Rep:Mod:xz7013
Introduction
Based on Born Oppenheimer approximation, the energy minimum structure of a molecule can be located on the potential energy surface (PES) generated by doing single point calculations and plotting all the possible structures.
The Cope Rearrangement (tutorial)
In this tutorial, the reaction pathway of Cope rearrangement of 1,5-hexadiene was studied. This reaction is a 3-3' sigma tropic rearrangement that obeys the Wardwood Hoffmann Rule and is thermally allowed. The total energy of the reactant/product was investigated and two possible transition structures, chair and boat, were located on the potential energy surface (PES) using various methods. The activation energies for both structures were calculated and compared with the experimental values. All of the calculations were done by GaussView 5.0.
Optimizing the Reactants and Products
Optimizing the 'anti' and 'gauche' 1,5-hexadiene
A 1,5-hexadiene molecule with anti linkage was drawn and optimized to minimum using HF/3-21G level of theory. After the job has finished, check file was opened and the Total Energy was record in summary under Results. The energy is -231.69097054 a.u.. The molecule has a symmetry and the point group is C1. This conformer is the anti4 in the Appendix 1. [1]
The total energy of the 'gauche' conformer is expected to be higher than 'anti' because of steric hindrance between the two π systems. A 1,5-hexadiene molecule with anti linkage was drawn and optimized to minimun using same level of theory. The Total Energy of the molecule is -231.69266121 a.u., which is lower than the optimized 'anti' conformer. This result was different from the expectation. The molecule has a symmetry and the point group is C1. This conformer is the gauche3 in the Appendix 1. [2]
Finding the lowest energy conformer
The lowest energy conformer of 1,5-hexadiene was expected to be the anti2 since the vinyl group are farthest away from each other. However this was not the case for 1,5-hexadiene. The difference of distance between two vinyl hydrogen atoms in 'anti' and 'gauche' forms is very small. Therefore the steric effect is not significant. The gauche3 structure that has already been drawn and optimized in the last step is the lowest conformer of 1,5-hexadiene.[3] This is because there is attractive interaction between the C=C double bond and the vinyl proton. The filled π orbital can donating electron density to the empty σ* orbital and stablise the molecule.[1]
(Re)optimizing the anti2 conformer
An anti2 1,5-hexadiene molecule was drawn and optimized using HF/3-21G level of method. The molecule has a symmetry and the point group is Ci. The Total Energy of the molecule is -231.69253528 a.u. and the reference value given in the table in Appendix 1 is -231.69254 a.u..[4] They are almost the same. So the structure has been drawn correctly. The HF/3-21G optimized anti2 1,5-hexadiene molecule was reoptimized by B3LYP/6-31G* level of theory. The overall geometry changed very little from the previous HF/3-21G optimized structure. However, the total energy of the B3LYP/6-31G* optimised structure is lower than the HF/3-21G optimized structure.
Frequency calculation on anti2 conformer
For the comparison of the final total energies of the output file and the experimental values, frequency calculation needs to be run. The B3LYP/6-31G* optimized anti2 1,5-hexadiene molecule was performed a frequency calculation and all the vibrational frequencies were positive (i.e. no imaginary frequencies). From the .log output file viewed in the visualizer, energies under different vibrational temperatures were noted down.
| sum of energies | in a.u. | in kcal/mol |
|---|---|---|
| electronic and zero-point energies | -234.469212 | -147131.683317 |
| electronic and thermal energies | -234.461856 | -147127.067357 |
| electronic and thermal enthalpies | -234.460912 | -147126.474988 |
| electronic and thermal free energies | -234.500821 | -147151.518269 |
Optimizing the 'Chair' and 'Boat' Transition Structures
It was proposed that the Cope rearrangement reaction can take place via two different transition states, chair and boat. Both transition state were optimized but with different methods.
Optimizing the 'Chair' transition structure (method 1)
The 'Chair' transition structure was located and optimized using the method of calculating the force constant. Two identical allyl fragments were drawn and optimized using the HF/3-21G level of theory. They were moved into an orientation that was close to the 'chair' transition structure with a distance of approximately 2.2 Å apart. Opt+Freq calculations were carried out with optimisation to TS (Berny) and calculate the force constant once. Opt=NoEigen was put in as additional keywords. The output molecule had only one negative vibration frequencies at -817.90 cm-1 and the rest were positive frequencies. The animation of this imaginary frequency was identified to be the Cope rearrangement as shown below. The RMS Gradient Norm of the molecule was 0.00001426 a.u., which was almost zero. IRC All of these factors indicated that it was the correct transition state of the reaction.
ANIMATION ADDED
The imaginary frequency represents a negative force constant k. Since for a 1D quantum harmonic oscillator, k is the second derivative of the energy (i.e. curvature). It would be negative at the energy maxima, where a transition state is on the potential energy surface. Therefore the vibrational frequency would be negative from the equation below.
Optimizing the 'Chair' transition structure (method 2)
The 'Chair' transition structure was optimized using the frozen coordinate method. Redundant Coord Editor was used to freeze the four carbon atoms that form/break bonds respectively and left the rest of the molecule free for minimisation. Here optimization to minimum was operated instead of optimization to TS (Berny). Force constant was not calculated. The output molecule showed a fixed distance of 2.2 Å between the bond form/break carbon atoms. Redundant Coord Editor was used again to unfreeze the carbon atoms and this time optimized to TS (Berny). Force constant was not calculated. The output molecule had only one negative vibration frequencies at -817.88 cm-1 and the rest were positive frequencies. The animation of this imaginary frequency was identified to be the Cope rearrangement as shown below. The RMS Gradient Norm of the molecule was 0.00002117 a.u., which was almost zero. IRC All of these factors indicated that it was the correct transition state of the reaction.
ANIMATION ADDED
The bond form/break bond lengths of the transition structures generated by the two methods were very similar with the one from method 2 slightly higher.
| Bond break/form C-C bond lengths (Å) | |
|---|---|
| method 1 | 2.02064 |
| method 2 | 2.02071 |
Optimizing the 'Boat' transition structure
The 'Boat' transition structure was located and optimized using the TS(QST2) method. In this method, the reactant and product were drawn out and given the same number to each atom. The calculation can work out the transition structure based on the numbering and orientation of the reactants and products. The HF/6-31G optimized anti2 1,5-hexadiene were used as both reactant and product and the two molecules were modified as below. It illustrated that from reactant to product the bond between C3 and C4 broke and between C1 and C6 formed. The double bond moved from between C1,C2 and C5,C6 to C2,C3 and C4,C5. The input molecules looked like this.
However in this case, the job failed. The TS(QTS2) method was not able to locate the correct transition state from this arrangement of reactant and product because it is too deviated from the actual boat transition state structure. So the input molecules were modified by changing the central C-C-C-C dihedral angle to 100° and the inside C-C-C angle to 100°. The numbering of each atom was kept unchanged. The molecules now looked like this.
.PNG)
This time the input molecule were close enough to the boat transition structure and it was successfully located. Again a negative frequency was shown at and the corresponding animation (below) indicated the Cope rearrangement reaction.
ANIMATION ADDED
This exercise showed that although the TS (QTS2) method was able to directly calculated the transition structure from just the reactant and product, it was only valid when the input reactant and product were modified in a way very close to the actual structure of the desired transition state. Otherwise the transition state could not be achieved.
Intrinsic reaction coordinate (IRC) method
Now the boat and chair transition structures were found out successfully. The conformer of 1,5-hexadiene that each transition structures connect to was difficult to predict. The IRC method were used to locate the locate minimum from the transition structure on the potential energy surface. This method was applied to the optimized chair transition structure. For calculation set up, only forward direction was chosen because the reaction coordinate was symmetrical. The reactant and product should have the same structure and therefore the same energy. Force constant calculation was chosen to be always and the points generated along the IRC path was set to be 50. The results were as shown below.
ANIMATION ADDED
It was realised that the chair transition structure connected to the gauche linkage of 1,5-hexadiene. Both the total energy in summary and the IRC plot showed that the minimum energy geometry had not been reached. (The total energy of gauche3 conformer is -231.69266 a.u.).[5] Therefore IRC method was improved to locate a further minimized structure. The last point on IRC plot, which was the minimum from last calculation, was taken to run a normal minimisation. The output geometry had a lower total energy than the input as shown in the table. Other possible improvement methods include increasing of the point on the IRC path and calculating the force constant on every step.
| total energy in a.u. | |
|---|---|
| input geometry | -231.69158 |
| output geometry | -231.69167 |
Activation energies
To activation energies of the reaction for both chair and boat transition structures, both chair and boat transition structures were reoptimized using B3LYP/6-31G* level of theory using the previously HF/3-21G optimized structures. As a result, the geometries of reactant and transition structure were very similar for each level of theory, yet the total energy difference was large. The activation energies were calculated at 0K and 298.15K from the raw data (in a.u.) collected in the .log file for both transition structures and compared to the experimental values. [6]
| HF/3-21G | HF/3.21G | B3LYP/6-31G* | B3LYP/6.31G* | Experimental values | |
|---|---|---|---|---|---|
| at 0 K | at 298.15 K | at 0 K | at 298.15 K | at 0 K | |
| ΔE in kcal/mol (Chair TS) | 38.4 | 37.3 | 34.1 | 33.2 | 33.5 ± 0.5 |
| ΔE in kcal/mol (Boat TS) | 48.2 | 47.4 | 42.0 | 41.3 | 44.7 ± 2.0 |
The comparison of results showed that the activation energies calculated from B3LYP-631G* is closer to the experimental values the ones from HG/3-21G. The values were almost in the range of the experimental data. This indicated that the Density Functional Theory (DFT) method was more accurate than the Hartree Fock (HF) method. This is because the DFT method includes the exchange-correlation term, which shows the correction to the kinetic energy due to interacting nature of electrons, as well as all of the nonclassic corrections to the repulsion energy between electrons. [2]
The Diels Alder Cycloaddition
The Diels Alder Cycloaddition is a type of pericyclic reaction in which an electron-rich cis diene react with an electron-deficient dienophile to form two new σ C-C bond.
Part 1: cis-butadiene + ethlyene
The reaction of cis butadiene and ethlyene was studied as a prototype cycloaddition reaction. The reaction obeys the Woodward Hoffmann Rule and is thermally allowed.
GaussView was used to build a cis butadiene and the molecule was optimized by Gaussian using AM 1 Semi-empirical and then B3LYP/6-31G* DFT method. The MOs of the molecule were visualized. The HOMO of cis butadiene is anti-symmetric and the LUMO is symmetric with respect to the plane (indicated by the black line in the middle).
It was stated that the reaction has a envelop shape transition structure. A guess transition structure with a guess distance of 2.2 Å between bond break/form C-C atoms was drawn and optimized by the method 1 (TS Berny) in the Cope Rearrangement tutorial. The output structure was confirmed to be the transition state because (i) it had one and only one negative (imaginary) vibrational frequency and the animation showed the bond formation. (ii) the RMS Gradient Norm was nearly zero. The formation of the two new σ C-C bonds are synchronous. This was to compare with the lowest positive vibrational frequency in which the two bonds formed asynchronously.
The geometry of the transition structure was shown below. The bond length of the partly formed C-C σ bond was 2.27228 Å. From the literature, the typical sp3 and sp2 C-C bond length are 1.54 Å and 1.46 Å[3] and the van der Waal radius of the carbon atom is 1.70 Å[4]. The bond length of partly formed C-C was longer than sp3 and sp2 bond length and even the van der Waal radius of the carbon atom. It can be concluded that at this stage the two substrate were not forming bond and only having interactions weaker than van der Waal force.
ANIMATION ADDED
The MOs of the transition structure were visualised. Both of the HOMO and LUMO were symmetric with respected to the plane (black line).
