ChemWiki - User contributions [en]

Mod:Hunt Research Group/calendar

2016-03-16T15:33:17Z

Aed12: /* Calendar */

Back to the main [https://www.ch.ic.ac.uk/wiki/index.php/Mod:Hunt_Research_Group wiki-page]
== Calendar ==

{| class="wikitable"
|-
! 1
! 2
! 3
|-
| Tricia (Not Done)
| Jan Szopinski (Done)
| Aimee Darke (Done)
|-
| Vincent Chen (Not Done)
| Ken Watson (Done)
| Rebecca Rowe (Done)
|-
|Anil Bijoy (Done)
| Richard Fogarty (Not Done)
|
|-
|-)
|}
Everyone should be away during the college closure dates, so you don't need to add your name on those days

Tricia maybe: Tricia may or may-not be in college i.e. working from home
{| class="wikitable" style="width: 100%"
!Mon
!Tue
!Wed
!Thur
!Fri
!Sat
!Sun

|-
|7th March
|8th
|9th
|10th
|11th
| style="background: grey;" |12th
| style="background: grey;" |13th

|-
|14th
|15th
|16th
|17th
|18th
| style="background: grey;" |19th
| style="background: grey;" |20th

|-
|21st
|22nd
|style="background: yellow;" |23rd Last day of easter term
Jan away

Ken at Burlington House
| style="background: lime;" |24th College Closure Start

| style="background: lime;" |25th

| style="background: grey;" |26th
| style="background: grey;" |27th

|-
| style="background: lime;" |28th

| style="background: lime;" |29th

| style="background: lime;" |30th College Closure End

| style="background: lime;" |31st
Jan away

| style="background: blue; color: white;" |1st April
Jan away

| style="background: grey;" |2nd
| style="background: grey;" |3rd

|-
|4th
Jan away

Ken away
|5th
Jan away
|6th
Jan away
|7th
|8th
| style="background: grey;" |9th
| style="background: grey;" |10th

|-
|11th
Anil away
|12th
Anil away
|13th
Anil away
|14th
Becky away
|15th
Becky away
| style="background: grey;" |16th
| style="background: grey;" |17th

|-
|18th
Becky away
|19th
Becky away
|20th
Becky away
|21st
Becky away
|22nd
Becky away
| style="background: grey;" |23rd
| style="background: grey;" |24th

|-
|style="background: yellow;" |25th 1st day of summer term
Aimee away
|26th
Aimee away
|27th
Aimee away
|28th
Aimee away
|29th
Aimee away
| style="background: grey;" |30th
| style="background: grey;" |1st May

|}

Mod:Hunt Research Group/calendar

2015-12-01T14:00:20Z

Aed12: /* Calendar */

Back to the main [https://www.ch.ic.ac.uk/wiki/index.php/Mod:Hunt_Research_Group wiki-page]
== Calendar ==

{| class="wikitable"
|-
! 1
! 2
! 3
|-
| Tricia (Done)
| Jan Szopinski (Done)
| Aimee Darke (Done)
|-
| Vincent Chen (Not Done)
| Ken Watson (Done)
| Rebecca Rowe (Done)
|-
|Anil Bijoy (Done)
| Richard Fogarty (Done)
|
|-
|-)
|}
Everyone should be away during the college closure dates, so you don't need to add your name on those days

Tricia maybe: Tricia may or may-not be in college i.e. working from home
{| class="wikitable" style="width: 100%"
!Mon
!Tue
!Wed
!Thur
!Fri
!Sat
!Sun

|-
|30th
| style="background: blue; color: white;" |1st December
Jan away
|2nd
|3rd
|4th
| style="background: grey;" |5th
| style="background: grey;" |6th

|-
|7th
|8th
|9th
|10th
|11th
| style="background: grey;" |12th
| style="background: grey;" |13th

|-
|14th
|15th
Jan away
|16th
Jan away
|17th
Jan away
| style="background: yellow;" |18th
End of term

Jan away
| style="background: grey;" |19th
| style="background: grey;" |20th

|-
|21st
Becky away

Ken away

Jan away

Anil away

Aimee away
|22nd
Tricia away

Becky away

Ken away

Jan away

Anil away

Aimee away
|23rd
Tricia away

Becky away

Ken away

Jan away

Anil Away

Richard F Away

Aimee away
| style="background: lime;" |24th
College closure

| style="background: lime;" |25th

| style="background: grey;" |26th
| style="background: grey;" |27th

|-
| style="background: lime;" |28th

| style="background: lime;" |29th

| style="background: lime;" |30th

| style="background: lime;" |31st

| style="background: blue; color: white;" |1st January

| style="background: grey;" |2nd
| style="background: grey;" |3rd

|-
|4th College opens
Tricia maybe

Becky away

Ken away

Anil away

Richard F Away

Aimee away
|5th
Tricia maybe

Becky away

Ken away

Anil Away

Richard F Away

Aimee away
|6th
Tricia maybe

Becky away

Anil Away

Richard F Away

Aimee away
|7th
Tricia maybe

Becky away

Richard F Away

Aimee away
|8th
Tricia away

Becky away

Richard F Away

Aimee away
| style="background: grey;" |9th
| style="background: grey;" |10th

|-
| style="background: yellow;" |11th
Start of term
|12th
|13th
|14th
|15th
| style="background: grey;" |16th
| style="background: grey;" |17th

|-
|18th
|19th
|20th
|21st
|22nd
| style="background: grey;" |23rd
| style="background: grey;" |24th

|-
|25th
|26th
|27th
|28th
|29th
| style="background: grey;" |30th
| style="background: grey;" |31st

|}

Rep:Mod:aed1210

2015-03-27T11:50:22Z

Aed12: /* Secondary Orbital Effects */

=Physical Module=

reaction is the point during the reaction at which the potential energy of the complex is highest. This point also corresponds to a saddle point on the potential energy surface of the reaction. Using the Gaussian software, the transition states of various reactions will be computed, optimised and certain qualities about them elucidated.
Several different transition states will be studied over the course of this experiment. Firstly, the chair and boat transition structures of the [3,3] Cope Rearrangement will be studied, along with the various conformations of reactants and the activation energy of the reaction. Then the transition states of two separate Diels-Alder reactions will be studied with the focus on the mixing of the molecular orbitals and the allowed and forbidden reactions, as well as the endo and exo conformations of the reaction between cyclohexa-1,3-diene and maleic anhydride.

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in the scheme below. The reaction mechanism has been debated and extensivley studied <ref>A. C. Cope, E. M. Hardy, J. Am. Chem. Soc, 1940, 62(2), pp 441-444.</ref>, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

<center>[[File:copeaed12.png|600px]]</center>

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the studied conformers, some have a gauche linkage for the central four atoms and the others have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

<center>[[File:Aed12_chair_ts.PNG|500px]]</center>

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below
<center>
<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

<center>[[File:Aed12_frozen_chair.PNG|400px]]</center>

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|1000px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. First described in 1928, it is very synthetically useful and so has been widely applied and studied ever since.
Here, the reaction studied in the following section is that between ethylene and cis-butadiene to form cyclohexene.

The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å.<ref>. P. Atkins, J. de Paula, Physical Chemistry, Oxford University Press, Oxford, 9th edition, 2010.</ref> The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The theory applied that allows the study of chemical reactions through analysis of their molecular orbital is Fronteir Molecular Orbital theory (FMO), we are able to use this theory to predict the outcomes of reactions, qualitatively, by considering the HOMO and LUMO orbitals of the reactants. The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

===Endo and Exo Transition States===

'''Endo'''

In order to create the endo transition state for the reaction, the reactants were optimised and placed in the endo formation before performing an optimisation and frequency calculation on the structure and the transition state saddle point is once again confirmed by the presence of a single negative frequency found.

The optimised transition state, HOMO and LUMO orbitals are all visualised below.

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_endo_oput.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_endo_lumo.png|400px]]
|
[[Image:Aed12_endo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

The same procedure was followed for the exo transition state as the endo form previously. Again, a saddle point was established and confirmed through the presence of only one negative frequency.

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_exo_opt_mol.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_exo_lumo.png|400px]]
|
[[Image:Aed12_exo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

For both structures, a comparison of the relative energies has been made. The endo conformation is of lower energy, rationalising why the endo conformation is the preferred product.

{| class="wikitable" border="1"
! Conformer !! Energy (Hartrees) !! Relative Energy (kcal/mol)
|-
| Exo|| -0.05041984 || 0.68
|-
| Endo|| -0.05150481 || 0
|}

A comparision of the bond lengths has also been made with respect to the labelled diagrams of each compound. The main difference brought to light here is the distance between the carbons where the new bonds will be formed. In the exo transition state, these are slightly longer than in the endo case. This elongation can be attributed to the greater steric repulsion in the exo transition state, making closer approach less favourable in this scenario.

<center>[[File:Aed12_endo_labelled.png|500px]]</center>

{| class="wikitable" border="2"
! Bond !! Endo Bond Lengths (Å) !! Exo Bond Lengths (Å)
|-
| C1-C2 || 1.49053 || 1.48975
|-
| C1-C4 || 1.52297 || 1.52209
|-
| C2-C9 || 1.39307 || 1.39439
|-
| C9-C11 || 1.39724 || 1.39677
|-
| C15-C16 || 1.40848 || 1.41013
|-
| C16-C17 || 1.48923 || 1.48820
|-
| C17-O23 || 1.40896 || 1.40963
|-
| C17-O21 || 1.22057 || 1.22054
|-
| C2-C16 || 2.16237 || 2.18027
|-
| C3-C15 || 2.16238 || 2.18054
|}

====Secondary Orbital Effects====

It has previously been proposed that a secondary orbital overlap effect, where by the pi system on the substituent of the dieneophile and the pi system of the diene have a favourable interaction in the endo form, give the reasons for the endo selectivity seen in the reaction. However, examination of the HOMO and LUMO orbitals only is not able to account for these effects in this case. It is a drawback of the frontier orbital model that this is based on that we cannot rationalise the selectivity seen from these visualised orbitals alone. Examination of the second highest filled orbital does show the effects we're looking for. Here, we see influence from the carbonyl pi system, that would show the favourable interation desired. As expected, this orbital interaction can only be seen in the endo form of the transition state, no such influence is seen in the exo form and thus no stabilising interaction would be possible, providing rational for the endo preference of the reaction.

==Conclusion==

The mechanisms and transition states of three reactions were studied with the Gaussian program; the cope rearrangement, and two forms of the Diels-Alder reaction. For the Cope rearrangment, several different conformations of the reactants were analysed, including one at two separate levels of theory, allowing comparison between the two. As expected, it was found that the higher level of theory B3LYP/6-31G* provdided a better optimisation of the molecule, however the HF/3-21G level does provide a fairly accurate optimsiation of the molecule and for simple molecules provides a very computationally efficient method that should not be discounted.

After this, two transition states of the cope rearrangement were studied through a variety of different methods to identify the transition state. The two forms of the TS for the reaction, the boat and chair, were optimised and their energies calculated, allowing for the activation energy of the reaction to be calculated.

The Diels-Alder reaction was studied in the second part of the experiment. Through use of Frontier Molecular Orbital theory, the HOMO and LUMO orbitals of each reaction were studied to make some qualitative deductions about the outcomes of the reactions. The downfalls of this method were also discussed when it comes to the endo preference of the reaction between maleic anhydride and cyclohexa-1,3-diene as the reaction preference cannot be rationalized through examination of these orbitals alone.

Overall, the use of the Gaussian program and generally, the effectiveness of computational methods into the study of the energy and potential energy surfaces of reactions has been demonstrated. With relative ease, we have been able to demonstrate complex physical phenomena and try to rationalise their origin.

==References==

<references/>

Rep:Mod:aed1210

2015-03-27T11:45:41Z

Aed12: /* References */

=Physical Module=

reaction is the point during the reaction at which the potential energy of the complex is highest. This point also corresponds to a saddle point on the potential energy surface of the reaction. Using the Gaussian software, the transition states of various reactions will be computed, optimised and certain qualities about them elucidated.
Several different transition states will be studied over the course of this experiment. Firstly, the chair and boat transition structures of the [3,3] Cope Rearrangement will be studied, along with the various conformations of reactants and the activation energy of the reaction. Then the transition states of two separate Diels-Alder reactions will be studied with the focus on the mixing of the molecular orbitals and the allowed and forbidden reactions, as well as the endo and exo conformations of the reaction between cyclohexa-1,3-diene and maleic anhydride.

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in the scheme below. The reaction mechanism has been debated and extensivley studied <ref>A. C. Cope, E. M. Hardy, J. Am. Chem. Soc, 1940, 62(2), pp 441-444.</ref>, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

<center>[[File:copeaed12.png|600px]]</center>

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the studied conformers, some have a gauche linkage for the central four atoms and the others have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

<center>[[File:Aed12_chair_ts.PNG|500px]]</center>

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below
<center>
<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

<center>[[File:Aed12_frozen_chair.PNG|400px]]</center>

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|1000px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. First described in 1928, it is very synthetically useful and so has been widely applied and studied ever since.
Here, the reaction studied in the following section is that between ethylene and cis-butadiene to form cyclohexene.

The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å.<ref>. P. Atkins, J. de Paula, Physical Chemistry, Oxford University Press, Oxford, 9th edition, 2010.</ref> The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The theory applied that allows the study of chemical reactions through analysis of their molecular orbital is Fronteir Molecular Orbital theory (FMO), we are able to use this theory to predict the outcomes of reactions, qualitatively, by considering the HOMO and LUMO orbitals of the reactants. The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

===Endo and Exo Transition States===

'''Endo'''

In order to create the endo transition state for the reaction, the reactants were optimised and placed in the endo formation before performing an optimisation and frequency calculation on the structure and the transition state saddle point is once again confirmed by the presence of a single negative frequency found.

The optimised transition state, HOMO and LUMO orbitals are all visualised below.

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_endo_oput.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_endo_lumo.png|400px]]
|
[[Image:Aed12_endo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

The same procedure was followed for the exo transition state as the endo form previously. Again, a saddle point was established and confirmed through the presence of only one negative frequency.

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_exo_opt_mol.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_exo_lumo.png|400px]]
|
[[Image:Aed12_exo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

For both structures, a comparison of the relative energies has been made. The endo conformation is of lower energy, rationalising why the endo conformation is the preferred product.

{| class="wikitable" border="1"
! Conformer !! Energy (Hartrees) !! Relative Energy (kcal/mol)
|-
| Exo|| -0.05041984 || 0.68
|-
| Endo|| -0.05150481 || 0
|}

A comparision of the bond lengths has also been made with respect to the labelled diagrams of each compound. The main difference brought to light here is the distance between the carbons where the new bonds will be formed. In the exo transition state, these are slightly longer than in the endo case. This elongation can be attributed to the greater steric repulsion in the exo transition state, making closer approach less favourable in this scenario.

<center>[[File:Aed12_endo_labelled.png|500px]]</center>

{| class="wikitable" border="2"
! Bond !! Endo Bond Lengths (Å) !! Exo Bond Lengths (Å)
|-
| C1-C2 || 1.49053 || 1.48975
|-
| C1-C4 || 1.52297 || 1.52209
|-
| C2-C9 || 1.39307 || 1.39439
|-
| C9-C11 || 1.39724 || 1.39677
|-
| C15-C16 || 1.40848 || 1.41013
|-
| C16-C17 || 1.48923 || 1.48820
|-
| C17-O23 || 1.40896 || 1.40963
|-
| C17-O21 || 1.22057 || 1.22054
|-
| C2-C16 || 2.16237 || 2.18027
|-
| C3-C15 || 2.16238 || 2.18054
|}

====Secondary Orbital Effects====

It has previously been proposed that a secondary orbital overlap effect, where by the pi system on the substituent of the dieneophile and the pi system of the diene have a favourable interaction in the endo form, give the reasons for the endo selectivity seen in the reaction. However, examination of the HOMO and LUMO orbitals only is not able to account for these effects in this case. It is a drawback of the frontier orbital model that this is based on that we cannot rationalise the selectivity seen from these visualised orbitals alone.

==Conclusion==

The mechanisms and transition states of three reactions were studied with the Gaussian program; the cope rearrangement, and two forms of the Diels-Alder reaction. For the Cope rearrangment, several different conformations of the reactants were analysed, including one at two separate levels of theory, allowing comparison between the two. As expected, it was found that the higher level of theory B3LYP/6-31G* provdided a better optimisation of the molecule, however the HF/3-21G level does provide a fairly accurate optimsiation of the molecule and for simple molecules provides a very computationally efficient method that should not be discounted.

After this, two transition states of the cope rearrangement were studied through a variety of different methods to identify the transition state. The two forms of the TS for the reaction, the boat and chair, were optimised and their energies calculated, allowing for the activation energy of the reaction to be calculated.

The Diels-Alder reaction was studied in the second part of the experiment. Through use of Frontier Molecular Orbital theory, the HOMO and LUMO orbitals of each reaction were studied to make some qualitative deductions about the outcomes of the reactions. The downfalls of this method were also discussed when it comes to the endo preference of the reaction between maleic anhydride and cyclohexa-1,3-diene as the reaction preference cannot be rationalized through examination of these orbitals alone.

Overall, the use of the Gaussian program and generally, the effectiveness of computational methods into the study of the energy and potential energy surfaces of reactions has been demonstrated. With relative ease, we have been able to demonstrate complex physical phenomena and try to rationalise their origin.

==References==

<references/>

Rep:Mod:aed1210

2015-03-27T11:45:02Z

Aed12: /* Optimizing the Reactants and Products */

=Physical Module=

reaction is the point during the reaction at which the potential energy of the complex is highest. This point also corresponds to a saddle point on the potential energy surface of the reaction. Using the Gaussian software, the transition states of various reactions will be computed, optimised and certain qualities about them elucidated.
Several different transition states will be studied over the course of this experiment. Firstly, the chair and boat transition structures of the [3,3] Cope Rearrangement will be studied, along with the various conformations of reactants and the activation energy of the reaction. Then the transition states of two separate Diels-Alder reactions will be studied with the focus on the mixing of the molecular orbitals and the allowed and forbidden reactions, as well as the endo and exo conformations of the reaction between cyclohexa-1,3-diene and maleic anhydride.

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in the scheme below. The reaction mechanism has been debated and extensivley studied <ref>A. C. Cope, E. M. Hardy, J. Am. Chem. Soc, 1940, 62(2), pp 441-444.</ref>, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

<center>[[File:copeaed12.png|600px]]</center>

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the studied conformers, some have a gauche linkage for the central four atoms and the others have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

<center>[[File:Aed12_chair_ts.PNG|500px]]</center>

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below
<center>
<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

<center>[[File:Aed12_frozen_chair.PNG|400px]]</center>

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|1000px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. First described in 1928, it is very synthetically useful and so has been widely applied and studied ever since.
Here, the reaction studied in the following section is that between ethylene and cis-butadiene to form cyclohexene.

The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å.<ref>. P. Atkins, J. de Paula, Physical Chemistry, Oxford University Press, Oxford, 9th edition, 2010.</ref> The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The theory applied that allows the study of chemical reactions through analysis of their molecular orbital is Fronteir Molecular Orbital theory (FMO), we are able to use this theory to predict the outcomes of reactions, qualitatively, by considering the HOMO and LUMO orbitals of the reactants. The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

===Endo and Exo Transition States===

'''Endo'''

In order to create the endo transition state for the reaction, the reactants were optimised and placed in the endo formation before performing an optimisation and frequency calculation on the structure and the transition state saddle point is once again confirmed by the presence of a single negative frequency found.

The optimised transition state, HOMO and LUMO orbitals are all visualised below.

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_endo_oput.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_endo_lumo.png|400px]]
|
[[Image:Aed12_endo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

The same procedure was followed for the exo transition state as the endo form previously. Again, a saddle point was established and confirmed through the presence of only one negative frequency.

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_exo_opt_mol.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_exo_lumo.png|400px]]
|
[[Image:Aed12_exo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

For both structures, a comparison of the relative energies has been made. The endo conformation is of lower energy, rationalising why the endo conformation is the preferred product.

{| class="wikitable" border="1"
! Conformer !! Energy (Hartrees) !! Relative Energy (kcal/mol)
|-
| Exo|| -0.05041984 || 0.68
|-
| Endo|| -0.05150481 || 0
|}

A comparision of the bond lengths has also been made with respect to the labelled diagrams of each compound. The main difference brought to light here is the distance between the carbons where the new bonds will be formed. In the exo transition state, these are slightly longer than in the endo case. This elongation can be attributed to the greater steric repulsion in the exo transition state, making closer approach less favourable in this scenario.

<center>[[File:Aed12_endo_labelled.png|500px]]</center>

{| class="wikitable" border="2"
! Bond !! Endo Bond Lengths (Å) !! Exo Bond Lengths (Å)
|-
| C1-C2 || 1.49053 || 1.48975
|-
| C1-C4 || 1.52297 || 1.52209
|-
| C2-C9 || 1.39307 || 1.39439
|-
| C9-C11 || 1.39724 || 1.39677
|-
| C15-C16 || 1.40848 || 1.41013
|-
| C16-C17 || 1.48923 || 1.48820
|-
| C17-O23 || 1.40896 || 1.40963
|-
| C17-O21 || 1.22057 || 1.22054
|-
| C2-C16 || 2.16237 || 2.18027
|-
| C3-C15 || 2.16238 || 2.18054
|}

====Secondary Orbital Effects====

It has previously been proposed that a secondary orbital overlap effect, where by the pi system on the substituent of the dieneophile and the pi system of the diene have a favourable interaction in the endo form, give the reasons for the endo selectivity seen in the reaction. However, examination of the HOMO and LUMO orbitals only is not able to account for these effects in this case. It is a drawback of the frontier orbital model that this is based on that we cannot rationalise the selectivity seen from these visualised orbitals alone.

==Conclusion==

The mechanisms and transition states of three reactions were studied with the Gaussian program; the cope rearrangement, and two forms of the Diels-Alder reaction. For the Cope rearrangment, several different conformations of the reactants were analysed, including one at two separate levels of theory, allowing comparison between the two. As expected, it was found that the higher level of theory B3LYP/6-31G* provdided a better optimisation of the molecule, however the HF/3-21G level does provide a fairly accurate optimsiation of the molecule and for simple molecules provides a very computationally efficient method that should not be discounted.

After this, two transition states of the cope rearrangement were studied through a variety of different methods to identify the transition state. The two forms of the TS for the reaction, the boat and chair, were optimised and their energies calculated, allowing for the activation energy of the reaction to be calculated.

The Diels-Alder reaction was studied in the second part of the experiment. Through use of Frontier Molecular Orbital theory, the HOMO and LUMO orbitals of each reaction were studied to make some qualitative deductions about the outcomes of the reactions. The downfalls of this method were also discussed when it comes to the endo preference of the reaction between maleic anhydride and cyclohexa-1,3-diene as the reaction preference cannot be rationalized through examination of these orbitals alone.

Overall, the use of the Gaussian program and generally, the effectiveness of computational methods into the study of the energy and potential energy surfaces of reactions has been demonstrated. With relative ease, we have been able to demonstrate complex physical phenomena and try to rationalise their origin.

==References==

</references>

Rep:Mod:aed1210

2015-03-27T11:43:17Z

Aed12: /* Ethylene and cis-Butadiene */

=Physical Module=

reaction is the point during the reaction at which the potential energy of the complex is highest. This point also corresponds to a saddle point on the potential energy surface of the reaction. Using the Gaussian software, the transition states of various reactions will be computed, optimised and certain qualities about them elucidated.
Several different transition states will be studied over the course of this experiment. Firstly, the chair and boat transition structures of the [3,3] Cope Rearrangement will be studied, along with the various conformations of reactants and the activation energy of the reaction. Then the transition states of two separate Diels-Alder reactions will be studied with the focus on the mixing of the molecular orbitals and the allowed and forbidden reactions, as well as the endo and exo conformations of the reaction between cyclohexa-1,3-diene and maleic anhydride.

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in the scheme below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

<center>[[File:copeaed12.png|600px]]</center>

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the studied conformers, some have a gauche linkage for the central four atoms and the others have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

<center>[[File:Aed12_chair_ts.PNG|500px]]</center>

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below
<center>
<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

<center>[[File:Aed12_frozen_chair.PNG|400px]]</center>

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|1000px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. First described in 1928, it is very synthetically useful and so has been widely applied and studied ever since.
Here, the reaction studied in the following section is that between ethylene and cis-butadiene to form cyclohexene.

The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å.<ref>. P. Atkins, J. de Paula, Physical Chemistry, Oxford University Press, Oxford, 9th edition, 2010.</ref> The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The theory applied that allows the study of chemical reactions through analysis of their molecular orbital is Fronteir Molecular Orbital theory (FMO), we are able to use this theory to predict the outcomes of reactions, qualitatively, by considering the HOMO and LUMO orbitals of the reactants. The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

===Endo and Exo Transition States===

'''Endo'''

In order to create the endo transition state for the reaction, the reactants were optimised and placed in the endo formation before performing an optimisation and frequency calculation on the structure and the transition state saddle point is once again confirmed by the presence of a single negative frequency found.

The optimised transition state, HOMO and LUMO orbitals are all visualised below.

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_endo_oput.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_endo_lumo.png|400px]]
|
[[Image:Aed12_endo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

The same procedure was followed for the exo transition state as the endo form previously. Again, a saddle point was established and confirmed through the presence of only one negative frequency.

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_exo_opt_mol.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_exo_lumo.png|400px]]
|
[[Image:Aed12_exo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

For both structures, a comparison of the relative energies has been made. The endo conformation is of lower energy, rationalising why the endo conformation is the preferred product.

{| class="wikitable" border="1"
! Conformer !! Energy (Hartrees) !! Relative Energy (kcal/mol)
|-
| Exo|| -0.05041984 || 0.68
|-
| Endo|| -0.05150481 || 0
|}

A comparision of the bond lengths has also been made with respect to the labelled diagrams of each compound. The main difference brought to light here is the distance between the carbons where the new bonds will be formed. In the exo transition state, these are slightly longer than in the endo case. This elongation can be attributed to the greater steric repulsion in the exo transition state, making closer approach less favourable in this scenario.

<center>[[File:Aed12_endo_labelled.png|500px]]</center>

{| class="wikitable" border="2"
! Bond !! Endo Bond Lengths (Å) !! Exo Bond Lengths (Å)
|-
| C1-C2 || 1.49053 || 1.48975
|-
| C1-C4 || 1.52297 || 1.52209
|-
| C2-C9 || 1.39307 || 1.39439
|-
| C9-C11 || 1.39724 || 1.39677
|-
| C15-C16 || 1.40848 || 1.41013
|-
| C16-C17 || 1.48923 || 1.48820
|-
| C17-O23 || 1.40896 || 1.40963
|-
| C17-O21 || 1.22057 || 1.22054
|-
| C2-C16 || 2.16237 || 2.18027
|-
| C3-C15 || 2.16238 || 2.18054
|}

====Secondary Orbital Effects====

It has previously been proposed that a secondary orbital overlap effect, where by the pi system on the substituent of the dieneophile and the pi system of the diene have a favourable interaction in the endo form, give the reasons for the endo selectivity seen in the reaction. However, examination of the HOMO and LUMO orbitals only is not able to account for these effects in this case. It is a drawback of the frontier orbital model that this is based on that we cannot rationalise the selectivity seen from these visualised orbitals alone.

==Conclusion==

The mechanisms and transition states of three reactions were studied with the Gaussian program; the cope rearrangement, and two forms of the Diels-Alder reaction. For the Cope rearrangment, several different conformations of the reactants were analysed, including one at two separate levels of theory, allowing comparison between the two. As expected, it was found that the higher level of theory B3LYP/6-31G* provdided a better optimisation of the molecule, however the HF/3-21G level does provide a fairly accurate optimsiation of the molecule and for simple molecules provides a very computationally efficient method that should not be discounted.

After this, two transition states of the cope rearrangement were studied through a variety of different methods to identify the transition state. The two forms of the TS for the reaction, the boat and chair, were optimised and their energies calculated, allowing for the activation energy of the reaction to be calculated.

The Diels-Alder reaction was studied in the second part of the experiment. Through use of Frontier Molecular Orbital theory, the HOMO and LUMO orbitals of each reaction were studied to make some qualitative deductions about the outcomes of the reactions. The downfalls of this method were also discussed when it comes to the endo preference of the reaction between maleic anhydride and cyclohexa-1,3-diene as the reaction preference cannot be rationalized through examination of these orbitals alone.

Overall, the use of the Gaussian program and generally, the effectiveness of computational methods into the study of the energy and potential energy surfaces of reactions has been demonstrated. With relative ease, we have been able to demonstrate complex physical phenomena and try to rationalise their origin.

==References==

</references>

Rep:Mod:aed1210

2015-03-27T11:41:08Z

Aed12: /* Conclusion */

=Physical Module=

reaction is the point during the reaction at which the potential energy of the complex is highest. This point also corresponds to a saddle point on the potential energy surface of the reaction. Using the Gaussian software, the transition states of various reactions will be computed, optimised and certain qualities about them elucidated.
Several different transition states will be studied over the course of this experiment. Firstly, the chair and boat transition structures of the [3,3] Cope Rearrangement will be studied, along with the various conformations of reactants and the activation energy of the reaction. Then the transition states of two separate Diels-Alder reactions will be studied with the focus on the mixing of the molecular orbitals and the allowed and forbidden reactions, as well as the endo and exo conformations of the reaction between cyclohexa-1,3-diene and maleic anhydride.

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in the scheme below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

<center>[[File:copeaed12.png|600px]]</center>

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the studied conformers, some have a gauche linkage for the central four atoms and the others have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

<center>[[File:Aed12_chair_ts.PNG|500px]]</center>

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below
<center>
<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

<center>[[File:Aed12_frozen_chair.PNG|400px]]</center>

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|1000px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. First described in 1928, it is very synthetically useful and so has been widely applied and studied ever since.
Here, the reaction studied in the following section is that between ethylene and cis-butadiene to form cyclohexene.

The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The theory applied that allows the study of chemical reactions through analysis of their molecular orbital is Fronteir Molecular Orbital theory (FMO), we are able to use this theory to predict the outcomes of reactions, qualitatively, by considering the HOMO and LUMO orbitals of the reactants. The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

===Endo and Exo Transition States===

'''Endo'''

In order to create the endo transition state for the reaction, the reactants were optimised and placed in the endo formation before performing an optimisation and frequency calculation on the structure and the transition state saddle point is once again confirmed by the presence of a single negative frequency found.

The optimised transition state, HOMO and LUMO orbitals are all visualised below.

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_endo_oput.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_endo_lumo.png|400px]]
|
[[Image:Aed12_endo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

The same procedure was followed for the exo transition state as the endo form previously. Again, a saddle point was established and confirmed through the presence of only one negative frequency.

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_exo_opt_mol.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_exo_lumo.png|400px]]
|
[[Image:Aed12_exo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

For both structures, a comparison of the relative energies has been made. The endo conformation is of lower energy, rationalising why the endo conformation is the preferred product.

{| class="wikitable" border="1"
! Conformer !! Energy (Hartrees) !! Relative Energy (kcal/mol)
|-
| Exo|| -0.05041984 || 0.68
|-
| Endo|| -0.05150481 || 0
|}

A comparision of the bond lengths has also been made with respect to the labelled diagrams of each compound. The main difference brought to light here is the distance between the carbons where the new bonds will be formed. In the exo transition state, these are slightly longer than in the endo case. This elongation can be attributed to the greater steric repulsion in the exo transition state, making closer approach less favourable in this scenario.

<center>[[File:Aed12_endo_labelled.png|500px]]</center>

{| class="wikitable" border="2"
! Bond !! Endo Bond Lengths (Å) !! Exo Bond Lengths (Å)
|-
| C1-C2 || 1.49053 || 1.48975
|-
| C1-C4 || 1.52297 || 1.52209
|-
| C2-C9 || 1.39307 || 1.39439
|-
| C9-C11 || 1.39724 || 1.39677
|-
| C15-C16 || 1.40848 || 1.41013
|-
| C16-C17 || 1.48923 || 1.48820
|-
| C17-O23 || 1.40896 || 1.40963
|-
| C17-O21 || 1.22057 || 1.22054
|-
| C2-C16 || 2.16237 || 2.18027
|-
| C3-C15 || 2.16238 || 2.18054
|}

====Secondary Orbital Effects====

It has previously been proposed that a secondary orbital overlap effect, where by the pi system on the substituent of the dieneophile and the pi system of the diene have a favourable interaction in the endo form, give the reasons for the endo selectivity seen in the reaction. However, examination of the HOMO and LUMO orbitals only is not able to account for these effects in this case. It is a drawback of the frontier orbital model that this is based on that we cannot rationalise the selectivity seen from these visualised orbitals alone.

==Conclusion==

The mechanisms and transition states of three reactions were studied with the Gaussian program; the cope rearrangement, and two forms of the Diels-Alder reaction. For the Cope rearrangment, several different conformations of the reactants were analysed, including one at two separate levels of theory, allowing comparison between the two. As expected, it was found that the higher level of theory B3LYP/6-31G* provdided a better optimisation of the molecule, however the HF/3-21G level does provide a fairly accurate optimsiation of the molecule and for simple molecules provides a very computationally efficient method that should not be discounted.

After this, two transition states of the cope rearrangement were studied through a variety of different methods to identify the transition state. The two forms of the TS for the reaction, the boat and chair, were optimised and their energies calculated, allowing for the activation energy of the reaction to be calculated.

The Diels-Alder reaction was studied in the second part of the experiment. Through use of Frontier Molecular Orbital theory, the HOMO and LUMO orbitals of each reaction were studied to make some qualitative deductions about the outcomes of the reactions. The downfalls of this method were also discussed when it comes to the endo preference of the reaction between maleic anhydride and cyclohexa-1,3-diene as the reaction preference cannot be rationalized through examination of these orbitals alone.

Overall, the use of the Gaussian program and generally, the effectiveness of computational methods into the study of the energy and potential energy surfaces of reactions has been demonstrated. With relative ease, we have been able to demonstrate complex physical phenomena and try to rationalise their origin.

==References==

</references>

Rep:Mod:aed1210

2015-03-27T11:38:23Z

Aed12: /* Conclusion */

=Physical Module=

reaction is the point during the reaction at which the potential energy of the complex is highest. This point also corresponds to a saddle point on the potential energy surface of the reaction. Using the Gaussian software, the transition states of various reactions will be computed, optimised and certain qualities about them elucidated.
Several different transition states will be studied over the course of this experiment. Firstly, the chair and boat transition structures of the [3,3] Cope Rearrangement will be studied, along with the various conformations of reactants and the activation energy of the reaction. Then the transition states of two separate Diels-Alder reactions will be studied with the focus on the mixing of the molecular orbitals and the allowed and forbidden reactions, as well as the endo and exo conformations of the reaction between cyclohexa-1,3-diene and maleic anhydride.

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in the scheme below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

<center>[[File:copeaed12.png|600px]]</center>

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the studied conformers, some have a gauche linkage for the central four atoms and the others have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

<center>[[File:Aed12_chair_ts.PNG|500px]]</center>

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below
<center>
<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

<center>[[File:Aed12_frozen_chair.PNG|400px]]</center>

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|1000px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. First described in 1928, it is very synthetically useful and so has been widely applied and studied ever since.
Here, the reaction studied in the following section is that between ethylene and cis-butadiene to form cyclohexene.

The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The theory applied that allows the study of chemical reactions through analysis of their molecular orbital is Fronteir Molecular Orbital theory (FMO), we are able to use this theory to predict the outcomes of reactions, qualitatively, by considering the HOMO and LUMO orbitals of the reactants. The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

===Endo and Exo Transition States===

'''Endo'''

In order to create the endo transition state for the reaction, the reactants were optimised and placed in the endo formation before performing an optimisation and frequency calculation on the structure and the transition state saddle point is once again confirmed by the presence of a single negative frequency found.

The optimised transition state, HOMO and LUMO orbitals are all visualised below.

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_endo_oput.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_endo_lumo.png|400px]]
|
[[Image:Aed12_endo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

The same procedure was followed for the exo transition state as the endo form previously. Again, a saddle point was established and confirmed through the presence of only one negative frequency.

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_exo_opt_mol.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_exo_lumo.png|400px]]
|
[[Image:Aed12_exo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

For both structures, a comparison of the relative energies has been made. The endo conformation is of lower energy, rationalising why the endo conformation is the preferred product.

{| class="wikitable" border="1"
! Conformer !! Energy (Hartrees) !! Relative Energy (kcal/mol)
|-
| Exo|| -0.05041984 || 0.68
|-
| Endo|| -0.05150481 || 0
|}

A comparision of the bond lengths has also been made with respect to the labelled diagrams of each compound. The main difference brought to light here is the distance between the carbons where the new bonds will be formed. In the exo transition state, these are slightly longer than in the endo case. This elongation can be attributed to the greater steric repulsion in the exo transition state, making closer approach less favourable in this scenario.

<center>[[File:Aed12_endo_labelled.png|500px]]</center>

{| class="wikitable" border="2"
! Bond !! Endo Bond Lengths (Å) !! Exo Bond Lengths (Å)
|-
| C1-C2 || 1.49053 || 1.48975
|-
| C1-C4 || 1.52297 || 1.52209
|-
| C2-C9 || 1.39307 || 1.39439
|-
| C9-C11 || 1.39724 || 1.39677
|-
| C15-C16 || 1.40848 || 1.41013
|-
| C16-C17 || 1.48923 || 1.48820
|-
| C17-O23 || 1.40896 || 1.40963
|-
| C17-O21 || 1.22057 || 1.22054
|-
| C2-C16 || 2.16237 || 2.18027
|-
| C3-C15 || 2.16238 || 2.18054
|}

====Secondary Orbital Effects====

It has previously been proposed that a secondary orbital overlap effect, where by the pi system on the substituent of the dieneophile and the pi system of the diene have a favourable interaction in the endo form, give the reasons for the endo selectivity seen in the reaction. However, examination of the HOMO and LUMO orbitals only is not able to account for these effects in this case. It is a drawback of the frontier orbital model that this is based on that we cannot rationalise the selectivity seen from these visualised orbitals alone.

==Conclusion==

The mechanisms and transition states of three reactions were studied with the Gaussian program; the cope rearrangement, and two forms of the Diels-Alder reaction. For the Cope rearrangment, several different conformations of the reactants were analysed, including one at two separate levels of theory, allowing comparison between the two. As expected, it was found that the higher level of theory B3LYP/6-31G* provdided a better optimisation of the molecule, however the HF/3-21G level does provide a fairly accurate optimsiation of the molecule and for simple molecules provides a very computationally efficient method that should not be discounted.

After this, two transition states of the cope rearrangement were studied through a variety of different methods to identify the transition state. The two forms of the TS for the reaction, the boat and chair, were optimised and their energies calculated, allowing for the activation energy of the reaction to be calculated.

The Diels-Alder reaction was studied in the second part of the experiment. Through use of Frontier Molecular Orbital theory, the HOMO and LUMO orbitals of each reaction were studied to make some qualitative deductions about the outcomes of the reactions. The downfalls of this method were also discussed when it comes to the endo preference of the reaction between maleic anhydride and cyclohexa-1,3-diene as the reaction preference cannot be rationalized through examination of these orbitals alone.

Overall, the use of the Gaussian program and generally, the effectiveness of computational methods into the study of the energy and potential energy surfaces of reactions has been demonstrated. With relative ease, we have been able to demonstrate complex physical phenomena and try to rationalise their origin.

Rep:Mod:aed1210

2015-03-27T11:22:16Z

Aed12: /* Diels Alder Cycloaddtions */

=Physical Module=

reaction is the point during the reaction at which the potential energy of the complex is highest. This point also corresponds to a saddle point on the potential energy surface of the reaction. Using the Gaussian software, the transition states of various reactions will be computed, optimised and certain qualities about them elucidated.
Several different transition states will be studied over the course of this experiment. Firstly, the chair and boat transition structures of the [3,3] Cope Rearrangement will be studied, along with the various conformations of reactants and the activation energy of the reaction. Then the transition states of two separate Diels-Alder reactions will be studied with the focus on the mixing of the molecular orbitals and the allowed and forbidden reactions, as well as the endo and exo conformations of the reaction between cyclohexa-1,3-diene and maleic anhydride.

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in the scheme below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

<center>[[File:copeaed12.png|600px]]</center>

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the studied conformers, some have a gauche linkage for the central four atoms and the others have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

<center>[[File:Aed12_chair_ts.PNG|500px]]</center>

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below
<center>
<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

<center>[[File:Aed12_frozen_chair.PNG|400px]]</center>

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|1000px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. First described in 1928, it is very synthetically useful and so has been widely applied and studied ever since.
Here, the reaction studied in the following section is that between ethylene and cis-butadiene to form cyclohexene.

The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The theory applied that allows the study of chemical reactions through analysis of their molecular orbital is Fronteir Molecular Orbital theory (FMO), we are able to use this theory to predict the outcomes of reactions, qualitatively, by considering the HOMO and LUMO orbitals of the reactants. The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

===Endo and Exo Transition States===

'''Endo'''

In order to create the endo transition state for the reaction, the reactants were optimised and placed in the endo formation before performing an optimisation and frequency calculation on the structure and the transition state saddle point is once again confirmed by the presence of a single negative frequency found.

The optimised transition state, HOMO and LUMO orbitals are all visualised below.

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_endo_oput.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_endo_lumo.png|400px]]
|
[[Image:Aed12_endo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

The same procedure was followed for the exo transition state as the endo form previously. Again, a saddle point was established and confirmed through the presence of only one negative frequency.

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_exo_opt_mol.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_exo_lumo.png|400px]]
|
[[Image:Aed12_exo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

For both structures, a comparison of the relative energies has been made. The endo conformation is of lower energy, rationalising why the endo conformation is the preferred product.

{| class="wikitable" border="1"
! Conformer !! Energy (Hartrees) !! Relative Energy (kcal/mol)
|-
| Exo|| -0.05041984 || 0.68
|-
| Endo|| -0.05150481 || 0
|}

A comparision of the bond lengths has also been made with respect to the labelled diagrams of each compound. The main difference brought to light here is the distance between the carbons where the new bonds will be formed. In the exo transition state, these are slightly longer than in the endo case. This elongation can be attributed to the greater steric repulsion in the exo transition state, making closer approach less favourable in this scenario.

<center>[[File:Aed12_endo_labelled.png|500px]]</center>

{| class="wikitable" border="2"
! Bond !! Endo Bond Lengths (Å) !! Exo Bond Lengths (Å)
|-
| C1-C2 || 1.49053 || 1.48975
|-
| C1-C4 || 1.52297 || 1.52209
|-
| C2-C9 || 1.39307 || 1.39439
|-
| C9-C11 || 1.39724 || 1.39677
|-
| C15-C16 || 1.40848 || 1.41013
|-
| C16-C17 || 1.48923 || 1.48820
|-
| C17-O23 || 1.40896 || 1.40963
|-
| C17-O21 || 1.22057 || 1.22054
|-
| C2-C16 || 2.16237 || 2.18027
|-
| C3-C15 || 2.16238 || 2.18054
|}

====Secondary Orbital Effects====

It has previously been proposed that a secondary orbital overlap effect, where by the pi system on the substituent of the dieneophile and the pi system of the diene have a favourable interaction in the endo form, give the reasons for the endo selectivity seen in the reaction. However, examination of the HOMO and LUMO orbitals only is not able to account for these effects in this case. It is a drawback of the frontier orbital model that this is based on that we cannot rationalise the selectivity seen from these visualised orbitals alone.

==Conclusion==

Rep:Mod:aed1210

2015-03-27T11:10:37Z

Aed12: /* Optimizing the "Chair" and "Boat" Transition Structures */

=Physical Module=

reaction is the point during the reaction at which the potential energy of the complex is highest. This point also corresponds to a saddle point on the potential energy surface of the reaction. Using the Gaussian software, the transition states of various reactions will be computed, optimised and certain qualities about them elucidated.
Several different transition states will be studied over the course of this experiment. Firstly, the chair and boat transition structures of the [3,3] Cope Rearrangement will be studied, along with the various conformations of reactants and the activation energy of the reaction. Then the transition states of two separate Diels-Alder reactions will be studied with the focus on the mixing of the molecular orbitals and the allowed and forbidden reactions, as well as the endo and exo conformations of the reaction between cyclohexa-1,3-diene and maleic anhydride.

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in the scheme below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

<center>[[File:copeaed12.png|600px]]</center>

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the studied conformers, some have a gauche linkage for the central four atoms and the others have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

<center>[[File:Aed12_chair_ts.PNG|500px]]</center>

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below
<center>
<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

<center>[[File:Aed12_frozen_chair.PNG|400px]]</center>

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|1000px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

===Endo and Exo Transition States===

'''Endo'''

In order to create the endo transition state for the reaction, the reactants were optimised and placed in the endo formation before performing an optimisation and frequency calculation on the structure and the transition state saddle point is once again confirmed by the presence of a single negative frequency found.

The optimised transition state, HOMO and LUMO orbitals are all visualised below.

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_endo_oput.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_endo_lumo.png|400px]]
|
[[Image:Aed12_endo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

The same procedure was followed for the exo transition state as the endo form previously. Again, a saddle point was established and confirmed through the presence of only one negative frequency.

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_exo_opt_mol.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_exo_lumo.png|400px]]
|
[[Image:Aed12_exo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

For both structures, a comparison of the relative energies has been made. The endo conformation is of lower energy, rationalising why the endo conformation is the preferred product.

{| class="wikitable" border="1"
! Conformer !! Energy (Hartrees) !! Relative Energy (kcal/mol)
|-
| Exo|| -0.05041984 || 0.68
|-
| Endo|| -0.05150481 || 0
|}

A comparision of the bond lengths has also been made with respect to the labelled diagrams of each compound. The main difference brought to light here is the distance between the carbons where the new bonds will be formed. In the exo transition state, these are slightly longer than in the endo case. This elongation can be attributed to the greater steric repulsion in the exo transition state, making closer approach less favourable in this scenario.

<center>[[File:Aed12_endo_labelled.png|500px]]</center>

{| class="wikitable" border="2"
! Bond !! Endo Bond Lengths (Å) !! Exo Bond Lengths (Å)
|-
| C1-C2 || 1.49053 || 1.48975
|-
| C1-C4 || 1.52297 || 1.52209
|-
| C2-C9 || 1.39307 || 1.39439
|-
| C9-C11 || 1.39724 || 1.39677
|-
| C15-C16 || 1.40848 || 1.41013
|-
| C16-C17 || 1.48923 || 1.48820
|-
| C17-O23 || 1.40896 || 1.40963
|-
| C17-O21 || 1.22057 || 1.22054
|-
| C2-C16 || 2.16237 || 2.18027
|-
| C3-C15 || 2.16238 || 2.18054
|}

====Secondary Orbital Effects====

It has previously been proposed that a secondary orbital overlap effect, where by the pi system on the substituent of the dieneophile and the pi system of the diene have a favourable interaction in the endo form, give the reasons for the endo selectivity seen in the reaction. However, examination of the HOMO and LUMO orbitals only is not able to account for these effects in this case. It is a drawback of the frontier orbital model that this is based on that we cannot rationalise the selectivity seen from these visualised orbitals alone.

==Conclusion==

Rep:Mod:aed1210

2015-03-27T11:09:27Z

Aed12: /* Physical Module */

=Physical Module=

reaction is the point during the reaction at which the potential energy of the complex is highest. This point also corresponds to a saddle point on the potential energy surface of the reaction. Using the Gaussian software, the transition states of various reactions will be computed, optimised and certain qualities about them elucidated.
Several different transition states will be studied over the course of this experiment. Firstly, the chair and boat transition structures of the [3,3] Cope Rearrangement will be studied, along with the various conformations of reactants and the activation energy of the reaction. Then the transition states of two separate Diels-Alder reactions will be studied with the focus on the mixing of the molecular orbitals and the allowed and forbidden reactions, as well as the endo and exo conformations of the reaction between cyclohexa-1,3-diene and maleic anhydride.

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in the scheme below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

<center>[[File:copeaed12.png|600px]]</center>

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the studied conformers, some have a gauche linkage for the central four atoms and the others have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|1000px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

===Endo and Exo Transition States===

'''Endo'''

In order to create the endo transition state for the reaction, the reactants were optimised and placed in the endo formation before performing an optimisation and frequency calculation on the structure and the transition state saddle point is once again confirmed by the presence of a single negative frequency found.

The optimised transition state, HOMO and LUMO orbitals are all visualised below.

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_endo_oput.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_endo_lumo.png|400px]]
|
[[Image:Aed12_endo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

The same procedure was followed for the exo transition state as the endo form previously. Again, a saddle point was established and confirmed through the presence of only one negative frequency.

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_exo_opt_mol.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_exo_lumo.png|400px]]
|
[[Image:Aed12_exo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

For both structures, a comparison of the relative energies has been made. The endo conformation is of lower energy, rationalising why the endo conformation is the preferred product.

{| class="wikitable" border="1"
! Conformer !! Energy (Hartrees) !! Relative Energy (kcal/mol)
|-
| Exo|| -0.05041984 || 0.68
|-
| Endo|| -0.05150481 || 0
|}

A comparision of the bond lengths has also been made with respect to the labelled diagrams of each compound. The main difference brought to light here is the distance between the carbons where the new bonds will be formed. In the exo transition state, these are slightly longer than in the endo case. This elongation can be attributed to the greater steric repulsion in the exo transition state, making closer approach less favourable in this scenario.

<center>[[File:Aed12_endo_labelled.png|500px]]</center>

{| class="wikitable" border="2"
! Bond !! Endo Bond Lengths (Å) !! Exo Bond Lengths (Å)
|-
| C1-C2 || 1.49053 || 1.48975
|-
| C1-C4 || 1.52297 || 1.52209
|-
| C2-C9 || 1.39307 || 1.39439
|-
| C9-C11 || 1.39724 || 1.39677
|-
| C15-C16 || 1.40848 || 1.41013
|-
| C16-C17 || 1.48923 || 1.48820
|-
| C17-O23 || 1.40896 || 1.40963
|-
| C17-O21 || 1.22057 || 1.22054
|-
| C2-C16 || 2.16237 || 2.18027
|-
| C3-C15 || 2.16238 || 2.18054
|}

====Secondary Orbital Effects====

It has previously been proposed that a secondary orbital overlap effect, where by the pi system on the substituent of the dieneophile and the pi system of the diene have a favourable interaction in the endo form, give the reasons for the endo selectivity seen in the reaction. However, examination of the HOMO and LUMO orbitals only is not able to account for these effects in this case. It is a drawback of the frontier orbital model that this is based on that we cannot rationalise the selectivity seen from these visualised orbitals alone.

==Conclusion==

File:Copeaed12.png

2015-03-27T11:09:22Z

Aed12:

Rep:Mod:aed1210

2015-03-26T23:52:31Z

Aed12: /* Endo and Exo Transition States */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|1000px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

===Endo and Exo Transition States===

'''Endo'''

In order to create the endo transition state for the reaction, the reactants were optimised and placed in the endo formation before performing an optimisation and frequency calculation on the structure and the transition state saddle point is once again confirmed by the presence of a single negative frequency found.

The optimised transition state, HOMO and LUMO orbitals are all visualised below.

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_endo_oput.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_endo_lumo.png|400px]]
|
[[Image:Aed12_endo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

The same procedure was followed for the exo transition state as the endo form previously. Again, a saddle point was established and confirmed through the presence of only one negative frequency.

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_exo_opt_mol.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_exo_lumo.png|400px]]
|
[[Image:Aed12_exo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

For both structures, a comparison of the relative energies has been made. The endo conformation is of lower energy, rationalising why the endo conformation is the preferred product.

{| class="wikitable" border="1"
! Conformer !! Energy (Hartrees) !! Relative Energy (kcal/mol)
|-
| Exo|| -0.05041984 || 0.68
|-
| Endo|| -0.05150481 || 0
|}

A comparision of the bond lengths has also been made with respect to the labelled diagrams of each compound. The main difference brought to light here is the distance between the carbons where the new bonds will be formed. In the exo transition state, these are slightly longer than in the endo case. This elongation can be attributed to the greater steric repulsion in the exo transition state, making closer approach less favourable in this scenario.

<center>[[File:Aed12_endo_labelled.png|500px]]</center>

{| class="wikitable" border="2"
! Bond !! Endo Bond Lengths (Å) !! Exo Bond Lengths (Å)
|-
| C1-C2 || 1.49053 || 1.48975
|-
| C1-C4 || 1.52297 || 1.52209
|-
| C2-C9 || 1.39307 || 1.39439
|-
| C9-C11 || 1.39724 || 1.39677
|-
| C15-C16 || 1.40848 || 1.41013
|-
| C16-C17 || 1.48923 || 1.48820
|-
| C17-O23 || 1.40896 || 1.40963
|-
| C17-O21 || 1.22057 || 1.22054
|-
| C2-C16 || 2.16237 || 2.18027
|-
| C3-C15 || 2.16238 || 2.18054
|}

====Secondary Orbital Effects====

It has previously been proposed that a secondary orbital overlap effect, where by the pi system on the substituent of the dieneophile and the pi system of the diene have a favourable interaction in the endo form, give the reasons for the endo selectivity seen in the reaction. However, examination of the HOMO and LUMO orbitals only is not able to account for these effects in this case. It is a drawback of the frontier orbital model that this is based on that we cannot rationalise the selectivity seen from these visualised orbitals alone.

==Conclusion==

File:Aed12 endo labelled.png

2015-03-26T23:51:28Z

Aed12:

Rep:Mod:aed1210

2015-03-26T23:51:04Z

Aed12: /* Intrinsic Reaction Coordinate (IRC) */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|1000px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Endo and Exo Transition States====

'''Endo'''

In order to create the endo transition state for the reaction, the reactants were optimised and placed in the endo formation before performing an optimisation and frequency calculation on the structure and the transition state saddle point is once again confirmed by the presence of a single negative frequency found.

The optimised transition state, HOMO and LUMO orbitals are all visualised below.

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_endo_oput.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_endo_lumo.png|400px]]
|
[[Image:Aed12_endo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

The same procedure was followed for the exo transition state as the endo form previously. Again, a saddle point was established and confirmed through the presence of only one negative frequency.

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_exo_opt_mol.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_exo_lumo.png|400px]]
|
[[Image:Aed12_exo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

For both structures, a comparison of the relative energies has been made. The endo conformation is of lower energy, rationalising why the endo conformation is the preferred product.

{| class="wikitable" border="1"
! Conformer !! Energy (Hartrees) !! Relative Energy (kcal/mol)
|-
| Exo|| -0.05041984 || 0.68
|-
| Endo|| -0.05150481 || 0
|}

A comparision of the bond lengths has also been made with respect to the labelled diagrams of each compound. The main difference brought to light here is the distance between the carbons where the new bonds will be formed. In the exo transition state, these are slightly longer than in the endo case. This elongation can be attributed to the greater steric repulsion in the exo transition state, making closer approach less favourable in this scenario.

<center>[[File:Aed12_endo_labelled.png|500px]]</center>

{| class="wikitable" border="2"
! Atoms !! Endo Bond Lengths (Å) !! Exo Bond Lengths (Å)
|-
| C1-C2 || 1.49053 || 1.48975
|-
| C1-C4 || 1.52297 || 1.52209
|-
| C2-C9 || 1.39307 || 1.39439
|-
| C9-C11 || 1.39724 || 1.39677
|-
| C15-C16 || 1.40848 || 1.41013
|-
| C16-C17 || 1.48923 || 1.48820
|-
| C17-O23 || 1.40896 || 1.40963
|-
| C17-O21 || 1.22057 || 1.22054
|-
| C2-C16 || 2.16237 || 2.18027
|-
| C3-C15 || 2.16238 || 2.18054
|}

====Secondary Orbital Effects====

It has previously been proposed that a secondary orbital overlap effect, where by the pi system on the substituent of the dieneophile and the pi system of the diene have a favourable interaction in the endo form, give the reasons for the endo selectivity seen in the reaction. However, examination of the HOMO and LUMO orbitals only is not able to account for these effects in this case. It is a drawback of the frontier orbital model that this is based on that we cannot rationalise the selectivity seen from these visualised orbitals alone.

==Conclusion==

Rep:Mod:aed1210

2015-03-26T23:50:39Z

Aed12: /* Optimizing the Reactants and Products */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|500px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Endo and Exo Transition States====

'''Endo'''

In order to create the endo transition state for the reaction, the reactants were optimised and placed in the endo formation before performing an optimisation and frequency calculation on the structure and the transition state saddle point is once again confirmed by the presence of a single negative frequency found.

The optimised transition state, HOMO and LUMO orbitals are all visualised below.

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_endo_oput.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_endo_lumo.png|400px]]
|
[[Image:Aed12_endo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

The same procedure was followed for the exo transition state as the endo form previously. Again, a saddle point was established and confirmed through the presence of only one negative frequency.

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_exo_opt_mol.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_exo_lumo.png|400px]]
|
[[Image:Aed12_exo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

For both structures, a comparison of the relative energies has been made. The endo conformation is of lower energy, rationalising why the endo conformation is the preferred product.

{| class="wikitable" border="1"
! Conformer !! Energy (Hartrees) !! Relative Energy (kcal/mol)
|-
| Exo|| -0.05041984 || 0.68
|-
| Endo|| -0.05150481 || 0
|}

A comparision of the bond lengths has also been made with respect to the labelled diagrams of each compound. The main difference brought to light here is the distance between the carbons where the new bonds will be formed. In the exo transition state, these are slightly longer than in the endo case. This elongation can be attributed to the greater steric repulsion in the exo transition state, making closer approach less favourable in this scenario.

<center>[[File:Aed12_endo_labelled.png|500px]]</center>

{| class="wikitable" border="2"
! Atoms !! Endo Bond Lengths (Å) !! Exo Bond Lengths (Å)
|-
| C1-C2 || 1.49053 || 1.48975
|-
| C1-C4 || 1.52297 || 1.52209
|-
| C2-C9 || 1.39307 || 1.39439
|-
| C9-C11 || 1.39724 || 1.39677
|-
| C15-C16 || 1.40848 || 1.41013
|-
| C16-C17 || 1.48923 || 1.48820
|-
| C17-O23 || 1.40896 || 1.40963
|-
| C17-O21 || 1.22057 || 1.22054
|-
| C2-C16 || 2.16237 || 2.18027
|-
| C3-C15 || 2.16238 || 2.18054
|}

====Secondary Orbital Effects====

It has previously been proposed that a secondary orbital overlap effect, where by the pi system on the substituent of the dieneophile and the pi system of the diene have a favourable interaction in the endo form, give the reasons for the endo selectivity seen in the reaction. However, examination of the HOMO and LUMO orbitals only is not able to account for these effects in this case. It is a drawback of the frontier orbital model that this is based on that we cannot rationalise the selectivity seen from these visualised orbitals alone.

==Conclusion==

Rep:Mod:aed1210

2015-03-26T23:49:06Z

Aed12: /* Endo and Exo Transition States */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69266
| 0.00
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Five</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68962
| 1.91
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2h
| -231.68907
| 2.25
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|500px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Endo and Exo Transition States====

'''Endo'''

In order to create the endo transition state for the reaction, the reactants were optimised and placed in the endo formation before performing an optimisation and frequency calculation on the structure and the transition state saddle point is once again confirmed by the presence of a single negative frequency found.

The optimised transition state, HOMO and LUMO orbitals are all visualised below.

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_endo_oput.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_endo_lumo.png|400px]]
|
[[Image:Aed12_endo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

The same procedure was followed for the exo transition state as the endo form previously. Again, a saddle point was established and confirmed through the presence of only one negative frequency.

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_exo_opt_mol.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_exo_lumo.png|400px]]
|
[[Image:Aed12_exo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

For both structures, a comparison of the relative energies has been made. The endo conformation is of lower energy, rationalising why the endo conformation is the preferred product.

{| class="wikitable" border="1"
! Conformer !! Energy (Hartrees) !! Relative Energy (kcal/mol)
|-
| Exo|| -0.05041984 || 0.68
|-
| Endo|| -0.05150481 || 0
|}

A comparision of the bond lengths has also been made with respect to the labelled diagrams of each compound. The main difference brought to light here is the distance between the carbons where the new bonds will be formed. In the exo transition state, these are slightly longer than in the endo case. This elongation can be attributed to the greater steric repulsion in the exo transition state, making closer approach less favourable in this scenario.

<center>[[File:Aed12_endo_labelled.png|500px]]</center>

{| class="wikitable" border="2"
! Atoms !! Endo Bond Lengths (Å) !! Exo Bond Lengths (Å)
|-
| C1-C2 || 1.49053 || 1.48975
|-
| C1-C4 || 1.52297 || 1.52209
|-
| C2-C9 || 1.39307 || 1.39439
|-
| C9-C11 || 1.39724 || 1.39677
|-
| C15-C16 || 1.40848 || 1.41013
|-
| C16-C17 || 1.48923 || 1.48820
|-
| C17-O23 || 1.40896 || 1.40963
|-
| C17-O21 || 1.22057 || 1.22054
|-
| C2-C16 || 2.16237 || 2.18027
|-
| C3-C15 || 2.16238 || 2.18054
|}

====Secondary Orbital Effects====

It has previously been proposed that a secondary orbital overlap effect, where by the pi system on the substituent of the dieneophile and the pi system of the diene have a favourable interaction in the endo form, give the reasons for the endo selectivity seen in the reaction. However, examination of the HOMO and LUMO orbitals only is not able to account for these effects in this case. It is a drawback of the frontier orbital model that this is based on that we cannot rationalise the selectivity seen from these visualised orbitals alone.

==Conclusion==

Rep:Mod:aed1210

2015-03-26T23:35:26Z

Aed12: /* Endo and Exo Transition States */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69266
| 0.00
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Five</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68962
| 1.91
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2h
| -231.68907
| 2.25
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|500px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Endo and Exo Transition States====

'''Endo'''

In order to create the endo transition state for the reaction, the reactants were optimised and placed in the endo formation before performing an optimisation and frequency calculation on the structure and the transition state saddle point is once again confirmed by the presence of a single negative frequency found.

The optimised transition state, HOMO and LUMO orbitals are all visualised below.

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_endo_oput.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_endo_lumo.png|400px]]
|
[[Image:Aed12_endo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

The same procedure was followed for the exo transition state as the endo form previously. Again, a saddle point was established and confirmed through the presence of only one negative frequency.

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_exo_opt_mol.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_exo_lumo.png|400px]]
|
[[Image:Aed12_exo_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

For both structures, a comparison of the relative energies has been made. The endo conformation is of lower energy, rationalising why the endo conformation is the preferred product.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

A comparision of the bond lengths has also been made with respect to the labelled diagrams of each compound. The main difference brought to light here is the distance between the carbons where the new bonds will be formed. In the exo transition state, these

====Secondary Orbital Effects====

It has previously been proposed that a secondary orbital overlap effect, where by the pi system on the substituent of the dieneophile and the pi system of the diene have a favourable interaction in the endo form, give the reasons for the endo selectivity seen in the reaction. However, examination of the HOMO and LUMO orbitals only is not able to account for these effects in this case. It is a drawback of the frontier orbital model that this is based on that we cannot rationalise the selectivity seen from these visualised orbitals alone.

==Conclusion==

File:Aed12 exo opt mol.mol

2015-03-26T23:31:16Z

Aed12:

File:Aed12 exo lumo.png

2015-03-26T23:29:24Z

Aed12:

File:Aed12 exo homo.png

2015-03-26T23:28:53Z

Aed12:

File:Aed12 endo homo.png

2015-03-26T23:27:15Z

Aed12:

File:Aed12 endo lumo.png

2015-03-26T23:26:32Z

Aed12:

File:Aed12 endo oput.mol

2015-03-26T23:25:54Z

Aed12:

Rep:Mod:aed1210

2015-03-25T16:38:46Z

Aed12: /* Cyclohexa-1,3-diene and Maleic Anhydride */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69266
| 0.00
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Five</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68962
| 1.91
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2h
| -231.68907
| 2.25
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|500px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Endo and Exo Transition States====

'''Endo'''

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|400px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

Rep:Mod:aed1210

2015-03-25T16:38:18Z

Aed12: /* Endo and Exo Transition States */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69266
| 0.00
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Five</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68962
| 1.91
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2h
| -231.68907
| 2.25
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|500px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|430px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Endo and Exo Transition States====

'''Endo'''

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|430px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|430px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

Rep:Mod:aed1210

2015-03-25T16:36:32Z

Aed12: /* Cyclohexa-1,3-diene and Maleic Anhydride */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69266
| 0.00
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Five</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68962
| 1.91
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2h
| -231.68907
| 2.25
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|500px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|430px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Endo and Exo Transition States====

'''Endo'''

<center>
<jmol><jmolApplet>
<title>Endo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|430px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

'''Exo'''

<center>
<jmol><jmolApplet>
<title>Exo Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|430px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

Rep:Mod:aed1210

2015-03-25T15:39:38Z

Aed12: /* Cyclohexa-1,3-diene and Maleic Anhydride */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69266
| 0.00
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Five</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68962
| 1.91
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2h
| -231.68907
| 2.25
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|500px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>Cyclohexa-1,3-diene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>Maleic Anhydride</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|430px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''LUMO''
| align="center" | ''HOMO''
|}

Rep:Mod:aed1210

2015-03-25T15:38:47Z

Aed12: /* Cyclohexa-1,3-diene and Maleic Anhydride */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69266
| 0.00
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Five</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68962
| 1.91
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2h
| -231.68907
| 2.25
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|500px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_chd_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_chd_LUMO.png|430px]]
|
[[Image:Aed12_chd_homo.png|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_MA_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Aed12_ma_lumo.png|430px]]
|
[[Image:Aed12_ma_homo.png|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

File:Aed12 chd opt.mol

2015-03-25T15:37:44Z

Aed12:

File:Aed12 chd LUMO.png

2015-03-25T15:34:53Z

Aed12:

File:Aed12 chd homo.png

2015-03-25T15:33:43Z

Aed12:

File:Aed12 ma lumo.png

2015-03-25T15:33:09Z

Aed12:

File:Aed12 ma homo.png

2015-03-25T15:32:52Z

Aed12:

File:Aed12 MA opt.mol

2015-03-25T15:31:49Z

Aed12:

Rep:Mod:aed1210

2015-03-25T15:15:52Z

Aed12: /* Ethylene and cis-Butadiene */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69266
| 0.00
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Five</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68962
| 1.91
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2h
| -231.68907
| 2.25
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|500px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

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

The Diels-Alder reaction between Cyclohexa-1,3-diene and Maleic Anhydride brings up another important issue of the transition states involved. With more complex reactants, there is the possibility of approach from two distinct directions, endo and exo. The dienophile substitute may face toward the diene (endo) or away (exo). With most cases under kinetic control, the endo product is usually preferred and a study of the orbitals involved shows, for the following reaction, why that is the case.

As before, the two reactants were first optimised and their HOMO and LUMOs visualised, the results of which are shown below.

====Cyclohexa-1,3-diene====

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

====Maleic Anhydride====

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

Rep:Mod:aed1210

2015-03-25T15:00:44Z

Aed12: /* Ethylene and cis-Butadiene */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69266
| 0.00
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Five</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68962
| 1.91
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2h
| -231.68907
| 2.25
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|500px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|400px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

Rep:Mod:aed1210

2015-03-25T14:58:10Z

Aed12: /* Ethylene and cis-Butadiene */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69266
| 0.00
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Five</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68962
| 1.91
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2h
| -231.68907
| 2.25
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|500px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

{| align="center"
|- align="center"
|
[[Image:Ts_lumo_aed12.PNG|40px]]
|
[[Image:Ts_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''TS LUMO - Symmetric w.r.t. Plane''
| align="center" | ''TS HOMO - Antisymmetric w.r.t. Plane''
|}

File:Ts lumo aed12.PNG

2015-03-25T14:56:44Z

Aed12:

File:Ts homo aed12.PNG

2015-03-25T14:56:34Z

Aed12:

Rep:Mod:aed1210

2015-03-25T14:23:03Z

Aed12: /* Ethylene and cis-Butadiene */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69266
| 0.00
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Five</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68962
| 1.91
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2h
| -231.68907
| 2.25
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|500px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Optimisation for the transition state leads to the below structure being formed. The presence of a single negative frequency confirms that the saddle point has been located which is also visualised below.

<center>
[[Image:Aed12_da_eb_optt.png|400px]]
<jmol>
<jmolApplet>
<title>Transition State Vibration</title><color>black</color><size>350</size>
<script>frame 53;vibration 1;
</script><uploadedFileContents>AED12_DA_EB_OPT.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

'''Bond Lengths'''

In order to analyse the bonding in the transition state, the bond lengths are examined.

{| class="wikitable"
|+ Bond Lengths
! Bond !! Length / Å
|-
| C1-C4 || 1.38192
|-
| C4-C6 || 1.39750
|-
| C1-C14 || 2.11935
|-
| C14-C11 || 1.38290
|}

Typical single C-C bond lengths in carbon are 1.54 Å for sp3 hybridisation and for sp2 are 1.46 Å. The bond lengths of the existing bonds of ethylene and cis-butadiene in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 Å so a distance of 2.12 Å between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.

The visualised frequency of the transition state infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.

'''Orbital Comparison'''

The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

File:Aed12 da eb optt.png

2015-03-25T14:15:37Z

Aed12: Aed12 uploaded a new version of "File:Aed12 da eb optt.png"

File:Aed12 da eb optt.png

2015-03-25T14:10:47Z

Aed12: Aed12 uploaded a new version of "File:Aed12 da eb optt.png"

File:Aed12 da eb optt.png

2015-03-25T14:10:01Z

Aed12:

File:Aed12 da eb opt.mol

2015-03-25T14:07:01Z

Aed12:

File:AED12 DA EB OPT.LOG

2015-03-25T14:06:34Z

Aed12:

Rep:Mod:aed1210

2015-03-25T13:57:10Z

Aed12: /* Ethylene and cis-Butadiene */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69266
| 0.00
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Five</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68962
| 1.91
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2h
| -231.68907
| 2.25
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|500px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state of the Diels-Alder reaction between the two was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Bond Lengths

Typical sp3 bond lengths in carbon are 1.54 and for sp2 are 1.46. The bond lengths in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 so a distance of 2.12 between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.
The visualised frequency of the transition state is visualised below and infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.
If the lowest positive frequency is compared, some twisting is observed but not in the plane of bond formation.
The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

Rep:Mod:aed1210

2015-03-25T13:51:49Z

Aed12: /* Ethylene and cis-Butadiene */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69266
| 0.00
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Five</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68962
| 1.91
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2h
| -231.68907
| 2.25
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|500px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|400px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Bond Lengths

Typical sp3 bond lengths in carbon are 1.54 and for sp2 are 1.46. The bond lengths in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 so a distance of 2.12 between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.
The visualised frequency of the transition state is visualised below and infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.
If the lowest positive frequency is compared, some twisting is observed but not in the plane of bond formation.
The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

Rep:Mod:aed1210

2015-03-25T13:49:04Z

Aed12: /* Ethylene and cis-Butadiene */

=Physical Module=

==The Cope Rearrangement Tutorial==

===Optimizing the Reactants and Products===

The Cope rearrangement is a [3,3]-sigmatropic rearrangement of 1,5-dienes, as seen in scheme 1 below. The reaction mechanism has been debated, it is now accepted that it occurs in a concerted fashion and that the transition state for the reaction is either a boat or chair structure. The objectives of the following exercises are to locate the low-energy minima and transition structures of 1,5-hexadiene Cope Rearrangement, therefore determining the preferred mechanism.

There are twelve possible conformers of the 1,5-hexadiene molecule, some of which have been created within the GaussView program. Of the ten studied conformers, six have a gauche linkage for the central four atoms and four have an anti-linkage. Each of these conformers was then optimised using Gaussian, using the Hartree Fock method and the 3-21G basis set. Shown below are the structures, point groups, and energy of the molecules, both in Hartrees and relative to the other molecules.

 

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Aed12_Gauche_one_321g_jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.68771607
| 3.10
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69166701
| 0.62
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69266
| 0.00
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69153035
| 0.71
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Five</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68962
| 1.91
|- align="center"
|
<jmol><jmolApplet>
<title>Gauche Six</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Gauche_six_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.68916019
| 2.20
|- align="center"
|
<jmol><jmolApplet>
<title>Anti One</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_one_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2
| -231.69260225
| 0.04
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Two</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_two_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| Ci
| -231.69253522
| 0.08
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Three</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>Gauche_two_321g_aed12.mol</uploadedFileContents>
</jmolApplet></jmol>
| C2h
| -231.68907
| 2.25
|- align="center"
|
<jmol><jmolApplet>
<title>Anti Four</title>
<color>black</color>
<size>200</size>
<uploadedFileContents>aed12_Anti_four_321g.mol</uploadedFileContents>
</jmolApplet></jmol>
| C1
| -231.69097055
| 1.06
|}

 

As can be seen from the above numbers, the conformer with the lowest energy and thus the one that would be seen in the highest proportions in an actual sample is the Gauche 3 conformer. The relative stability of the Gauche 3 conformer can be attributed to a favourable interaction between the vinyl proton and the C=C π orbital. The highest energy conformer is the Gauche 1 conformer, at a relative energy of 3.10 kcal/mol higher than the Gauche 3 conformation. The reason why the Gauche 1 conformer is of the highest energy is due to the proximity of the two vinyl groups. These are electron rich and so it is unfavourable to have them so close together. The remaining sequence for the gauche conformers is determined by the distance between these two groups, specifically the distance between the protons on each group, as a general trend, the energy decreases with increasing distance between these two protons. The energy of the anti-conformers can be explained as a trend between the distance of the vinyl protons and the protons on the β carbon.

====Comparison of Theories====

Two levels of computational theory will be applied to calculations throughout this experiment, HF/3-21G and B3LYP/6-31G*. In order to compare the two theories, both are applied to the optimisation of the same molecule and the bond lengths and angles obtained in the optimised molecule are compared. In this case, the Anti 2 molecule has been reoptimised at the B3LYP/6-31G* for comparison to the previously calculated HF/3-21G value. Upon inspection, there is no visible difference between the optimised molecules, however closer investigation shows some differences between the two structures.

Angles and bond lengths are described with reference to the numbered molecule below.

[[File:aed12_Anti2_labelled_angles_lengths.PNG|500px]]

{| class="wikitable"
|+ Bond Lengths
! Bond !! HF Value / Å !! B3LYP Value / Å
|-
| C2-C3 || 1.31623 || 1.33824
|-
| C2-C1 || 1.50890 || 1.50721
|-
| C1-C6 || 1.55278 || 1.55505
|}

The B3LYP method produces a longer bond length for the carbon-carbon double bond, slightly shorter but almost identical bond between the sp2 and sp3 hybridised centres and a shorter bond length for the sp3-sp3 single bond than the Hartree-Fock method. For the double bond, the literature value is calculated to be 1.33, and so is more accurately modelled by the B3LYP model. As the value for the other bonds are fairly similar, and as such the extra computational power may not be deemed worth it for simple calculations.

{| class="wikitable"
|+ Bond Angles
! Dihedral !! HF Value / deg !! B3LYP Value / deg
|-
| C3-C2-C1-C6 || 114.69527 || 118.77555
|-
| C2-C1-C6-C5 || 179.98991 || 180.00000
|}

The dihedral angle between the carbons in the different environments also varies between methods. For the C3-C2-C1-C6 angle, the B3LYP method returns a larger angle than the Hartree-Fock method. We see a slight variation in angle for the central dihedral (between C2-C1-C6-C5) but this very small between the two methods (<0.011 units).

====Frequency Analysis====

Frequency analysis can also be performed on the optimised structure we have calculated. The frequency obtained is related to the second derivative of the potential energy surface with respect to the geometry of the atoms involved. The output obtained allows further information about the stationary point on the PES that was obtained from the optimisation to be gathered.

The [[Media:AED12_BH3_OPT_631G_DP.LOG| log file]] obtained from the frequency analysis confirms that the structure has been optimised to a minimum, due to the lack of imaginary frequencies. It also provides some thermochemical data in the form of energies and enthalpies, as a displayed below.

Sum of electronic and zero-point Energies= -234.416255
Sum of electronic and thermal Energies= -234.408963
Sum of electronic and thermal Enthalpies= -234.408019
Sum of electronic and thermal Free Energies= -234.447872

===Optimizing the "Chair" and "Boat" Transition Structures===

The chair transition state is optimised first. An allyl fragment is first optimised and then two of these fragments are placed, staggered with a C2h point group, 2.2 A apart. This structure was optimised using the TS (Berny) method as we are dealing with a transition state, and a frequency analysis was performed on the optimised structure using the HF method and the 3-21G basis set. The resulting optimised transition state is visualised below.

[[File:Aed12_chair_ts.PNG|200px]]

Examination of the log file obtained from the optimisation gives the chair transition state as hoped, and the singular negative frequency present at 818 cm-1 confirms the presence of a minimum, the transition state, and can be thought of as a saddle point on the PES.

The vibration at -818 cm-1 is visualised below

<jmol>
<jmolApplet>
<title>Transition State Vibration -818cm-1</title><color>black</color><size>250</size>
<script>frame 19;vibration 1;
</script><uploadedFileContents>CHAIR_TS_AED12.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

====Frozen Coordinate Chair Optimisation====

An alternate method of optimising the chair transition state, but still using the Hartree-Fock method and 3-21G basis set is the frozen coordinate optimisation. In this method, the same procedure of setting the distance between the fragments (2.2A) was used, but with the additional constraint of fixing the ends of the molecule to a distance of 2.2 Å for the duration of the optimisation. The molecule is optimised twice in total, first with frozen groups, then re-optimised with the groups free. Although the product of the first optimisation looks like the transition state produced from the method above, a single negative frequency of -765 cm-1 is obtained, confirming that the optimisation has not been fully completed. The molecule produced after the second optimisation is shown below.

[[File:Aed12_frozen_chair.PNG|400px]]

The difference between the two structures obtained is marginal and cannot be seen from examination of the structure displayed above, but there is a difference. In the frozen coordinate optimisation, the bond lengths are slightly longer (2.02074 vs 2.02032), and the frequency is slightly less negative (-817.88 vs -817.92). The difference must arise from the differences in method applied as the calculation results are directly comparable due to being performed with the same basis set. The desired frequency (-818) is more closely achieved by the first method of optimisation, and bond lengths are closer to 2.2 Å, potentially pointing to this being the more reliable method of the two.

===QTS2 Optimisation===

For optimisation of the boat transition state, the QST2 method was employed. This involves giving both the products and the reactants for the reaction and allowing the simulation to try and find the transition state. The products and reactants thus have to be numbered accordingly to allow the method to identify the pathway between them. The first input for this method is shown below.

{| align="center"
|- align="center"
|
[[Image:Boat_reactant1_aed12.png|400px]]
|
[[Image:Boat_product1_aed12.png|400px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

The calculation fails under this input, as the reaction mapped between the two involved the cleavage of the central bond, followed by translation of the allyl fragment. This gives a chair transition state and so the input and output must be modified to produce the correct modelling.

{| align="center"
|- align="center"
|
[[Image:Boat_react2_aed12.PNG|400px]]
|
[[Image:Boat_prod2_aed12.PNG|422px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

This time, the angles of the reactant and product molecules are modified to give the structures seen above, a structure closer to the Gauche 2 conformation. Here, the reactant and product are closer to the transition state and so the saddle point is found during the scale of the calculation. The boat transition state is visualised below.

<center>
<jmol><jmolApplet>
<title>Boat Transition State</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_boat_2.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

Upon examination of the vibrational modes, a single negative vibration is found at -840 cm-1, corresponding to the transition state vibrations. The presence of this vibration also confirms the maximum (saddle point) in this pathway has been found.

<center>
<jmol>
<jmolApplet>
<title>Reaction Pathway Vibration</title><color>black</color><size>350</size>
<script>frame 41;vibration 1;
</script><uploadedFileContents>AED12_BOAT_2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
</center>

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

It is very difficult to predict the conformers that are produced from the transition states, however there is a Gaussian method (IRC) that allows the mapping of the minimum energy path from the transition structure to its local minimum on the PES. It can be thought of the path a ball with zero kinetic energy would follow if placed on the PES. Generation is done by following the curvature of the PES stepwise and is useful for us to see which conformers of the reactants and products are connected by the transition state in question. The size of step selected for the generation of the IRC has a direct effect on the accuracy of the output as a larger step size can accrue significant errors during the course of calculation. To determine the conformer produced from the transition state, the forward reaction path is calculated. The resulting IRC plot is shown below.

<center>[[File:IRC_total_plot_aed12.PNG|500px]]</center>

The total energy graph shows the drop in energy as the molecule comes down from the high energy transition state to settle in the low energy well of the product conformation. From the plot above we can see that as the energy has yet to reach a plateau, a minimum geometry has not yet been found. The minimum geometry can be found in a number of ways but for ease it was chosen to perform an optimisation on the lowest energy conformation from the low energy plot, which can be visualised by opening the checkpoint file. The structure obtained after this optimsation is very similar to the Gauche 2 conformer pointing to this conformer being the one formed from this transition state.

===Activation Energies===

In order to calculate the activation energies of the reaction pathways, we need optimised forms of both the transition states, to compare with the energies of the reactant. These can be performed using both the HF/3-21G and the B3LYP/6-31G* levels of theory and so the two transition states were optimised at the B3LYP/6-31G* level.
At the higher level of theory, transition states are still found, geometries are visualised below…
By subtracting the energy of the reactant away from the transition state energies, the activation energies are found, as shown below.

 

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556985
| align="center" | -234.414909
| align="center" | -234.408968
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402341
| align="center" | -234.396007
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611709
| align="center" | -234.469208
| align="center" | -234.461860
|}

 Where 1 hartree = 627.509 kcal mol-1 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.94
| align="center" | 44.69
| align="center" | 34.07
| align="center" | 33.19
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 56.31
| align="center" | 54.76
| align="center" | 43.06
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

Compared to experiment, the higher level of theory gives a more accurate result than the HF/3-21G level, as expected. Compared across the two temperatures, 0K and 298.15K it can be seen that the activation energy is largest for the 0K case. This is due to the presence of thermal energy at 298.15K, it can be thought of that with the additional thermal energy the molecule begins the reaction process from a place closer to the transition state maximum than without the thermal energy, thus the barrier to the transition state is relatively smaller. The activation energy for the chair pathway is lower in both cases, and is therefore the state of preference.

==Diels Alder Cycloaddtions==

===Ethylene and cis-Butadiene===

The Diels-Alder reaction is a well-documented pericyclic reaction between a diene and a dienophile that forms a cyclic structure. It is very useful, synthetically and so has been widely applied and studied.
The reaction studied in the following section is that between ethylene and cis-butadiene.
The optimised structures allow us to visualise the orbitals of the compound, and so the HOMO and LUMO orbitals of both compounds are shown below.

<center>
<jmol><jmolApplet>
<title>cis-Butadiene</title>
<color>black</color>
<size>350</size>
<uploadedFileContents>Aed12_butadiene_am1_opt.mol</uploadedFileContents>
</jmolApplet></jmol>
</center>

{| align="center"
|- align="center"
|
[[Image:Butadiene_lumo_aed12.PNG|430px]]
|
[[Image:Butadiene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''cis-Butadiene LUMO''
| align="center" | ''cis-Butadiene HOMO''
|}

{| align="center"
|- align="center"
|
[[Image:Ethylene_lumo_aed12.PNG|430px]]
|
[[Image:Ethylene_homo_aed12.PNG|400px]]
|- align="center"
| align="center" | ''Ethylene LUMO''
| align="center" | ''Ethylene HOMO''
|}

As before with the previous example, the transition state was optimised from an input of the two optimised reactants at a fixed distance from one another. Initially, a distance of 2.2 was used as in the previous example but this distance resulted in a failure to optimise the transition structure and so a shorter distance of 2.15 was used to gain a successful optimisation.

Bond Lengths

Typical sp3 bond lengths in carbon are 1.54 and for sp2 are 1.46. The bond lengths in the transition state suggest a delocalised, partial double bond character for the C-C bonds in both the reactants. The Van der Waals radius of carbon is 1.70 so a distance of 2.12 between the carbon atoms that will share a sigma bond in the product suggests an attractive interaction between them.
The visualised frequency of the transition state is visualised below and infers some information about the mechanism of bond formation. The two points at which the new sigma bonds are to be created move synchronously, as expected from the concerted, pericyclic reaction.
If the lowest positive frequency is compared, some twisting is observed but not in the plane of bond formation.
The HOMO and LUMO of the transition state are shown below, when compared with the HOMOs and LUMOs of the reactants, some similarities can be seen. Namely, the HOMO of the transition state appears to be a blend of the butadiene HOMO and the ethylene LUMO and the transition state LUMO a blend of the butadiene LUMO and ethylene HOMO. As the HOMO and LUMO of the reactants must be well matched in energy and of the right symmetry to allow the reaction to proceed, the mixing of the two to give the transition state orbitals shows that the reaction is an allowed pathway.

File:Ethylene homo aed12.PNG

2015-03-25T13:47:59Z

Aed12: