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Rep:Module 3 - ceg09

2012-02-17T11:37:48Z

Ceg09: /* Transition State Optimization */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1732 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1735 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1733 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.0 || -234.55698 || -234.61179 || 0.05481 ||34.4 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.4 || -234.54309 || -234.61179 || 0.0687 || 43.1 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using the DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed below. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".

===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable. Having said this, from the data shown above the DFT HOMO can be said to be a combination of the butadiene LUMO and ethene HOMO. This combination of reactant orbitals has already been used to form the transition state LUMO and hence would not in fact be available.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Further MO analysis would need to be conducted in order to explained the aforementioned discrepancy in LCAO analysis.
|}
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei<ref>{{DOI|10.1098/rspa.1924.0082}}</ref>. This model still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction is found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation. It is the region in which the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer distinct C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured<ref>{{DOI|10.1002/1521-3773(20020517)41:10}}</ref> and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | -0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride HOMO calculated using the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report of a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09_Exo_Structure.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09_Endo_Structure.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structure a nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the right, shows that the system is asymmetric and so the IRC must be initially calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
Interestingly, both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.

The product formed via the kinetic/endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the right, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===Conclusion===
Once again computational analysis has successfully predicted the optimal structures of reactants, products and transition states. The so-called "endo-rule" has been shown by examining the relative energies of endo- and exo-transition states. Some interesting results concerning endo- and exo-products have been obtained and would perhaps provide a good basis for further computational exploration.

===References===
<references/>

File:Ceg09 Endo Structure.mol

2012-02-17T11:37:23Z

Ceg09:

File:Ceg09 Exo Structure.mol

2012-02-17T11:37:23Z

Ceg09:

Rep:Module 3 - ceg09

2012-02-17T11:24:54Z

Ceg09: /* Overall Activation Energies */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1732 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1735 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1733 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.0 || -234.55698 || -234.61179 || 0.05481 ||34.4 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.4 || -234.54309 || -234.61179 || 0.0687 || 43.1 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using the DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed below. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".

===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable. Having said this, from the data shown above the DFT HOMO can be said to be a combination of the butadiene LUMO and ethene HOMO. This combination of reactant orbitals has already been used to form the transition state LUMO and hence would not in fact be available.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Further MO analysis would need to be conducted in order to explained the aforementioned discrepancy in LCAO analysis.
|}
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei<ref>{{DOI|10.1098/rspa.1924.0082}}</ref>. This model still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction is found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation. It is the region in which the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer distinct C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured<ref>{{DOI|10.1002/1521-3773(20020517)41:10}}</ref> and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | -0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride HOMO calculated using the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report of a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structure a nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the right, shows that the system is asymmetric and so the IRC must be initially calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
Interestingly, both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.

The product formed via the kinetic/endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the right, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===Conclusion===
Once again computational analysis has successfully predicted the optimal structures of reactants, products and transition states. The so-called "endo-rule" has been shown by examining the relative energies of endo- and exo-transition states. Some interesting results concerning endo- and exo-products have been obtained and would perhaps provide a good basis for further computational exploration.

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T11:24:15Z

Ceg09: /* Frequency Analysis */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1732 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1735 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1733 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using the DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed below. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".

===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable. Having said this, from the data shown above the DFT HOMO can be said to be a combination of the butadiene LUMO and ethene HOMO. This combination of reactant orbitals has already been used to form the transition state LUMO and hence would not in fact be available.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Further MO analysis would need to be conducted in order to explained the aforementioned discrepancy in LCAO analysis.
|}
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei<ref>{{DOI|10.1098/rspa.1924.0082}}</ref>. This model still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction is found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation. It is the region in which the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer distinct C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured<ref>{{DOI|10.1002/1521-3773(20020517)41:10}}</ref> and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | -0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride HOMO calculated using the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report of a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structure a nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the right, shows that the system is asymmetric and so the IRC must be initially calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
Interestingly, both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.

The product formed via the kinetic/endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the right, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===Conclusion===
Once again computational analysis has successfully predicted the optimal structures of reactants, products and transition states. The so-called "endo-rule" has been shown by examining the relative energies of endo- and exo-transition states. Some interesting results concerning endo- and exo-products have been obtained and would perhaps provide a good basis for further computational exploration.

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T11:21:20Z

Ceg09: /* Intrinsic Reaction Coordinate */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using the DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed below. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".

===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable. Having said this, from the data shown above the DFT HOMO can be said to be a combination of the butadiene LUMO and ethene HOMO. This combination of reactant orbitals has already been used to form the transition state LUMO and hence would not in fact be available.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Further MO analysis would need to be conducted in order to explained the aforementioned discrepancy in LCAO analysis.
|}
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei<ref>{{DOI|10.1098/rspa.1924.0082}}</ref>. This model still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction is found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation. It is the region in which the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer distinct C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured<ref>{{DOI|10.1002/1521-3773(20020517)41:10}}</ref> and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | -0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride HOMO calculated using the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report of a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structure a nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the right, shows that the system is asymmetric and so the IRC must be initially calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
Interestingly, both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.

The product formed via the kinetic/endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the right, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===Conclusion===
Once again computational analysis has successfully predicted the optimal structures of reactants, products and transition states. The so-called "endo-rule" has been shown by examining the relative energies of endo- and exo-transition states. Some interesting results concerning endo- and exo-products have been obtained and would perhaps provide a good basis for further computational exploration.

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T11:20:37Z

Ceg09: /* Intrinsic Reaction Coordinate */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using the DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed below. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".

===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable. Having said this, from the data shown above the DFT HOMO can be said to be a combination of the butadiene LUMO and ethene HOMO. This combination of reactant orbitals has already been used to form the transition state LUMO and hence would not in fact be available.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Further MO analysis would need to be conducted in order to explained the aforementioned discrepancy in LCAO analysis.
|}
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei<ref>{{DOI|10.1098/rspa.1924.0082}}</ref>. This model still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction is found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation. It is the region in which the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer distinct C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured<ref>{{DOI|10.1002/1521-3773(20020517)41:10}}</ref> and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | -0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride HOMO calculated using the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report of a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structure a nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the right, shows that the system is asymmetric and so the IRC must be initially calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
Interestingly, both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.

The product formed via the kinetic/endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the right, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===Conclusion===
Once again computational analysis has successfully predicted the optimal structures of reactants, products and transition states. The so-called "endo-rule" has been shown by examining the relative energies of endo- and exo-transition states. Some interesting results concerning endo- and exo-products have been obtained and would perhaps provide a good basis for further computational exploration.

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T11:15:27Z

Ceg09: /* Further MO Analysis */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using the DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed below. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".

===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable. Having said this, from the data shown above the DFT HOMO can be said to be a combination of the butadiene LUMO and ethene HOMO. This combination of reactant orbitals has already been used to form the transition state LUMO and hence would not in fact be available.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Further MO analysis would need to be conducted in order to explained the aforementioned discrepancy in LCAO analysis.
|}
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei<ref>{{DOI|10.1098/rspa.1924.0082}}</ref>. This model still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction is found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation. It is the region in which the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer distinct C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured<ref>{{DOI|10.1002/1521-3773(20020517)41:10}}</ref> and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | -0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride HOMO calculated using the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report of a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structure a nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T11:14:55Z

Ceg09: /* Further MO Analysis */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using the DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed below. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".

===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable. Having said this, from the data shown above the DFT HOMO can be said to be a combination of the butadiene LUMO and ethene HOMO. This combination of reactant orbitals has already been used to form the transition state LUMO and hence would not in fact be available.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Further MO analysis would need to be conducted in order to explained the aforementioned discrepancy in LCAO analysis.
|}
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei<ref>{{DOI|10.1098/rspa.1924.0082}}</ref>. This model still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction is found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation. It is the region in which the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer distinct C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured<ref>{{DOI|10.1002/1521-3773(20020517)41:10}}</ref> and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | -0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride HOMO calculated using the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report of a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T11:14:31Z

Ceg09: /* Further MO Analysis */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using the DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed below. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".

===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable. Having said this, from the data shown above the DFT HOMO can be said to be a combination of the butadiene LUMO and ethene HOMO. This combination of reactant orbitals has already been used to form the transition state LUMO and hence would not in fact be available.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Further MO analysis would need to be conducted in order to explained the aforementioned discrepancy in LCAO analysis.
|}
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei<ref>{{DOI|10.1098/rspa.1924.0082}}</ref>. This model still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction is found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation. It is the region in which the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer distinct C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured<ref>{{DOI|10.1002/1521-3773(20020517)41:10}}</ref> and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | -0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride HOMO calculated using the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report of a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1 style="float: right"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T11:12:07Z

Ceg09: /* Reactant Optimization */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using the DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed below. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".

===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable. Having said this, from the data shown above the DFT HOMO can be said to be a combination of the butadiene LUMO and ethene HOMO. This combination of reactant orbitals has already been used to form the transition state LUMO and hence would not in fact be available.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Further MO analysis would need to be conducted in order to explained the aforementioned discrepancy in LCAO analysis.
|}
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei<ref>{{DOI|10.1098/rspa.1924.0082}}</ref>. This model still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction is found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation. It is the region in which the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer distinct C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured<ref>{{DOI|10.1002/1521-3773(20020517)41:10}}</ref> and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | -0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride HOMO calculated using the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report of a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T11:10:46Z

Ceg09: /* Reactant Optimization */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using the DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed below. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".

===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable. Having said this, from the data shown above the DFT HOMO can be said to be a combination of the butadiene LUMO and ethene HOMO. This combination of reactant orbitals has already been used to form the transition state LUMO and hence would not in fact be available.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Further MO analysis would need to be conducted in order to explained the aforementioned discrepancy in LCAO analysis.
|}
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei<ref>{{DOI|10.1098/rspa.1924.0082}}</ref>. This model still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction is found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation. It is the region in which the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer distinct C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured<ref>{{DOI|10.1002/1521-3773(20020517)41:10}}</ref> and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride HOMO calculated using the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report of a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T11:09:15Z

Ceg09: /* Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using the DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed below. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".

===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable. Having said this, from the data shown above the DFT HOMO can be said to be a combination of the butadiene LUMO and ethene HOMO. This combination of reactant orbitals has already been used to form the transition state LUMO and hence would not in fact be available.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Further MO analysis would need to be conducted in order to explained the aforementioned discrepancy in LCAO analysis.
|}
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei<ref>{{DOI|10.1098/rspa.1924.0082}}</ref>. This model still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction is found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation. It is the region in which the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer distinct C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured<ref>{{DOI|10.1002/1521-3773(20020517)41:10}}</ref> and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T11:07:14Z

Ceg09: /* Transition State Optimization */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using the DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed below. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".

===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable. Having said this, from the data shown above the DFT HOMO can be said to be a combination of the butadiene LUMO and ethene HOMO. This combination of reactant orbitals has already been used to form the transition state LUMO and hence would not in fact be available.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Further MO analysis would need to be conducted in order to explained the aforementioned discrepancy in LCAO analysis.
|}
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei<ref>{{DOI|10.1098/rspa.1924.0082}}</ref>. This model still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction is found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation. It is the region in which the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer distinct C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T11:04:18Z

Ceg09: /* Transition State Optimization */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using the DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed below. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".

===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable. Having said this, from the data shown above the DFT HOMO can be said to be a combination of the butadiene LUMO and ethene HOMO. This combination of reactant orbitals has already been used to form the transition state LUMO and hence would not in fact be available.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Further MO analysis would need to be conducted in order to explained the aforementioned discrepancy in LCAO analysis.
|}
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei<ref>{{DOI|10.1098/rspa.1924.0082}}</ref>. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T11:01:43Z

Ceg09: /* Transition State Optimization */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using the DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed below. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".

===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable. Having said this, from the data shown above the DFT HOMO can be said to be a combination of the butadiene LUMO and ethene HOMO. This combination of reactant orbitals has already been used to form the transition state LUMO and hence would not in fact be available.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Further MO analysis would need to be conducted in order to explained the aforementioned discrepancy in LCAO analysis.
|}
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T10:54:49Z

Ceg09: /* Reactant Optimization */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using the DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed below. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".

===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T10:49:04Z

Ceg09: /* Optimization of Chair & Boat Transition States */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway, perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state, an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. This imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking and the formation of bonds.It is clear that whilst two different methods were used to optimize the chair transition structure they both yield the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC takes the previously found transition state structure and follows the minimum reaction pathway along the system's reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product moving away from the transition state.

In this case, the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated methods for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially, the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect it only translates the structures. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===
This section has demonstrated the power of computational chemistry to quantitatively explain observed experimental and theoretical trends. The differences between two methods of differing levels of theory have been highlighted showing the B3LYP/6-31G* method to be consistently more accurate. In addition, IRC analysis has allowed us to model the relaxation of the transition state into products and also to identify the products' structures.

Clearly a great deal of valuable information can be obtained via computational techniques. The methods used in this section will now be further applied to more complex systems.

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T10:34:45Z

Ceg09: /* Frequency Analysis */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2<sup>nd</sup> derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry or minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state has been found. All of the frequencies calculated for all 3 conformers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods. The imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking of one bond and the formation of another.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T10:32:51Z

Ceg09: /* Optimization of Reactants & Products */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization, the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change in the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared, it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method builds upon the HF method, combining it with an additional calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>http://www.ch.ic.ac.uk/local/organic/conf/</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is, with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app configurations whereas the gauche 3 conformer has two angles that are slightly lower than this optimal 180° angle. This slight deviation from the optimal app geometry means that the stabilization from the σ<sub>C-C</sub>/σ*<sub>C-H</sub> and σ<sub>C-H</sub>/σ*<sub>C-C</sub> interactions will be lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor orbitals and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction (this type of interaction will be discussed further in later sections). Closer examination of this effect could be carried out with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods. The imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking of one bond and the formation of another.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T10:24:19Z

Ceg09: /* Optimization of Reactants & Products */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all, the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method combines the HF theory with an additonal calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods. The imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking of one bond and the formation of another.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T10:23:36Z

Ceg09:

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method combines the HF theory with an additonal calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods. The imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking of one bond and the formation of another.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T10:22:17Z

Ceg09:

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method combines the HF theory with an additonal calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
===Chair ===
====Optimization====
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods. The imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking of one bond and the formation of another.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

====Intrinsic Reaction Coordinate====
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

===Boat ===
====Optimization====
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

====Intrinsic Reaction Coordinate====
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-17T10:21:03Z

Ceg09:

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

A range of computational methods will be used throughout and their differences highlighted as part of this report.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method combines the HF theory with an additonal calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
==Chair ==
===Optimization===
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods. The imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking of one bond and the formation of another.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

===Intrinsic Reaction Coordinate===
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

==Boat ==
===Optimization===
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

===Intrinsic Reaction Coordinate===
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-16T23:18:50Z

Ceg09: /* Optimization */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method combines the HF theory with an additonal calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
==Chair ==
===Optimization===
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods. The imaginary frequency directly relates to the Cope rearrangement effectively showing the breaking of one bond and the formation of another.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

===Intrinsic Reaction Coordinate===
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

==Boat ==
===Optimization===
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

===Intrinsic Reaction Coordinate===
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-16T23:16:30Z

Ceg09: /* Transition State Optimization */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method combines the HF theory with an additonal calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
==Chair ==
===Optimization===
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

===Intrinsic Reaction Coordinate===
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

==Boat ==
===Optimization===
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

===Intrinsic Reaction Coordinate===
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-16T23:15:52Z

Ceg09: /* Transition State Optimization */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method combines the HF theory with an additonal calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
==Chair ==
===Optimization===
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

===Intrinsic Reaction Coordinate===
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

==Boat ==
===Optimization===
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

===Intrinsic Reaction Coordinate===
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however could be used to examine the product of the reaction.
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is noted that further optimization of the exo-transition state was conducted using the B3LYP/6-31G* method however it
|}

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-16T23:14:59Z

Ceg09: /* Overall Activation Energies */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method combines the HF theory with an additonal calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
==Chair ==
===Optimization===
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

===Intrinsic Reaction Coordinate===
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

==Boat ==
===Optimization===
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

===Intrinsic Reaction Coordinate===
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however ....
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is noted that further optimization of the exo-transition state was conducted using the B3LYP/6-31G* method however it
|}

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-16T23:14:26Z

Ceg09: /* Intrinsic Reaction Coordinate */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method combines the HF theory with an additonal calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
==Chair ==
===Optimization===
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

===Intrinsic Reaction Coordinate===
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

==Boat ==
===Optimization===
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

===Intrinsic Reaction Coordinate===
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref> and seen to be equivalent to the gauche 3 conformer.
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| The activation energies could also have been calculated at 0K as was done for the products.MORE?
|}

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however ....
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is noted that further optimization of the exo-transition state was conducted using the B3LYP/6-31G* method however it
|}

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-16T23:12:57Z

Ceg09: /* Intrinsic Reaction Coordinate */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method combines the HF theory with an additonal calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
==Chair ==
===Optimization===
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

===Intrinsic Reaction Coordinate===
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2.

==Boat ==
===Optimization===
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

===Intrinsic Reaction Coordinate===
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref>:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

MORE...

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| The activation energies could also have been calculated at 0K as was done for the products.MORE?
|}

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however ....
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is noted that further optimization of the exo-transition state was conducted using the B3LYP/6-31G* method however it
|}

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-16T23:12:15Z

Ceg09: /* Energies */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method combines the HF theory with an additonal calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
#'''Sum of electronic and zero point energies''' - increase
#'''Sum of electronic and thermal Energies''' - increase
# '''Sum of electronic and thermal Enthalpies''' - increase c.f. H = E + RT
#'''Sum of electronic and thermal Free Energies''' - decrease c.f. G = H - TS
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
==Chair ==
===Optimization===
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

===Intrinsic Reaction Coordinate===
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2. MORE?

==Boat ==
===Optimization===
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

===Intrinsic Reaction Coordinate===
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref>:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

MORE...

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| The activation energies could also have been calculated at 0K as was done for the products.MORE?
|}

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however ....
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is noted that further optimization of the exo-transition state was conducted using the B3LYP/6-31G* method however it
|}

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-16T23:09:26Z

Ceg09: /* Frequency Analysis */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method combines the HF theory with an additonal calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
*FINISH???
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
==Chair ==
===Optimization===
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

===Intrinsic Reaction Coordinate===
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2. MORE?

==Boat ==
===Optimization===
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

===Intrinsic Reaction Coordinate===
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref>:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

MORE...

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| The activation energies could also have been calculated at 0K as was done for the products.MORE?
|}

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however ....
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is noted that further optimization of the exo-transition state was conducted using the B3LYP/6-31G* method however it
|}

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-16T23:08:42Z

Ceg09: /* Optimization of Reactants & Products */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The DFT (B3LYP) method combines the HF theory with an additonal calculated exchange energy and so is much more accurate. In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}
The calculated vibrations match well to those in the literature<ref>???</ref>, being within the ????? error range for infra-red analysis.

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
*FINISH???
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
==Chair ==
===Optimization===
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

===Intrinsic Reaction Coordinate===
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2. MORE?

==Boat ==
===Optimization===
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

===Intrinsic Reaction Coordinate===
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref>:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

MORE...

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| The activation energies could also have been calculated at 0K as was done for the products.MORE?
|}

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however ....
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is noted that further optimization of the exo-transition state was conducted using the B3LYP/6-31G* method however it
|}

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-16T23:07:12Z

Ceg09:

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states. Transition state modelling is very important as it is often extremely hard to analyze transition states experimentally and it allows us to obtain clear information about the nature of a reaction pathway.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The HF method is a ???????optimization technique that unfortunately does not...The DFT (B3LYP) method on the other hand combines the HF theory with a calculated exchange energy and so id much more accurate. MORE? In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}
The calculated vibrations match well to those in the literature<ref>???</ref>, being within the ????? error range for infra-red analysis.

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
*FINISH???
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
==Chair ==
===Optimization===
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

===Intrinsic Reaction Coordinate===
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2. MORE?

==Boat ==
===Optimization===
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

===Intrinsic Reaction Coordinate===
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref>:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

MORE...

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| The activation energies could also have been calculated at 0K as was done for the products.MORE?
|}

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however ....
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is noted that further optimization of the exo-transition state was conducted using the B3LYP/6-31G* method however it
|}

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-16T22:49:08Z

Ceg09: /* Optimization */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The HF method is a ???????optimization technique that unfortunately does not...The DFT (B3LYP) method on the other hand combines the HF theory with a calculated exchange energy and so id much more accurate. MORE? In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}
The calculated vibrations match well to those in the literature<ref>???</ref>, being within the ????? error range for infra-red analysis.

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
*FINISH???
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
==Chair ==
===Optimization===
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods.

The terminal C-C bond lengths can also be said to be attractive, this is discussed further in section 5.2.
|}

===Intrinsic Reaction Coordinate===
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2. MORE?

==Boat ==
===Optimization===
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

===Intrinsic Reaction Coordinate===
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref>:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

MORE...

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| The activation energies could also have been calculated at 0K as was done for the products.MORE?
|}

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however ....
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is noted that further optimization of the exo-transition state was conducted using the B3LYP/6-31G* method however it
|}

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-16T22:48:07Z

Ceg09: /* Optimization */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The HF method is a ???????optimization technique that unfortunately does not...The DFT (B3LYP) method on the other hand combines the HF theory with a calculated exchange energy and so id much more accurate. MORE? In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}
The calculated vibrations match well to those in the literature<ref>???</ref>, being within the ????? error range for infra-red analysis.

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
*FINISH???
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
==Chair ==
===Optimization===
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods.TALK ABOUT VAND DER WAALS RADII AND INSERT DIAGRAM
|}

===Intrinsic Reaction Coordinate===
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2. MORE?

==Boat ==
===Optimization===
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}

The terminal C-C bond length is seen to be attractive, this is discussed further in section 5.2.

===Intrinsic Reaction Coordinate===
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref>:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

MORE...

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| The activation energies could also have been calculated at 0K as was done for the products.MORE?
|}

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however ....
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is noted that further optimization of the exo-transition state was conducted using the B3LYP/6-31G* method however it
|}

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-16T22:45:45Z

Ceg09: /* Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The HF method is a ???????optimization technique that unfortunately does not...The DFT (B3LYP) method on the other hand combines the HF theory with a calculated exchange energy and so id much more accurate. MORE? In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}
The calculated vibrations match well to those in the literature<ref>???</ref>, being within the ????? error range for infra-red analysis.

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
*FINISH???
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
==Chair ==
===Optimization===
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods.TALK ABOUT VAND DER WAALS RADII AND INSERT DIAGRAM
|}

===Intrinsic Reaction Coordinate===
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2. MORE?

==Boat ==
===Optimization===
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}
===Intrinsic Reaction Coordinate===
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref>:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

MORE...

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| The activation energies could also have been calculated at 0K as was done for the products.MORE?
|}

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however ....
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation. It is also important to note that the exo-transition state/product has a higer torsional strain due to the more compressed dihedral angles, as discussed in module 1<ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Module_1_-_ceg09#Dimerisation</ref>.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is noted that further optimization of the exo-transition state was conducted using the B3LYP/6-31G* method however it
|}

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>

Rep:Module 3 - ceg09

2012-02-16T22:42:03Z

Ceg09: /* Transition State Optimization */

This project aims to demonstrate the power of computational chemistry in predicting the optimal structures of not only reagents and products but also those of transition states.

==Cope Rearrangement==
===Introduction===
[[Image:Ceg09CopeReaction.png|thumb|150px|The Cope Reaction ]]
[[Image:Ceg09TransitionStates.png|thumb|150px|The two possible transition states ]]

The cope rearrangement is a [3,3]-sigmatropic shift reaction first developed by Arther Cope in the 1940's<ref>{{DOI|10.1021/ja01859a055}}</ref>. In this section we will explore such a reaction between two 1,5-hexadiene molecules using computational methods. The reaction involves the concerted breaking of 2 bonds and formation of a new σ-bond, passing through either a so-called "chair" or "boat" transition state. Studies have shown<ref>{{DOI|10.1021/ja00101a078}}</ref> that the "boat" structure to be higher in energy at the B3LYP/6-31G* level of theory. Similar computational methods will be employed here hoping to align with the results from previous studies.

===Optimization of Reactants & Products===
First of all the minimal energy conformations were found using the Hartree-Fock method with a 3-21G basis set . The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From HF/3-21G Method
! Structure!! Corresponding Appendix Structure!! Image!! Energy (Hartrees) !! Relative Energy (kcal/mol)!! Point Group
|-
| Gauche 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 4 ||[[File:Ceg09Gauche 1.jpg|150px]] ||-231.69153 || 0.71 || C<sub>2</sub>
|-
| Gauche 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 2 ||[[File:Ceg09Gauche 2.jpg|150px]] ||-231.69167 || 0.62 || C<sub>2</sub>
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12471</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 3 ||[[File:Ceg09Gauche 3.jpg|150px]] || -231.69266 || 0.00 || C<sub>1</sub>
|-
| Gauche 4 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 1 ||[[File:Ceg09Gauche 4.jpg|150px]] ||-231.68772 || 3.10 || C<sub>2</sub>
|-
| Gauche 5 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 5 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 6 ||[[File:Ceg09Gauche 5.jpg|150px]] ||-231.68962 || 2.20 || C<sub>1</sub>
|-
| Gauche 6 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 6 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| Gauche 5 ||[[File:Ceg09Gauche 6.jpg|150px]] ||-231.68962 || 1.91 || C<sub>1</sub>
|-
| Anti 1 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| rowspan="4"| As appendix ||[[File:Ceg09Anti 1.jpg|150px]] ||-231.69260 || 0.04 || C<sub>2</sub>
|-
| Anti 2 - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 2.jpg|150px]] ||-231.69254 || 0.08 || C<sub>i</sub>
|-
| Anti 3<ref>http://hdl.handle.net/10042/to-12473</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 3 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 3.jpg|150px]] ||-231.68907 || 2.25 || C<sub>2h</sub>
|-
| Anti 4<ref>http://hdl.handle.net/10042/to-12474</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 4 Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>|| [[File:Ceg09Anti 4.jpg|150px]] ||-231.69097 || 1.06 || C<sub>1</sub>
|-
|}

It can be seen that the most stable structure is gauche 3 followed by anti 1 and anti 2 respectively. These three minimized structures were then further optimized using a higher level DFT B3LYP method and basis set 6-31G(d).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Energies From B3LYP/6-31G Method
! Structure!! Energy (Hartrees) !! Relative Energy (kcal/mol) !! Point Group
|-
| Gauche 3<ref>http://hdl.handle.net/10042/to-12475</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Gauche 3 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61133 ||0.29 || C<sub>1</sub>
|-
| Anti 1<ref>http://hdl.handle.net/10042/to-12476</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 1 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61179 ||0.00 || C<sub>2</sub>
|-
| Anti 2<ref>http://hdl.handle.net/10042/to-12477</ref> - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Anti 2 DFT Opt.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white</script>
</jmolAppletLink>
</jmol>||-234.61171 || 0.05 || C<sub>i</sub>
|-
|}
After further optimization the geometries remain extremely similar however the energy values and relative order of the three conformers has changed. The change is the energy values themselves is simply a consequence of using a different optimization method. Whilst values calculated using the same method can be compared it is not possible to compare numerical values between two calculation methods. However, the relative energies can be compared and yield interesting results.

Under the B3LYP/6-31G* level of theory the anti 1 conformer is seen to be lowest in energy compared to the HF/3-21G level which calculates the gauche 3 conformer to be the most stable. This discrepancy is most likely due to the differences in accuracy between the two methods. The HF method is a ???????optimization technique that unfortunately does not...The DFT (B3LYP) method on the other hand combines the HF theory with a calculated exchange energy and so id much more accurate. MORE? In addition, the 3-21G basis set is of low accuracy especially when compared to the more advanced 6-31G* set. All in all it is fair to say that the B3LYP/6-31G* method is much more accurate and indeed often aligns well with corresponding experimental results.
[[Image:Ceg09App Energy Picture.png|right|thumb|150px|Stabilizing interaction energy profile ]]
[[Image:Ceg09App Orbital Overlap Picture.png|right|thumb|150px|Stabilizing orbital interaction for a simple ethane molecule ]]
The relative energies found under the B3LYP method can be explained by considering the intramolecular interactions involved in stabilizing the anti and gauche conformers. It is known that<ref>????</ref> interactions between σ and σ*-orbitals lead to a stabilizing effect as shown in the diagram to the right. These interactions are maximal when the σ and σ*-orbitals are in an anti-periplanar (app) configuration, that is with a dihedral angle of 180°. The table below summarizes such interactions for both the gauche 3 and anti 1 conformer.

{| class="wikitable" border="1" style="text-align:center"
|+ Qualitative Explanation of Conformer Stabilization
!colspan="2"| Gauche 3!!colspan="2" | Anti 1
|-
| colspan="2"| [[Image:Ceg09Gauche-Picture.png|200px|Gauche 3 structure]] || colspan="2"| [[Image:Ceg09Anti-Picture.png|200px|Anti 1 structure]]
|-
| '''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)''' ||'''Stabilizing Orbital Interaction''' || '''Dihedral Angle(s) (°)'''
|-
|2 x σ<sub>C-C</sub>/σ*<sub>C-H</sub> ||rowspan="2"| 171 and 173 ||rowspan="2"|4 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> ||rowspan="2"|178 and 178
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-C</sub>
|-
|2 x σ<sub>C-H</sub>/σ*<sub>C-H</sub> || 179 || 2 x σ<sub>C-C</sub>/σ*<sub>C-C</sub> ||177
|
|}

As you can see the anti 1 σ/σ* interactions are all very close to perfectly app whereas the gauche 3 conformer two angles that are slightly lower than the optimal 180°angle. This slight deviation from the optimal app geometry means that the stabilization from the σC-C/σ*C-H and σC-H/σ*C-C interactions will lessened. In addition, the energy profile for these types of interactions, shown above, shows that the E2/stabilization energy is maximized when the σ-σ* energy gap is reduced. This can be achieved when you have a good σ-donor (high in energy) and a good σ*-acceptor (low in energy). All in all, the anti 1 conformer has the best combination of donor-acceptor interactions and so will have the largest total E2 value and hence be the most stable.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is known that a second stabilizing effect of van der Waals dispersion forces between two non-bonded atoms can in fact favour the gauche conformation. This is a longer ranging effect often seen between hydrogens at non-bonding distances and is maximal at the sum of the van der Waals radii for the two interacting atoms, 2.4Å for H...H interaction. Closer examination of this effect could be carried with the gauche conformer expected to have more favourable contacts than the anti conformer.
|}

===Frequency Analysis===
Frequency analysis works by taking the 2nd derivative of the potential energy surface of the system. If all of the frequencies generated by the frequency analysis are positive the optimized geometry/ minimum on the potential energy surface has been found, however if one of the frequency values is negative a maximum point/transition state on the potential energy surface has been found. All of the frequencies calculated for all 3 confomers were positive and the C=C vibrations are shown below.
{| class="wikitable" border="1" style="text-align:center"
|+ Frequency Analysis
! Structure!! Frequency (cm<sup>-1</sup>) !! Intensity !! Description !! Image/Animation
|-
|rowspan="2"|Anti 1<ref>http://hdl.handle.net/10042/to-12478</ref> - Click [[Media:Ceg09Anti 1 IR.jpg|here]] to visualize IR spectrum || 1731.96 || 4.73 ||Symmetric C=C stretch ||[[File:Ceg09Anti 1 Symm.gif|150px]]
|-
| 1734.86 ||13.55 ||Asymmetric C=C stretch ||[[File:Ceg09Anti 1 Asymm.gif|150px]]
|-
|rowspan="2"|Anti 2<ref>http://hdl.handle.net/10042/to-12479</ref> - Click [[Media:Ceg09Anti 2 IR.jpg|here]] to visualize IR spectrum ||1731.07 || 0.00 ||Symmetric C=C stretch ||[[File:Ceg09Anti 2 Symm.gif|150px]]
|-
| 1734.31 || 18.1 || Asymmetric C=C stretch ||[[File:Ceg09Anti 2 Asymm.gif|150px]]
|-
|rowspan="2"|Gauche 3<ref>http://hdl.handle.net/10042/to-12480</ref> - Click [[Media:Ceg09Gauche 3 IR.jpg|here]] to visualize IR spectrum ||1731.86 || 6.89 ||Symmetric C=C stretch ||[[File:Ceg09Gauche 3 Symm.gif|150px]]
|-
| 1732.97 || 6.13 ||Asymmetric C=C stretch ||[[File:Ceg09Gauche 3 Asymm.gif|150px]]
|-
|}
The calculated vibrations match well to those in the literature<ref>???</ref>, being within the ????? error range for infra-red analysis.

===Energies===
Computational frequency analysis is not only a powerful tool for accurately predicting infra-red vibration frequencies but also for providing key energetic data from the system. Thermochemical data from the frequency analysis provides us with 4 key pieces of information:
#'''Sum of electronic and zero point energies''' - the sum of the system's potential energy at 0K and the system's zero-point vibrational energy (E<sub>elec</sub> + ZPE)
#'''Sum of electronic and thermal Energies''' - the energy of the system under standard conditions, 298.15K and 1 atm. This includes the vibrational, translational and rotational energies also under standard conditions (E + E<sub>vib</sub> + E<sub>rot</sub> +E<sub>trans</sub>)
# '''Sum of electronic and thermal Enthalpies''' - the energy including an enthalpic RT correction
#'''Sum of electronic and thermal Free Energies''' - the entropic contribution to the free energy

The results for standard conditions along with the results from frequency analysis conducted at 0K (in fact, 0.00001K) are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Energies From B3LYP/6-31G Vibrational Analysis
! Anti 1<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12482</ref> !! Anti 2<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12483</ref> !! Gauche 3<ref>Frequency Analysis at 0K - http://hdl.handle.net/10042/to-12481</ref>
|-
|colspan="3"| At 293.15K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469298
Sum of electronic and thermal Energies= -234.461966
Sum of electronic and thermal Enthalpies= -234.461022
Sum of electronic and thermal Free Energies= -234.500862
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.469204
Sum of electronic and thermal Energies= -234.461857
Sum of electronic and thermal Enthalpies= -234.460913
Sum of electronic and thermal Free Energies= -234.500777
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468693
Sum of electronic and thermal Energies= -234.461464
Sum of electronic and thermal Enthalpies= -234.460520
Sum of electronic and thermal Free Energies= -234.500105
</pre>
|-
|colspan="3"| At 0K
|-
|<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468861
Sum of electronic and thermal Energies= -234.468861
Sum of electronic and thermal Enthalpies= -234.468861
Sum of electronic and thermal Free Energies= -234.468861
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468766
Sum of electronic and thermal Energies= -234.468766
Sum of electronic and thermal Enthalpies= -234.468766
Sum of electronic and thermal Free Energies= -234.468766
</pre>||<pre style="white-space: pre-wrap">
Sum of electronic and zero-point Energies= -234.468256
Sum of electronic and thermal Energies= -234.468256
Sum of electronic and thermal Enthalpies= -234.468256
Sum of electronic and thermal Free Energies= -234.468256
</pre>
|-
|}

As you can see at 0K/absolute zero there is no thermal contribution, as is the definition of absolute zero, and all of the energies are equal to the sum of electronic and zero-point energy.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| This type of frequency analysis can be conducted under many difference conditions of temperature, pressure etc. Further study of the effects of varying the temperature and pressure would prove interesting. On raising the temperature one would expect the following results:
*FINISH???
|}

== Optimization of Chair & Boat Transition States==
So far we have demonstrated the use of computational chemistry in determining optimal structures, vibrational data and energetic information for individual reactant systems. Another perhaps more useful computational technique is the modelling and analysis of transition states. Transition states are extremely hard to isolate/observe experimentally as they correspond to a maximum on the system's potential energy surface hence are very unstable. For this reason it is clearly very useful to be able to model such structures in order to gain valuable information about the reaction pathway perhaps helping to improve and better understand synthetic methods.

The possible boat and chair transition structures of the hexadiene cope rearrangement will be optimized and analyzed in order to explain the relative stability of the chair structure as discussed in section 1.1.
==Chair ==
===Optimization===
[[Image:Ceg09Allyl Fragment.jpg|thumb|150px|Initial optimized allyl fragment ]]
[[Image:Ceg09Initial Chair Optimization.jpg|thumb|150px|General optimized chair geometry ]]
In order to generate the optimized chair transition state an initial allyl fragment was first optimized using the HF/3-21G method, the structure of which can been seen on the right. Two of these fragments were then combined and optimized in 2 different fashions which are explained in detail below. The initial "guessed" structure is displayed to the right.

{| class="wikitable" border="1" style="text-align:center"
|+ Optimization of the Chair Transition State Using HF/3-21G
! colspan="2" rowspan="2"|Method Number!!rowspan="2"| 3D Visualization!! rowspan="2"|Method !! colspan="2"| Terminal C-C Bond Distance (Å)
|-
| C14-C6 || C9-C1
|-
| colspan="2"|1||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Initial Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> ||<pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Resulting imaginary frequency = -817.9cm<sup>-1</sup>
Additional parameters = Force constants calculated once</pre> ||2.02 ||2.02
|-
|rowspan="2"| 2||a||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Coord Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization
Optimization to = Minimum
Additional parameters = redundant coordinates use to freeze the terminal C-C distances to 2.2Å</pre>||2.20 ||2.20
|-
| b||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Redundant Deriv Chair Optimization.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white; measure 1 9; measure 14 6</script>
</jmolAppletButton>
</jmol> || <pre style="white-space: pre-wrap">
Type = Optimization and frequency
Optimization to = TS (Berny)
Additional parameters = Optimized structure from part a taken and redundant coordinates use to set the derivative of the terminal C-C distances. Force constants never calculated</pre>||2.02 ||2.02
|}
{|
|-
|[[File:Ceg09Chair -818 Animation.gif|100px|thumb|Imaginary frequency vibration of the chair transition structure]] ||The presence of a negative vibration/imaginary vibration shows that a transition state has been found, to see the IR spectrum click [[Media:File:Ceg09Chair IR.jpg |here]]. It is clear that whilst two different methods were used to optimize the chair transition structure they both yielded the same terminal C-C bond lengths and so can be said to be reliable methods.TALK ABOUT VAND DER WAALS RADII AND INSERT DIAGRAM
|}

===Intrinsic Reaction Coordinate===
Another useful computational tool is the calculation of the intrinsic reaction coordinate (IRC). The IRC take the previously found transition state structure and follows the minimum reaction pathway along the systems reaction coordinate until a local minimum is found. This is very useful when trying to predict the most likely product when moving away from the transition state.

In this case the reaction pathway is symmetrical and so the IRCs will only be calculated in the forwards direction. Several different parameters were tested to find which set of conditions gave the most accurate results.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Chair Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12487</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09NormalIRCMovie.gif|150px]]||[[Image:Ceg09EnergyNormalIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSNormalIRC.jpg|500px]]
|-
| rowspan="2"| 100<ref>http://hdl.handle.net/10042/to-12489</ref> || rowspan="2"| once ||rowspan="2"| [[File:Ceg09100ptsIRCMovie.gif|150px]]||[[Image:Ceg09Energy100ptsFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMS100ptsFCIRC.jpg|500px]]
|-
| rowspan="2"| 50<ref>http://hdl.handle.net/10042/to-12488</ref> || rowspan="2"| always || rowspan="2"| [[File:AlwaysFCIRCMovie.gif|150px]]||[[Image:Ceg09EnergyAlwaysFCIRC.jpg|500px]]
|-
|[[Image:Ceg09RMSAlwaysFCIRC.jpg|500px]]
|}
The structure of the molecule at the end of the IRC is the minimum energy structure moving away from the transition state towards products and so in theory should be the reaction product. This final structure was optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12493</ref>, the results of which are shown below:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 2 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09 Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69167|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 2.jpg|150px]]||-231.69167
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 2 Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

As you can see the energy of the final IRC structure is identical to the energy of the Gauche 2 conformer found in section 1.2. MORE?

==Boat ==
===Optimization===
[[Image:ceg09Before_Opt.png|thumb|250px|Reactant and product leading to an unsuccessful optimized structure ]]
[[Image:Ceg09Before_Opt_Bent.png|thumb|250px|Reactant and product leading to a successful optimized structure ]]
In order to optimize the boat transition state the QST2 method will be used. Unlike the demonstrated method for the chair optimization, in which a "guessed" transition state structure must be used, the QST2 method takes the reactants and products and finds a transition state between them. It is noted that the atomic labels of the reactants and products must be identical for this method to work.

Initially the optimized anti 2 structure was used as the basis for the reactant and product structures, shown to the right. When the QST2 transition state was optimized<ref>http://hdl.handle.net/10042/to-12484</ref> straight from this structure the job failed. This is because the QST2 method only linearly interpolates between the reactant and product and does not rotate any of the bonds. In effect all it has done is translated the structure. In order to overcome this problem, the central C9-C1-C4-C7 dihedral was set to 0° and the two internal C9-C1-C4 and C7-C4-C1 angles set to 100°.

Unfortunately, the structure (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09Unsuccessful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize) calculated by this method was seen to be asymmetric and so suggested that the true minimal chair transition state structure had not been found. Observation of the frequency log file<ref>http://hdl.handle.net/10042/to-12485</ref> shows that two imaginary/negative frequencies were calculated showing that neither a minimum nor a maximum point on the potential energy surface was found. These imaginary frequencies are shown below:

<pre>
Low frequencies --- -681.9496 -339.2038 -66.2980 -60.6073 -36.1093 0.0002
Low frequencies --- 0.0007 0.0008 208.6869
</pre>

This problem arises when no force constants are calculated as the computational process effectively guesses a value that can sometimes
be wrong. The optimization was carried out for a second time this time specifying to calculate the force constants once and indeed a successful optimization structure was obtained (click <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09successful Chair.mol</uploadedFileContents>
<text>here</text>
<script>background=white</script>
</jmolAppletLink>
</jmol> to visualize).

{| class="wikitable" border="1" style="text-align:center"
|+ Optimized Boat Transition Structure Data<ref>http://hdl.handle.net/10042/to-12486</ref>
! Image!! Terminal C-C Bond Length Å!! Imaginary Frequency (cm<sup>-1</sup>)- click [[Media:Ceg09Boat_IR.jpg|here]] to visualize IR spectrum!! Motion of Imaginary Frequency
|-
| [[Image:Ceg09Good boat Structure.jpg|150px]] || 2.14 || -839.92|| [[File:Ceg09Imaginary_Motion_Of_Boat.gif‎|150px]]
|}
===Intrinsic Reaction Coordinate===
The same IRC analysis was conducted for the boat transition structure however the force constants were always calculated in both cases.
{| class="wikitable" border="1" style="text-align:centre"
|+ IRC Analysis for The Boat Transition State
! Number of Points Along IRC!! Calculate Force Constants!! IRC Animation!! IRC Data!! Final Structures
|-
| rowspan="2"| 50 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 50.gif |150px]]||[[Image:Ceg09IRC Energy 50.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 50.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 50.jpg|500px]]
|-
| rowspan="2"| 100 || rowspan="2"| always ||rowspan="2"| [[File:Ceg09Boat IRC 100.gif |150px]]||[[Image:Ceg09IRC Energy 100.jpg |500px]]||rowspan="2" | [[Image:Ceg09Final IRC Boat 100.jpg |150px]]
|-
|[[Image:Ceg09IRC RMS 100.jpg |500px]]
|}

The final IRC structure was again optimized using HF/3-21G<ref>http://hdl.handle.net/10042/to-12492</ref>:
{|style="text-align:centre"
!Final IRC Structure !! Energy (Hartrees)!! !! Gauche 3 Structure !! Energy (Hartrees)
|-
|[[Image:Ceg09Final IRC Opt Structure.jpg|150px]]|| rowspan="2"|-231.69266|| rowspan="2"| c.f.|| [[Image:Ceg09Gauche 3.jpg|150px]]||-231.69266
|-
|<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Final Boat IRC Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09Gauche 3 OPT.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>
|}

MORE...

=== Overall Activation Energies===
Shown below are the calculated activation energies of the cope rearrangement via both the chair and boat transition states. The results reflect the reported relative stabilities of the two transition states where the boat structure, under both levels of theory, was found to have a higher activation energy and be less stable than the chair structure.

{| class="wikitable" border="1" style="text-align:center"
|+ Activation Energies Using Hartree-Fock and DFT Methods
! !!colspan="4"| HF/3-21G!! colspan="4" | B3LYP/6-31G*!!rowspan="3" | Literature Activation Energy (kcal/mol)
|-
|rowspan="2"| '''Structure''' || rowspan="2"| '''Energy''' '''(Hartrees)''' || '''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''|| rowspan="2"| '''Energy (Hartrees)'''||'''Corresponding Hexadiene Conformer Energy (Hartrees)''' || rowspan="2"| '''Activation Energy (Hartrees)'''|| rowspan="2"| '''Activation Energy (kcal/mol)'''
|-
|'''Gauche 3''' || '''Anti 1'''
|-
|Chair||-231.61932 || -231.69266 ||0.07334 || 46.021555 || -234.55698 || -234.61179 || 0.05481 ||34.3938 || 33.5±0.5
|-
|Boat|| -231.60280 || -231.69266 ||0.08986 ||56.388013 || -234.54309 || -234.61179 || 0.0687 || 43.1099 || 44.7±2.0
|-
|}

As you can see the B3LYP/6-31G* method yielded results that were very similar to the experimental values whereas the values calculated under the HF/3-21G method were not so comparable. This is most likely due to the lower accuracy of both the HF method and 3-21G basis set as discussed previously.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| The activation energies could also have been calculated at 0K as was done for the products.MORE?
|}

===Conclusion===

==Diels-Alder Cycloaddition - Butadiene + Ethene==
[[Image:Ceg09Sigmatropic.png|thumb|right|200px|The Diels-Alder reaction between butadiene and ethene ]]

The Diels-Alder reaction is an example of a [4s+2s] cycloaddition and is a very important synthetic tool. It also provides an interesting system for computational analysis.

First of all a relatively simple system of butadiene and ethene will be examined.

===Reactant Optimization===
Both butadiene and ethene were optimized first using the semi-empirical AM1 method then using a DFT B3LYT/6-31G* method. The results of which along with the HOMO and LUMO molecular orbitals (MOs) are displayed. It is noted that, in terms of MO symmetry, '''''s''''' refers to a symmetric orbital with respect to the plane and '''''a''''' to an asymmetric orbital.
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Butadiene||rowspan="2" | 0.48797||rowspan="2" | 0.0000175||HOMO || [[Image:Ceg09AM1 But HOMO.jpg|100px]]||a||rowspan="2" | -155.98595||rowspan="2" | 0.0000567||HOMO || [[Image:DFT But HOMO.jpg|100px]]||a
|-
|LUMO||[[Image:AM1 But LUMO.jpg|100px]]||s||LUMO||[[Image:DFT But LUMO.jpg|100px]]||s
|-
| rowspan="2" |Ethene||rowspan="2" | 0.02619 ||rowspan="2" | 0.0000252 ||HOMO || [[Image:Ceg09AM1 HOMO.jpg|100px ]]||s||rowspan="2" | -78.58746 ||rowspan="2" | 0.0001445 ||HOMO || [[Image:Ceg09DFT HOMO.jpg |100px]]||s
|-
|LUMO||[[Image:Ceg09AM1 LUMO.jpg|100px]]||a||LUMO||[[Image:Ceg09DFT LUMO.jpg|100px]]||a
|}

You can that there is no visible difference between the MOs generated using AM1 and DFT which is encouraging. This, however, is not always the case. Here we are dealing with a relatively simple system hence the continuity of the results. For more complex the systems we will see later, this agreement is not always so clear and should be expected when using two different computational methods.

In addition, it is clear that the LUMO of ethene and the HOMO of butadiene both have the same symmetry hence there is a favorable overlap between the two leading to a "reaction".
===Transition State Optimization===
Following the optimization of the reactants, their optimized structures were then aligned once again in a guessed transition state geometry in order to optimize to the transition state. The fragments were aligned so that the two forming σ-bonds were approximately 2Å. The system was optimized to a TS(Berny) with the force constants calculated once, the results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*!!rowspan="2"| Frequency Analysis
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" | 0.11165||rowspan="2" | 0.00002973||HOMO || [[Image:Ceg09TS AM1 HOMO.jpg|100px]]||a||rowspan="2" | -234.54389655||rowspan="2" |0.00000826||HOMO || [[Image:Ceg09TS DFT HOMO.jpg|100px]]||s||rowspan="2"| Both of the methods showed one imaginary frequency at -956.42 and -524.81 for AM1 and B3LYP respectively

[[Image:Ceg09AM1_TS_Imaginary_Freq_Animation.gif‎|150px]]

(animation does not always run, please click to visualize)

It is noted that the imaginary frequency corresponds to the synchronous formation of the two new σ-bonds in the product
|-
|LUMO||[[Image:Ceg09TS AM1 LUMO.jpg|100px]]||s||LUMO||[[Image:Ceg09TS DFT LUMO.jpg|100px]]||s
|-
|}

{| class="wikitable" style="border: 1px solid darkgray;" style="float: left;" "text-align:centre"
|+LCAO Analysis
!rowspan="2"| MO of the Transition State!!colspan="2"|Reactant MO Combination:
|-
|Butadiene || Ethene
|-
|LUMO || LUMO ||HOMO
|-
|HOMO (AM1)|| HOMO ||LUMO
|}

At first HOMOs generated by the two different computational methods appear to be different. Further analysis of the MOs shows that in fact several sets of degenerate orbitals were found and in all of the examined cases these degenerate pairs simply flipped moving between the two different methods. This is perhaps more clearly demonstrated visually below. The three sets of degenerate orbitals are highlighted in yellow in the energy diagram.

{| class="wikitable" border="1" style="text-align:center"
|+ Comparison of MO Calulated Using AM1 and B3LYP/6-31G* Methods
|-
|Relative MO Energy Levels|| Method || HOMO-5 || HOMO-4 || HOMO-3|| HOMO-2|| HOMO-1 || HOMO || LUMO
|-
|rowspan="2"| [[Image:Ceg09TS DFT Enrgy Diagram.jpg|150px|Relative energy levels for MOs]]|| AM1||[[Image:Ceg09TS AM1 HOMO-5.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO-4.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-3.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-2.jpg |115px|]] ||[[Image:Ceg09TS AM1 HOMO-1.jpg|115px|]] ||[[Image:Ceg09TS AM1 HOMO.jpg|115px|]] ||[[Image:Ceg09TS AM1 LUMO.jpg|115px|]]
|-
|DFT||[[Image:Ceg09TS DFT HOMO-5.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO-4.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-3.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-2.jpg|115px|]] ||[[Image:Ceg09TS DFT HOMO-1.jpg |115px|]] ||[[Image:Ceg09TS DFT HOMO.jpg |115px|]] ||[[Image:Ceg09TS DFT LUMO.jpg|115px|]]
|}
This discrepancy is due to the different ways in which each method calculated MOs and in their differing accuracies. The B3LYP/6-31G* method is much more accurate hence it would be reasonable to assume that its results are more reliable.
[[Image:Ceg09Lennard-Jones Potential.png |thumb|right|200px| A general Lennard-Jones potential]]
Another interesting approach to analyzing transition state structures is to consider the distance between the atoms forming new bonds in the product. In 1924 John Lennard-Jones first proposed a potential energy surface showing the interaction between two atoms/nuclei. This model is still holds true and can be used to explain how molecular orbitals overlap resulting in a stabilizing effect. A simple Lennard-Jones potential is displayed to the right and shows that, most simply, there are 4 key regions on the potential energy curve:
# Repulsive region - at very small internuclear distances the nuclei are so close together that they repel one another and the energy tends towards infinity.
#Equilibrium bond distance - this is the minimum on the potential energy curve where the most stable interaction if found. This is equal the length of the bond between two atoms at equilibrium.
# Dissociation limit - this is the region in which the two atoms are too far apart to interact and are essentially two separate atoms.

Finally, there is the 4<sup>th</sup> region in which the atoms move closer and closer until finally a bond is formed. This is the region in which attractive van der Waals interactions stabilize the system and may eventually lead to bond formation and is where the transition state structure is found. Van der Waals interactions are observed between atoms that are separated by a distance equal to the sum of their van der Waaals radii. This is because it is at this distance that the electron densities/MOs of each atom begin to overlap. The van der Waals radius for a carbon atom it approximately 1.6Å and so attractive interactions will be felt when the C-C internuclear distance is <3.2Å. These terminal C-C contacts for the transition state are shown below along with other C-C bond lengths.

{| class="wikitable" border="1" style="text-align:center"
|+ Bond Lengths (Å) and Angles
! Method!! Ethene C=C !! Diene C=Cs !! Diene C-C !! Terminal C-C distance/bond being formed
|-
| AM1- <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09AM1 Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.38||1.38|| 1.40||2.12
|-
| B3LYP/6-31G* - <jmol>
<jmolAppletLink>
<uploadedFileContents>Ceg09DFT Structure.mol</uploadedFileContents>
<text>visualize</text>
<script>background=white; measure 7 1; measure 10 12; measure 1 4; measure 12 14</script>
</jmolAppletLink>
</jmol>|| 1.39||1.38||1.41||2.27
|}

As you can see the terminal C-C distance is considerably lower than 3.2Å proving that a new σ-bond is indeed being formed in the transition state. The C-C bonds are also all very similar in length i.e. there are no longer C-C single and C=C double bonds. This again fits with a delocalized transition state structure in which bonds are being broken and formed.
{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| Intrinsic reaction coordinate analysis was not conducted here however ....
|}

==Diels-Alder Cycloaddition - Hexadiene + Maleic Anhydride==
[[Image:Ceg09MAReaction.png|thumb|right|200px|The Diels-Alder reaction between hexadiene and maleic anhydride ]]
A more complex Diels-Alder reaction will now be examined, that of cyclohexa-1,3-diene and maleic anhydride. An interesting characteristic of many Diels-Alder reactions is the selectivity between endo- and exo-products. The kinetically formed endo-product is often favoured and can be explained by considering the molecular orbitals involved in the formation of the transition state. MO analysis will be conducted here in order to support this observation.
===Reactant Optimization===
Once again the reactants were optimized using both AM1 and B3LYP/6-31G* methods. The results are shown below:
{| class="wikitable" border="1" style="text-align:center"
|+ Optimization Results
! rowspan="2"| Molecule!! colspan="5"| AM1!!colspan="5"| B3LYP/6-31G*
|-
| Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry ||Energy (Hartrees)|| RMS || colspan="2"| MOs||MO Symmetry
|-
| rowspan="2" |Cyclohexadiene||rowspan="2" | 0.02771<ref>http://hdl.handle.net/10042/to-12508</ref>||rowspan="2" | 0.000005||HOMO || [[Image:Ceg09 CHD HOMO AM1.jpg|100px]]||a||rowspan="2" | -233.419<ref>http://hdl.handle.net/10042/to-12507</ref>||rowspan="2" | 0.000032||HOMO || [[Image:Ceg09 CHD HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 CHD LUMO AM1.jpg |100px]]||s||LUMO||[[Image:Ceg09 CHD LUMO DFT.jpg|100px]]||s
|-
| rowspan="2" |Maleic Anhydride||rowspan="2" |-0.12182<ref>http://hdl.handle.net/10042/to-12509</ref>|| rowspan="2" | 0.000037 ||HOMO || [[Image:Ceg09 MA HOMO AM1.jpg |100px ]]||s||rowspan="2" | -379.2895<ref>http://hdl.handle.net/10042/to-12510</ref> ||rowspan="2" | 0.000141 ||HOMO || [[Image:Ceg09 MA HOMO DFT.jpg |100px]]||a
|-
|LUMO||[[Image:Ceg09 MA LUMO AM1.jpg |100px]]||a||LUMO||[[Image:Ceg09 MA LUMO DFT.jpg |100px]]||a
|}
The HOMO and LUMOs show the correct symmetry for favourable interactions with the exception of the maleic anhydride calculated usind the B3LYP method. This was again found to be due to a "flipping" of degenerate MOs between the two methods however the results have not been shown in order to make this report a reasonable size!

===Transition State Optimization===
The optimized reactants were then combined and the frozen coordinates method, explained in section 3.1, used to set the distance between the two sets of carbon atoms forming new bonds to 2Å.
{| class="wikitable" border="1" style="text-align:center"
|rowspan="2"| '''Transition State Structure'''||colspan="4"|'''AM1 Optimization Results''' ||Colspan="4"|'''B3LYP/6-31G* Optimization Results'''||rowspan="2"| '''Frequency Analysis'''
|-
| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''|| '''Energy (Hartrees)'''|| '''RMS''' || colspan="2"| '''MOs'''
|-
| rowspan="2" |Exo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.050420||rowspan="2" | 0.000020||HOMO || [[Image:Ceg09 MA Exo HOMO.jpg|100px]] ||rowspan="2" |-612.67931095<ref>http://hdl.handle.net/10042/to-12499</ref> ||rowspan="2" |0.00001137 ||HOMO || [[Image:Ceg09 MA Exo HOMO DFT.jpg|100px]] ||rowspan="2"|A single imaginary frequency was found at -812.15cm<sup>-1</sup> and 448.22cm<sup>-1</sup>for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09Exo_TS_Imaginary_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Exo LUMO.jpg |100px]] || LUMO||[[Image:Ceg09 MA Exo LUMO DFT.jpg|100px]]
|-
| rowspan="2" |Endo<jmol>
<jmolAppletButton>
<uploadedFileContents></uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol>||rowspan="2" | -0.051504<ref>http://hdl.handle.net/10042/to-12503</ref> ||rowspan="2" | 0.000006 ||HOMO || [[Image:Ceg09 MA Endo HOMO.jpg |100px ]] ||rowspan="2" |-612.68339677<ref>http://hdl.handle.net/10042/to-12498</ref> ||rowspan="2" |0.00000460 ||HOMO || [[Image:Ceg09 MA Endo HOMO DFT.jpg|100px ]] || rowspan="2"|A single imaginary frequency was found at -806.36cm<sup>-1</sup> and -447.03cm<sup>-1</sup> for AM1 and B3LYP/6-31G* respectively, proving that a transition state structure was found.
[[Image:Ceg09_MA_Endo_Imagin_Freq.gif|150px]]
|-
|LUMO||[[Image:Ceg09 MA Endo LUMO.jpg |100px]] ||LUMO||[[Image:Ceg09 MA Endo LUMO DFT.jpg |100px]]
|}

Once again some variation is seen between the two methods however the MOs are all relatively similar. The optimization results clearly support the relatively stabilization of the endo-transition state with the endo- being 0.0011 Hartrees lower in energy than the exo.

{| class="wikitable style="border: 2px solid gold;"
|+ style="text-align:left"|'''Improvements:'''
| It is noted that further optimization of the exo-transition state was conducted using the B3LYP/6-31G* method however it
|}

===Further MO Analysis===
The endo-selectivity commonly observed for Diels-Alder reactions can be explained by secondary orbital overlap. Shown below are the LUMO+1 and LUMO+2 for both the endo- and exo-transition states. It can be seen that whilst their is a clear favourable interaction between the -(C=O)-O-(C=O)- fragment and the cyclohexadiene orbitals in the endo-structure, in the exo-structurea nodal plane is observed. This missing secondary orbital overlap stabilization in the exo-transition state explains why the endo-structure is more stable and hence favoured.
{| class="wikitable" border="1"
|+ Secondary Orbital Overlap Comparison
! !!Endo !! Exo
|-
| LUMO+2||[[Image:Ceg09 MA Endo LUMO+2.jpg |200px]] || [[Image:Ceg09 MA Exo LUMO+2.jpg |200px]]
|-
| LUMO+1 || [[Image: Ceg09 MA Endo LUMO+1.jpg|200px]]||[[Image:Ceg09 MA Exo LUMO+1.jpg |200px]]
|}

===Intrinsic Reaction Coordinate===
[[File:Ceg09-Energy_Profile.jpg|thumb|right|250px|General Diels-Alder energy profile.]]
IRC analysis was conducted for both the endo- and exo-transition states, the results of which are shown below. A simple energy profile, shown to the left, shows that the system is asymmetric and so the IRC must be initally calculated in both the forwards and reverse directions. It is clear from the IRC energy profiles as well as the final forwards and backwards structures that for the exo-transition state the IRC runs backwards in order to form products and the endo- forwards. It can also be seen that IRC when run away from the products takes the transition state back to the reactants as is expected.

Interestingly, the both of the final "products" look very similar hence further IRC analysis and optimization was used to examine the true nature of these structures.
{| class="wikitable" border="1" style="text-align:centre"
|+ Forwards and Backwards IRC Analysis for The Endo and Exo Transition States
! rowspan="2"|Structure!! rowspan="2"|Number of Points Along IRC!!rowspan="2"| Calculate Force Constants!! rowspan="2"|IRC Animation!! rowspan="2"|IRC Data!! colspan="2"|Final Structures
|-
|Forwards ||Backwards
|-
| rowspan="2"|Endo||rowspan="2"|10<ref>http://hdl.handle.net/10042/to-12502</ref> || rowspan="2"|Once ||rowspan="2"| [[File:Ceg09_IRC_Endo.gif |150px]]||[[Image:Ceg09 Endo IRC F&B Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09 Endo IRC Final Image Forwards.jpg|150px]]||rowspan="2"|[[Image:Ceg09 Endo IRC Final Image Backwards.jpg|150px]]
|-
|[[Image:Ceg09 Endo IRC F&B RMS.jpg |500px]]
|-
| rowspan="2"|Exo|| rowspan="2"| 50<ref>??</ref>|| rowspan="2"|Once ||rowspan="2"| [[File:Ceg09Exo IRC F and B.gif |150px]]||[[Image:Ceg09 MA Exo IRC Energy.jpg |500px]]||rowspan="2" | [[Image:Ceg09ExoForwardsIRCFinal.jpg |150px]]||rowspan="2"|[[Image:Ceg09ExoBackwardsIRCFinal.jpg|150px]]
|-
|[[Image:Ceg09 MA Exo IRC RMS.jpg |500px]]
|}
[[Image:Ceg09MSEnergy Profile.png|thumb|right|250px|Reviewed reaction energy profile ]]
The extended IRC and optimization analysis, shown below, yielded some very interesting results.

The product formed via the kinetic, endo-pathway is expected to be higher in energy than the thermodynamic/exo-product. However, the IRC produced two identical products with the same energies suggesting that both transition states lead to the same product. A revised reaction profile has been drawn, shown to the left, however further analysis of the products using a higher level of theory is needed to distinguish between the two structures and to draw a more definite picture of the overall reaction pathway.
{| class="wikitable" border="1" style="text-align:centre"
|+ Final IRC and Optimization Data for The Endo and Exo Transition States
!Structure!!IRC Data<ref>Endo - http://hdl.handle.net/10042/to-12500 & Exo - http://hdl.handle.net/10042/to-12501</ref>!!Energy of Final Optimized Structure (Hartrees)!! Image of Final Structure
|-
|rowspan="2"|Endo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09EndoFinal_IRC_Structure_Opt.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09 IRC Endo Forwards Enegry.jpg |150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12497</ref>||rowspan="2"|[[File:Ceg09_Endo_Final_IRC_Structure_Opt.jpg |150px]]
|-
|[[File:Ceg09 IRC Forwards RMS.jpg |150px]]
|-
|rowspan="2"|Exo - <jmol>
<jmolAppletButton>
<uploadedFileContents>Ceg09ExoIRCBackwardsFinal.mol</uploadedFileContents>
<text>Visualise</text>
<script>background=white</script>
</jmolAppletButton>
</jmol> ||[[File:Ceg09_IRC_Back_Energy.jpg|150px]] ||rowspan="2"|0.16<ref>http://hdl.handle.net/10042/to-12496</ref>||rowspan="2"|[[File: Ceg09Exo IRC Structure.jpg |150px]]
|-
|[[File:Ceg09_IRC_Back_RMS.jpg |150px]]
|}

===References===
<references/>