be investigated.

===Optimising Hexadiene===

1,5 Hexadiene has several possible conformations, with varying energies. A selection of conformers were optimised using the 3-21G Hartree-Frock (HF) method in order to investigate the relative energies.

====3-21G Hartree-Frock Optimisations====

The various conformations were obtained by drawing a molecule of 1,5 hexadiene in an orientation close to the desired conformer, using the clean tool to adjust the geometry, and then performing the optimisation. The point group tool was then used to investigate the symmetry of the molecule. Energies and point groups for the four conformers investigated are summarised in the table below.

{| class="wikitable"
|'''Conformer'''
|'''Energy (hartrees)'''
|'''Geometry'''
|'''Point Group'''
|'''Log File'''
|-
|Anti 1
| -231.69260235
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZAnti1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|C2
|[https://wiki.ch.ic.ac.uk/wiki/images/5/52/PLZANTI1.LOG Log file]
|-
|Anti 2
| -231.69253528
||<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZAnti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|Ci
|[https://wiki.ch.ic.ac.uk/wiki/images/c/c5/PLZANTI2T.LOG Log file]
|-
|Gauche 3
| -231.69266121
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZGauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|C1
|[https://wiki.ch.ic.ac.uk/wiki/images/d/d6/PLZGAUCHE3T.LOG Log file]
|-
|Gauche 4
| -231.69153032
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZGauche4.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|C2
|[https://wiki.ch.ic.ac.uk/wiki/images/8/8e/PLZGAUCHE4.LOG Log file]
|}

As can be seen from the table that the gauche 3 conformer is the lowest energy conformation. Considering steric interactions alone, this is unexpected as the antiperiplanar conformations have less strain and therefore would be expected to have a lower energy. Examiniation of the molecular orbitals (MOs) of the gauche 3 conformer shows a favourable interaction between the π orbitals on the terminal carbon atoms, leading to a lower overall energy.

<jmol>
<jmolApplet>
<size>300</size><color>white</color>
<script>isosurface color orange purple "images/a/a8/PLZhexadieneMO.jvxl" translucent;</script>
<uploadedFileContents>PLZhexadieneMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>

====6-31G* Density Functional Theory Optimisations====

The anti 2 conforner were then optimised using Denisty Functional Theory (DFT) with a 6-31G* basis set. As this differs from the 3-21G basis set used with the HF optimisation, the energies obtained cannot be meaninfully compared. As the 6-31G* basis set is more sophisticated than 3-21G (adding polarisation functions to atoms beyond H), it would be expected that the energies obtained are more accurate. Geometries of the two different optimisations have been examined and displayed in the table below in order to provide a level of comparison between the basis sets. The DFT method models the system as a series of functions of spatial electron density <ref name="DFT"/>, compared to a series of single electron wavefunctions in HF <ref name="HF"/>. [https://wiki.ch.ic.ac.uk/wiki/images/f/f9/ANTI2TB3LYP_631G_D.LOG Log file]

{| class="wikitable"
|rowspan="2" |'''Calculation Type'''
|colspan="3" |'''Bond Length (Å)'''
|'''Dihedral angle'''
|-
|C1-C2
|C2-C3
|C3-C4
|C1-C2-C3-C4
|-
|HF 3-21G
|1.316
|1.509
|1.553
| -144.66
|-
|DFT B3LYP 6-31G*
|1.334
|1.504
|1.548
| -118.58
|}

The data show that there is not a particularly large difference in the geometries, illustrating the role of less accurate basis sets in providing a good approximation of molecules in a shorter space of time.

=====Frequency Calculations=====

A frequency calculation was performed on the anti 2 conformer, this allowed the IR spectrum to be produced and thermochemistry data to be determined. This is summarised in the table below.
{| class="wikitable"
|'''Quantity'''
|'''Energy (Hartrees)'''
|-
|Zero-point Energy
|0.142497
|-
|Sum of electronic and zero-point Energies
| -234.469215
|-
|Sum of electronic and thermal Energies
| -234.461866
|-
|Sum of electronic and thermal Enthalpies
| -234.460922
|-
|Sum of electronic and thermal Free Energies
| -234.500800
|}

The Zero-point energy is the absolute minimum energy for the system. This is combined with the electronic energy (at 0K), then the thermal energy (the energy of translations, vibrations and rotations at 298.15K), an enthalpic adjustment is then made, and finally a free energy adjustment is made.

===Optimising a Transition State===

A transition state was optimised using three different methods, a IRC was run and activation energies were calculated.

Transition states do not occur at energy minimums and so optimisations can be difficult as they can proceed towards nonsensical geometries. In order to reduce this risk, optimisations can be started from a "best guess" geometry that is close to the true transition state, a frozen co-ordinate method, where sections of the molecule are optimised individually, or the QST2 method, which specifies the products and reactants and attempts to find an intermediate transition state. The Cope Rearrangement has both a chair and a boat transition state, the chair transition state was determined using the "best guess" and frozen co-ordinate methods, and the boat transition state was determined using the QST2 method.

Numerical results from these optimisations (such as energy and imaginary vibration frequencies are summarised at the end.

===="Best Guess" Type Method====

Transition states can be determined by calculating a Hessian matrix that contains one negative eigenvalue, which is then continuously updated as the optimisation proceeds towards the transition state. In this optimisation, a CH<sub>2</sub>CHCH<sub>2</sub> allyl fragment was drawn and optimised using a HF method with a 3-21G basis set. Two of these were then positioned roughly 2.2 Å apart, and orientated so that they formed a transition state-like structure. [https://wiki.ch.ic.ac.uk/wiki/images/5/53/PLSCOPEGUESS.LOG Log file]

<jmol>
<jmolApplet>
<title>Chair TS Vibration</title>
<color>white</color>
<size>250</size>
<script>frame 1.25;vibration 1;</script>
<uploadedFileContents>PLSCOPEGUESS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Co-ordinate Method====

This approach seeks to eliminate some of the degrees of freedom of the system by freezing the reaction co-ordinate and optimising the rest of the molecule in order to produce a structure closer to the transition state. The reaction co-ordinate is then unfrozen and a transition state is found by differentiating down the reaction co-ordinate. This optimisation using the same starting orientation as the "best guess" method, with the addition of the frozen co-ordinates. [https://wiki.ch.ic.ac.uk/wiki/images/1/17/PLZCOPEFROZE.LOG Log file]

<jmol>
<jmolApplet>
<title>Chair TS Vibration</title>
<color>white</color>
<size>250</size>
<script>frame 1.21;vibration 1;</script>
<uploadedFileContents>PLZCOPEFROZE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====QST2 Method====

This method used adjusted versions of the previously optimised anti 2 conformer of 1,5 hexadiene. Atom labeling was used to specify the location of each atom in the reactant and product, and allow a transition state to be formed. The geometries were also altered in order to produce structures closer to the transition state, as the original linear structures were not close enough to produce a transition state from optimisation. [https://wiki.ch.ic.ac.uk/wiki/images/3/3a/PLZCOPEQT.LOG Log file]

<jmol>
<jmolApplet>
<title>Boat TS Vibration</title>
<color>white</color>
<size>250</size>
<script>frame 1.43;vibration 1;</script>
<uploadedFileContents>PLZCOPEQT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====IRC====

While transition states have been determined, it is not possible to tell which conformers of 1,5 hexadiene they link. Therefore an IRC was run to profile the reaction. This was performed on the chair transition state produced by the frozen co-ordinate method. The IRC calculation follows the steepest gradient down the potential energy surface in order to locate the resultant geometry. The results of the IRC calculation are summarised below, and the resultant conformer is gauche 4 (based on the final energies reached in the IRC path). [https://wiki.ch.ic.ac.uk/wiki/images/c/cf/PLZCOPEIRC.LOG Log file]

{|
|[[File:PLZIRCen.png|none|thumb|500px|Fig14. IRC plot for the chair TS]]
|[[File:PLZIRCgrad.png|none|thumb|500px|Fig15. RMS plot for the chair TS]]
|}

====Activation Energies and Other Numerical Data====

The chair and boat transition states were repeated at the DFT B3LYP 6-31G* level of theory, and the results are summarised here. [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/PLZCOPECHAIRDFT.LOG Chair log file] [https://wiki.ch.ic.ac.uk/wiki/images/1/1c/PLZCOPEBOATDFT.LOG Boat log file]

{| class="wikitable"
|
|colspan="3"|'''HF 3-21G'''
|colspan="3"|'''DFT BLYP 6-31G*'''
|-
|
|Electronic Energy (Hartrees)
|Sum of Electronic and Zero-Point Energies (Hartrees)
|Sum of Electronic and Thermal Energies at 298.15K (Hartrees)
|Electronic Energy (Hartrees)
|Sum of Electronic and Zero-Point Energies (Hartrees)
|Sum of Electronic and Thermal Energies at 298.15K (Hartrees)
|-
|Chair TS
| -231.619322
| -231.466698
| -231.461339
| -234.610701
| -234.468248
| -234.460958
|-
|Boat TS
| -231.602802
| -231.450930
| -231.445302
| -234.611329
| -234.468693
| -234.461464
|-
|Anti 2 Conformer
| -231.692535
| -231.539540
| -231.532566
| -234.611711
| -234.469215
| -234.461866
|}

A summary of activation energies is provided below (kcal/mol). These appear to suggest that the HF method produces vastly more difference in energies than the DFT method.
{| class="wikitable"
|
|colspan="2"|'''HF 3-21G'''
|colspan="2"|'''DFT BLYP 6-31G*'''
|-
|
|0K
|298.15K
|0K
|298.15K
|-
|ΔE Chair
|45.709010578
|44.695583543
|0.606801203
|0.569778172
|-
|ΔE Boat
|55.60357249
|54.758945376
|0.327559698
|0.252258618
|}

==Diels-Alder Cycloaddition==

The Diels-Alder reaction is a reaction between a conjugated diene and a dienophile, with the concerted formation of two sigma bonds and the breaking of two pi bonds with a cyclic transition state.

Two examples of this reaction were examined, the reaction of ethene and cis-butadiene, and the reaction of malaeic anhydride and cyclohexa-1,3-diene.

The calculations performed all used the semi-empirical AM1 method (semi-empirical refers to the use of some empirical data to make approximations on the HF method, allowing calculations to be run at a greater speed but with reduced accuracy). The AM1 method is an adjusted version of the Neglect of Diatomic Differential Overlap method, which ignores certain integrals such as two electron replusion integrals in order to save time on calculations.

===The Reaction of Ethene and cis-Butadiene===

Ethene was drawn and optimised, and the HOMO and LUMO are displayed below. It can be seen that the HOMO is symmetric and the LUMO is antisymmetric. [https://wiki.ch.ic.ac.uk/wiki/images/4/4c/PLZETHENE.LOG Log file]
{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/c/cf/PLZetheneHOMO.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZetheneHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/b/b1/PLZetheneLUMO.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZetheneLUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}
This process was repeated with cis-butadiene with a dihedral angle of 0<sup>o</sup>, the antisymmetric HOMO and symmetric LUMO are displayed below. [https://wiki.ch.ic.ac.uk/wiki/images/2/20/PLZBUTADIENE.LOG Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/7/72/PLZbutadieneHOMO.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZbutadieneHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/a/a1/Cisbutdiene_opt_mo12.cub.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZbutadieneHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}
The transition state of the reaction was then calculated using the frozen co-ordinate method. An imaginary frequency was found and the vibration corresponding to the transition state is shown below. The synchronised motion in the vibration is indicative of a concerted reaction. The first positive vibration has also been shown to illustrate that it is not involved in the formation of the transition state. [https://wiki.ch.ic.ac.uk/wiki/images/0/07/PLZda1.LOG Log file]

<jmol>
<jmolApplet>
<title>Imaginary Vibration</title><color>white</color><size>300</size>
<script>frame 1.7;vibration 1</script>
<uploadedFileContents>PLZda1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
<jmol>
<jmolApplet>
<title>Lowest Positive Vibration</title><color>white</color><size>300</size>
<script>frame 1.8;vibration 1</script>
<uploadedFileContents>PLZda1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
MO data has been calculated for the transition state and is displayed below.

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/0/01/PLZda1HOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZda1HOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/4/4a/PLZda1LUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZda1LUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}

As can be seen from these 6 MOs, the HOMO of the transition state is comprised of the HOMO from cis-butadiene and the LUMO from ethene, and is also antisymmetric. Likewise the LUMO of the transition state is comprised of the LUMO from cis-butadiene and the HOMO from ethene and is also symmetric.

The transition structure is shown below with a forming bond length of 2.21 Å. Typical bond lengths for sp<sup>2</sup> and sp<sup>3</sup> hybridised carbon carbon bond are 1.34 Å and 1.54 Å respectively <ref name="Clen"/>. The Wan der Waals radius of a carbon atom is 1.70 Å <ref name="Cwaal"/>. The bond lengths in the transition state are between two Wan der Waals radii and a normal carbon-carbon bond, suggesting a bond is in the process of forming.

===The Reaction of Maleic Anhydride and Cyclohexa-1,5-diene===

Maleic Anydride was first drawn and optimised, and MOs were calculated. [https://wiki.ch.ic.ac.uk/wiki/images/e/ef/PLZMAL.LOG Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/7/70/PLZMALHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZMALHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/0/08/PLZMALLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZMALLUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>

|}

The same was done for Cyclohexa-1,5-diene. [https://wiki.ch.ic.ac.uk/wiki/images/7/7b/PLZCYCLO.LOG Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/d/d2/PLZCYHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZCY.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/d/da/PLZCYLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZCY.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}

The endo and exo transition states were determined seperately, through the frozen co-ordinate method.

====Endo State====

The energy of the transition state is ?, the MOs and imaginary vibration are displayed below. [https://wiki.ch.ic.ac.uk/wiki/images/3/36/PLZENDO.log Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/0/0b/PLZENDOHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZENDOHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/3/3e/PLZENDOLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZENDOLUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Vibration</title><color>white</color><size>300</size>
<script>frame 1.7;vibration 1</script>
<uploadedFileContents>PLZENDO.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

====Exo State====

The energy of the transition state is ?, the MOs and imaginary vibration are displayed below. [https://wiki.ch.ic.ac.uk/wiki/images/0/0a/PLZEXO.LOG Log file]
{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/e/e4/PLZEXOHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZEXOHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/9/9c/PLZEXOLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZEXOHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Vibration</title><color>white</color><size>300</size>
<script>frame 1.9;vibration 1</script>
<uploadedFileContents>PLZEXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}

==References==
<references>
<ref name="Cope def"> S. J. Rhoads, N. R. Raulins. <i>Organic Reactions</i>. 1975(22): 1–252. DOI:10.1002/0471264180.or022.01</ref>
<ref name="DFT"> R. G. Parr, Electron Distributions and the Chemical Bond, 1982, 95–100.</ref>
<ref name="HF"> 2 J. C. Slater, Physical Review, 1951, 81, 385–390. </ref>
<ref name="Clen">E. Lippert. The Strengths of Chemical Bonds. Butterworths Publications Ltd., London 1958. </ref>
<ref name="Cwaal"> A. Bondi. van der Waals Volumes and Radii. J Phys Chem. 1964;68(3):441–51. </ref>
</references>

Rep:Mod:PLZ

File:PLZENDOLUMO.xyz

2016-03-11T07:48:02Z

Oc113:

2016-03-11T05:48:34Z

2016-03-11T05:44:36Z

Oc113: /* Optimising a Transition State */

The Cope Rearrangement

?intro

==Optimising Hexadiene==

1,5 Hexadiene has several possible conformations, with varying energies. A selection of conformers were optimised using the 3-21G Hartree-Frock (HF) method in order to investigate the relative energies. ?and then at a higher level etc.

===3-21G Hartree-Frock Optimisations===

The various conformations were obtained by drawing a molecule of 1,5 hexadiene in an orientation close to the desired conformer, using the clean tool to adjust the geometry, and then performing the optimisation. The point group tool was then used to investigate the symmetry of the molecule. Energies and point groups for the four conformers investigated are summarised in the table below.

{| class="wikitable"
|'''Conformer'''
|'''Energy (hartrees)'''
|'''Geometry'''
|'''Point Group'''
|'''Log File'''
|-
|Anti 1
| -231.69260235
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZAnti1.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|C2
|-
|Anti 2
| -231.69253528
||<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZAnti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|Ci
|-
|Gauche 3
| -231.69266121
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZGauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|C1
|-
|Gauche 4
| -231.69153032
|<jmol>
<jmolApplet>
<color>white</color>
<size>300</size>
<uploadedFileContents>PLZGauche4.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|C2
|}

As can be seen from the table that the gauche 3 conformer is the lowest energy conformation. Considering steric interactions alone, this is unexpected as the antiperiplanar conformations have less strain and therefore would be expected to have a lower energy. Examiniation of the molecular orbitals (MOs) of the gauche 3 conformer shows a favourable interaction between the π orbitals on the terminal carbon atoms, leading to a lower overall energy.

<jmol>
<jmolApplet>
<size>300</size><color>white</color>
<script>isosurface color orange purple "images/a/a8/PLZhexadieneMO.jvxl" translucent;</script>
<uploadedFileContents>PLZhexadieneMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>

===6-31G* Density Functional Theory Optimisations===

The anti 2 conforner were then optimised using Denisty Functional Theory (DFT) with a 6-31G* basis set. As this differs from the 3-21G basis set used with the HF optimisation, the energies obtained cannot be meaninfully compared. As the 6-31G* basis set is more sophisticated than 3-21G (adding polarisation functions to atoms beyond H), it would be expected that the energies obtained are more accurate. Geometries of the two different optimisations have been examined and displayed in the table below in order to provide a level of comparison between the basis sets. [https://wiki.ch.ic.ac.uk/wiki/images/f/f9/ANTI2TB3LYP_631G_D.LOG Log file]

{| class="wikitable"
|rowspan="2" |'''Calculation Type'''
|colspan="3" |'''Bond Length (Å)'''
|'''Dihedral angle'''
|-
|C1-C2
|C2-C3
|C3-C4
|C1-C2-C3-C4
|-
|HF 3-21G
|1.316
|1.509
|1.553
| -144.66
|-
|DFT B3LYP 6-31G*
|1.334
|1.504
|1.548
| -118.58
|}

The data show that there is not a particularly large difference in the geometries, illustrating the role of less accurate basis sets in providing a good approximation of molecules in a shorter space of time.

====Frequency Calculations====

A frequency calculation was performed on the anti 2 conformer, this allowed the IR spectrum to be produced and thermochemistry data to be determined. This is summarised in the table below.
{| class="wikitable"
|'''Quantity'''
|'''Energy (Hartrees)'''
|-
|Zero-point Energy
|0.142497
|-
|Sum of electronic and zero-point Energies
| -234.469215
|-
|Sum of electronic and thermal Energies
| -234.461866
|-
|Sum of electronic and thermal Enthalpies
| -234.460922
|-
|Sum of electronic and thermal Free Energies
| -234.500800
|}

The Zero-point energy is the absolute minimum energy for the system. This is combined with the electronic energy (at 0K), then the thermal energy (the energy of translations, vibrations and rotations at 298.15K), an enthalpic adjustment is then made, and finally a free energy adjustment is made.

==Optimising a Transition State==

A transition state was optimised using three different methods, a IRC was run and activation energies were calculated.

Transition states do not occur at energy minimums and so optimisations can be difficult as they can proceed towards nonsensical geometries. In order to reduce this risk, optimisations can be started from a "best guess" geometry that is close to the true transition state, a frozen co-ordinate method, where sections of the molecule are optimised individually, or the QST2 method, which specifies the products and reactants and attempts to find an intermediate transition state. The Cope Rearrangement has both a chair and a boat transition state, the chair transition state was determined using the "best guess" and frozen co-ordinate methods, and the boat transition state was determined using the QST2 method.

Numerical results from these optimisations (such as energy and imaginary vibration frequencies are summarised at the end.

==="Best Guess" Type Method===

Transition states can be determined by calculating a Hessian matrix that contains one negative eigenvalue, which is then continuously updated as the optimisation proceeds towards the transition state. In this optimisation, a CH<sub>2</sub>CHCH<sub>2</sub> allyl fragment was drawn and optimised using a HF method with a 3-21G basis set. Two of these were then positioned roughly 2.2 Å apart, and orientated so that they formed a transition state-like structure. [https://wiki.ch.ic.ac.uk/wiki/images/5/53/PLSCOPEGUESS.LOG Log file]

<jmol>
<jmolApplet>
<title>Chair TS Vibration</title>
<color>white</color>
<size>250</size>
<script>frame 1.25;vibration 1;</script>
<uploadedFileContents>PLSCOPEGUESS.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

===Frozen Co-ordinate Method===

This approach seeks to eliminate some of the degrees of freedom of the system by freezing the reaction co-ordinate and optimising the rest of the molecule in order to produce a structure closer to the transition state. The reaction co-ordinate is then unfrozen and a transition state is found by differentiating down the reaction co-ordinate. This optimisation using the same starting orientation as the "best guess" method, with the addition of the frozen co-ordinates. [https://wiki.ch.ic.ac.uk/wiki/images/1/17/PLZCOPEFROZE.LOG Log file]

<jmol>
<jmolApplet>
<title>Chair TS Vibration</title>
<color>white</color>
<size>250</size>
<script>frame 1.21;vibration 1;</script>
<uploadedFileContents>PLZCOPEFROZE.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

===QST2 Method===

This method used adjusted versions of the previously optimised anti 2 conformer of 1,5 hexadiene. Atom labeling was used to specify the location of each atom in the reactant and product, and allow a transition state to be formed. The geometries were also altered in order to produce structures closer to the transition state, as the original linear structures were not close enough to produce a transition state from optimisation. [https://wiki.ch.ic.ac.uk/wiki/images/3/3a/PLZCOPEQT.LOG Log file]

<jmol>
<jmolApplet>
<title>Boat TS Vibration</title>
<color>white</color>
<size>250</size>
<script>frame 1.43;vibration 1;</script>
<uploadedFileContents>PLZCOPEQT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

===IRC===

While transition states have been determined, it is not possible to tell which conformers of 1,5 hexadiene they link. Therefore an IRC was run to profile the reaction. This was performed on the chair transition state produced by the frozen co-ordinate method. The IRC calculation follows the steepest gradient down the potential energy surface in order to locate the resultant geometry. The results of the IRC calculation are summarised below, and the resultant conformer is gauche 4 (based on the final energies reached in the IRC path). [https://wiki.ch.ic.ac.uk/wiki/images/c/cf/PLZCOPEIRC.LOG Log file]

{|
|[[File:PLZIRCen.png|none|thumb|500px|Fig14. IRC plot for the chair TS]]
|[[File:PLZIRCgrad.png|none|thumb|500px|Fig15. RMS plot for the chair TS]]
|}

===Activation Energies and Other Numerical Data===

The chair and boat transition states were repeated at the DFT B3LYP 6-31G* level of theory, and the results are summarised here. [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/PLZCOPECHAIRDFT.LOG Chair log file] [https://wiki.ch.ic.ac.uk/wiki/images/1/1c/PLZCOPEBOATDFT.LOG Boat log file]

{| class="wikitable"
|
|colspan="3"|'''HF 3-21G'''
|colspan="3"|'''DFT BLYP 6-31G*'''
|-
|
|Electronic Energy (Hartrees)
|Sum of Electronic and Zero-Point Energies (Hartrees)
|Sum of Electronic and Thermal Energies at 298.15K (Hartrees)
|Electronic Energy (Hartrees)
|Sum of Electronic and Zero-Point Energies (Hartrees)
|Sum of Electronic and Thermal Energies at 298.15K (Hartrees)
|-
|Chair TS
| -231.619322
| -231.466698
| -231.461339
| -234.556983
| -234.414932
| -234.409011
|-
|Boat TS
| -231.602802
| -231.450930
| -231.445302
| -232.878395
| -232.727998
| -232.722104
|-
|Anti 2 Conformer
|-
|
| -234.611711
| -234.469215
| -234.461866
|}

==Diels-Alder Cycloaddition==

The Diels-Alder reaction is a reaction between a conjugated diene and a dienophile, with the concerted formation of two sigma bonds and the breaking of two pi bonds with a cyclic transition state.

Two examples of this reaction were examined, the reaction of ethene and cis-butadiene, and the reaction of malaeic anhydride and cyclohexa-1,3-diene.

The calculations performed all used the semi-empirical AM1 method (semi-empirical refers to the use of some empirical data to make approximations on the HF method, allowing calculations to be run at a greater speed but with reduced accuracy). The AM1 method is an adjusted version of the Neglect of Diatomic Differential Overlap method, which ignores certain integrals such as two electron replusion integrals in order to save time on calculations.

===The Reaction of Ethene and cis-Butadiene===

Ethene was drawn and optimised, and the HOMO and LUMO are displayed below. [https://wiki.ch.ic.ac.uk/wiki/images/4/4c/PLZETHENE.LOG Log file]
{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/c/cf/PLZetheneHOMO.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZetheneHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/b/b1/PLZetheneLUMO.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZetheneLUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}
This process was repeated with cis-butadiene with a dihedral angle of 0<sup>o</sup>, the HOMO and LUMO are displayed below. [https://wiki.ch.ic.ac.uk/wiki/images/2/20/PLZBUTADIENE.LOG Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/7/72/PLZbutadieneHOMO.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZbutadieneHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/a/a1/Cisbutdiene_opt_mo12.cub.jvxl" translucent;rotate y 90</script>
<uploadedFileContents>PLZbutadieneHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}
The transition state of the reaction was then calculated using the frozen co-ordinate method. An imaginary frequency was found and the vibration corresponding to the transition state is shown below. The synchronised motion in the vibration is indicative of a concerted reaction. [https://wiki.ch.ic.ac.uk/wiki/images/0/07/PLZda1.LOG Log file]

<jmol>
<jmolApplet>
<color>white</color><size>300</size>
<script>frame 1.7;vibration 1</script>
<uploadedFileContents>PLZda1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

MO data has been calculated for the transition state and is displayed below.

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/0/01/PLZda1HOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZda1HOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/4/4a/PLZda1LUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZda1LUMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|}

As can be seen from these 6 MOs, the HOMO of the transition state is comprised of the HOMO from cis-butadiene and the LUMO from ethene. Likewise the LUMO of the transition state is comprised of the LUMO from cis-butadiene and the HOMO from ethene.

?bond lengths etc.

===The Reaction of Maleic Anhydride and Cyclohexa-1,5-diene===

Maleic Anydride was first drawn and optimised, and MOs were calculated. [https://wiki.ch.ic.ac.uk/wiki/images/e/ef/PLZMAL.LOG Log file]

?

The same was done for Cyclohexa-1,5-diene. [https://wiki.ch.ic.ac.uk/wiki/images/7/7b/PLZCYCLO.LOG Log file]

?

The endo and exo transition states were determined seperately, through the frozen co-ordinate method.

====Endo State====

The energy of the transition state is ?, the MOs and imaginary vibration are displayed below. [https://wiki.ch.ic.ac.uk/wiki/images/3/36/PLZENDO.log Log file]

{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/0/0b/PLZENDOHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZENDOHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/4/4a/PLZda1LUMO?.jvxl" translucent;</script>
<uploadedFileContents>PLZda1LUMO.xyz?</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Vibration</title><color>white</color><size>300</size>
<script>frame 1.7;vibration 1</script>
<uploadedFileContents>PLZENDO.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}

====Exo State====

The energy of the transition state is ?, the MOs and imaginary vibration are displayed below. [https://wiki.ch.ic.ac.uk/wiki/images/0/0a/PLZEXO.LOG Log file]
{|
|<jmol>
<jmolApplet>
<title>HOMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/e/e4/PLZEXOHOMO.jvxl" translucent;</script>
<uploadedFileContents>PLZEXOHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>LUMO</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/9/9c/PLZEXOLUMO.jvxl" translucent;</script>
<uploadedFileContents>PLZEXOHOMO.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Vibration</title><color>white</color><size>300</size>
<script>frame 1.9;vibration 1</script>
<uploadedFileContents>PLZEXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|}