Module 3: The Transition State

Introduction

Tutorial

In order to become familiar with the techniques which were to be used and the outputs they generated a tutorial exercise was undertaken before the main body of the computational study was done.

This tutorial involved a study of 1,5-cyclohexadiene. 1,5-cyclohexadiene can exist principly in two different forms, the anti and the gauche conformers. These two differ in the dihedral angle around the central four carbons with the anti having a bond angle of 180° with the groups anti-periplanar to one another while the gauche has a bond angle of 0° with the groups completely eclipsed. As such we would not be unreasonable to suggest that the anti form would be lower in energy

To begin with the anti form was drawn in GaussView and optimised. This was done using the Hartree Fock approximation and a 3-21G basis set. The Hartree Fock approximation models LCAO, this is the linear combination of atomic orbitals and so the molecular orbitals are built up of simple sums of the atomic orbitals. This is a widely used approximation and so is a useful place to start. The 3-21G basis set is also a widely used basis set. This is a basic example of a split valence basis set. In this the different numbers describe the number of functions which are summed to make up the approximation for a particular type of orbital, e.g. the 3 is the number of functions used to make up the approximation for the inner orbitals.

The optimisation was set up in GaussView and then run in Gaussian. This resulted in the below results:

Anti Form


Point Group = C₂
Energy = -231.69260234

In a similar fashion the same was done for the Gauche conformer. Again results are tabulated:

Gauche Form


Point Group = C₂
Energy = -231.69166702

We can compare the results of these two conformers. We initially see the two are quite similar, they both have a point group of C₂. The main, and most important difference is the energy of the two conformers. We see that the anti isomer is in fact the lower in energy this is by a value of 2.46kJ mol^-1, which is comparable to RT. This means that we will see some conversion between forms at room temperature.

What we do notice from this is that the most stable conformer is where the two C=C bonds are as far away from each other as possible. We can envisage a further more stable conformer where the two groups are even further apart. It is important that we find the lowest energy conformer as this is the ground state with respect to which we quote relevent energies such as the activation energy and the enthalpies. This is tabulated below:

Lowest Energy Gauche Form


Point Group = C₁
Energy = -231.69266120

It is possible to compare these structures generated with reference structures given^[1]. In doing so we see that the anti structure generated corresponds to "anti1" while the gauche structure corresponds to "gauche2". Finally the lowest energy structure generated is given as "gauche3". We can also compare the energies we have obtained with reference values, these are pleasing results as the two agree, indicating that the computation done has been successful. The final observation we make is that we have indeed identified the lowest energy structure.

Moving forward we can look into conformers with different point groups. As such we can model and optimise the C_i structure. In this we optimised firstly to with a 3-21G basis set and then with a more accurate 6-31G basis set, this basis set uses more functions with which to build up the orbitals being approximated and so can be seen to be more accurate. We have also changed the method used and have now moved to using DFT, density functional theory, with the B3LYP method. This is again more advanced than Hartree Fock used before. This resulted in the following output:

C_i Point Group Optimisation

Energy = -231.68029061


Point Group = C_i
Energy = -234.61112035

What we notice immediately is the much lower energy of the result at this level of theory. This is a big reduction in energy and so is a significant factor. Overall we see this reduction in energy but no real noticeable change in the geometry or structure. We see identical bond lengths, angles and other parameters. This implies that the higher level optimisation has enabled us to better calculate the energy of the structure.

A further advancement to the calculations we have performed is to take into account the frequency and vibrations of the molecule. This is done by selecting 'Frequency' as the job type in GaussView. In doing this for the C_i point group we generate the below IR Spectrum:

The crucial observation we make here is that we see no imaginary vibrations only real ones. The significance of this is that we can now infer that we have found a minimum energy structure, if we had found imaginary vibrations we would have been at a transition state structure.

Further to this we could look further into the file and do some analysis of the structure and its relative energy. In doing so we see the following lines:

We can tabulate these results and compare them to reference^[2] data. This is done below:

Comparison Of Thermochemistry Data
Quantity	Energy	Physical Meaning
Sum of Electronic and Zero-Point Energies	-234.469339	E = E_elec + ZPE
Sum of Electronic and Thermal Energies	-234.463595	E = E + E_vib + E_rot + E_trans
Sum of Electronic and Thermal Enthalpies	-234.462651	H = E + RT
Sum of Electronic and Thermal Free Energies	-234.498545	G = H - TS

We can compare these with the reference data and we find a both a good correlation and a good fit of the computational results with the literature, this leads us to have confidence in the value of the computational modelling we have done thus far.

From this we are now ready to begin to move forward and begin to model some transition state structures.

Optimising the Chair and Boat Structures

The techniques established from the tutorial section were then used to model the chair and boat structures of the Cope Rearrangement. This was done by first drawing the allyl fragment CH₂CHCH₂ and optimising this with a HF approximation and a 3-21G basis set. This was the basis for the fragments which were then modelled as combining through both a chair and boat combination.

Chair Modelling

Firstly the chair transition state was guessed. This was then optimised to the transition state by selecting this from the GaussView calculation menu. The initial optimisation failed, this was due to a poor guess of the structure and so the input structure was then redrawn more carefully to look as close as possible to the predicted transition state. We also gave an additional command in the keywords box by adding Opt=NoEigen. This was to circumvent the calculation crashing if we found more than one imaginary frequency during the optimisation.

The results of all of this was the below structure with the given vibrations.

Optimisation Of Transition State


Energy = -231.61932244
Key Vibration = -817.995

The bond lengths for the bonds being made and broken can be found in looking at this optimised structure, in both cases they correspond to 2.02Å.

Optimisation Of Transition State 2


Energy = -231.61518525

Optimisation Of Transition State


Energy = -231.61932239
Key Vibration = -817.759

Again we find the lengths of the bonds made and broken to be 2.02Å.

Boat Optimisation

yap

Optimisation Of Transition State


Energy = -231.602820248
Key Vibration = -839.91

Following The Reaction Path

there is a method implemented in Gaussian which allows you to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This is called the Intrinisic Reaction Coordinate or IRC method. This creates a series of points by taking small geometry steps in the direction where the gradient or slope of the energy surface is steepest.

This was first done with 50 steps, with the results given below:

Following the Reaction Path

Input Structure	Output Structure

In looking at the output file we can see this has failed to reach a minimum. We have a number of options here. Firstly we can take the final point and optimise from here. This option was explored and resulted in the below details:

Optimisation Of Transition State 2


Energy = -230.17405957

A further and more accurate option would be to restart the calculation but with more points. This was also done with the number of points doubled to take into account 100 points.

Again results are tabulated:

Following the Reaction Path 2

Input Structure	Output Structure

We can now take this structure and optimise this to ensure that we have indeed culminated at the lowest energy structure:

Optimisation Of Transition State 2


Energy = -230.17405959

Finding the Activation Energy

Summary of activation energies (in kcal/mol)

Comparison of Activation Energies
	HF/3-21G at 298.15 K	B3LYP/6-31G* at 298.15 K	Expt. at 0 K
?E (Chair)	45.31		33.5 ± 0.5
?E (Boat)	55.67		44.7 ± 2.0

Diels-Alder Cycloaddition

The most basic Diels-Alder reaction imaginable is the [4s + 2s] cycloaddition of cis butadiene and ethylene. The basic mechanism is given below:

It is the intention to model this computationally to study the orbital interactions in more detail. The initial step in this was to optimise both of the reactant molecules, the summary of this is tabulated below:

Optimising the Cycloaddition Reactants
cis-Butadiene

Ethylene

We could then look at the molecular orbitals of the two molecules involved to gain insight into the key interactions and overlaps which allow reaction to proceed.

cis-Butadiene Molecular Orbitals

HOMO	LUMO
Plane Reflection: Antisymmetric	Plane Reflection: Symmetric
Axis Rotation: Symmetric	Axis Rotation: Antisymmetric

Likewise we could use Gaussview to plot the Molecular orbitals of ethylene pictorially.

Ethylene Molecular Orbitals

HOMO	LUMO
Plane Reflection: Symmetric	Plane Reflection: Antisymmetric
Axis Rotation: Antisymmetric	Axis Rotation: Symmetric

It is well defined that the principal orbital interactions are those of the HOMO-LUMO pairs which overlap with each other to form the new bonds in the product.

Previous studies have shown results for the orbitals which agree with those which we observe for our computational study and so we can link the results obtained with previous experimental details

It was also possible to take this further and model the transition state of the diels alder reaction. This was done by first freezing the bonds formed and optimising to a minimum and then taking this forward by then optimising to a transition state a forming bond derivatives.

Optimising Diels Alder to a Minimum


Energy = -231.48586105

Optimising the Diels Alder Transition State


Energy = -231.48963220
Key Vibration = -1067.23

Comparing the Exo and Endo Forms
Exo

Energy = -605.71873525
Endo

Energy = -605.72132078

These results give a observation which from first principles would be unexpected. It is widely accepted that this reaction gives the exo product however when we look at the relative energies of the two forms we see the exo form is in fact almost 7kJ mol^-1 less stable than the endo form. From this we must conclude that this reaction is under kinetic control. This has direct effects on our study of transition state structures as the kinetic product is that which is formed by the lowest energy. From this we would expect that the exo transition state will be more stable than that for the endo structure.

We can look into the transition states of the two conformers to see which is in fact lower energy. This is done in a similar way to that done before by using the frozen coordinate method. As before we created the two different isomers and first froze the bonds while optimising. The checkpoint files were then opened and the Redundant Coodinate Editor was used to create coordinates at the two bonds made in the reaction to form bond derivatives.

We then ran an optimisation to the transition state with frequency analysis. The results for this process are given below: