Rep:Mod3:md308
Mark Driver
Module 3
Introduction
In these experiemnts the transition state structure of two pericyclic reactions will be determined computaionally. The calculations will use molecular orbital theory to describe the process of bond formation and dissociation. By numerically solving the Schrödinger equation, it is possible to scan the potential energy surface and locate potential transition states. From this we can gather information about the transition state structures and energy differences. This allows us to follow a reaction pathway and study the mechanism and kinetics of a reaction.
Tutorial - Cope Rearrangement
The Cope rearrangement is a [3,3] sigmatropic shift reaction. This involves the migration of a C-C bond in a 1,5-hexadiene structure. This process occurs in a concerted fashion via either a "chair" or "boat" transition state. In this exercise these transition states will be modelled using computational methods. This will allow for calculation of the activation energies for this rearrangement.
Geometry Optimisation and Conformational Analysis of 1,5-hexadiene
The simplest system which undergoes the Cope rearrangment is 1,5-hexadiene. In this case rearrangement does not result in a chemical change due to the symmetry of this molecule. However, this is a suitable model for studying this type of reactivity as it is fairly simple and so should be computationally less demanding than alternative molecules. A good starting point for finding a stable geometry is the dihedral angle of the C-CH2-CH2-C linkage. Varying this angle shows that the staggered conformations are of lower energy than the eclipsed structures. Therefore the geometry of 1,5-hexadiene likely has either a gauche or an anti linkage as these correspond to energy minima. By varying the orientation of the alkenes it can be shown that there are 10 possible conformations of 1,5-hexadiene. In order to make accurate comparisons with the literature all of these conformers were modelled and their energies determined.
Each structure was optimised at the HF/3-21G level and the energies and symmetry of the structures was determined. This is a low level calculation but should give a good approximation to the structures. The optimisation was confirmed by vibrational analysis in each case.
| Conformer | Point Group | Energy / Hartree | Relative energy / kcal mol-1 |
|---|---|---|---|
|
|
C2 | -231.68772 | 3.10 |
|
|
C2 | -231.69167 | 0.62 |
|
|
C1 | -231.69266 | 0.00 |
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|
C2 | -231.69153 | 0.71 |
|
|
C1 | -231.68962 | 1.91 |
|
|
C1 | -231.68916 | 2.20 |
|
|
C2 | -231.68916 | 0.04 |
|
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Ci | -231.69254 | 0.08 |
|
|
C2h | -231.68907 | 2.25 |
|
|
C1 | -231.69097 | 1.06 |
In general the anti conformers are most stable. This is due to relief of steric strain introduced in the gauche conformation by adjacent alkenes. This destabilises the gauche conformers relative to the anti conformers. However, The lowest energy conformer is predicted to be the gauche 3 conformer. This is somewhat unexpected and suggests that some effect other than sterics is at work here. It has been suggested[1] that this is due to a favourable π - σ* orbital interaction. The conformers are all very similar in energy, so assuming that the barrier to interconversion is not too large then all of the conformers are likely to be accessible.
The anti 2 conformer was further optimised using the B3LYP/6-31G(d) method. This is a higher level calculation and should give a more accurate result.
The structure of the molecule is very similar for the B3LYP/6-31G(d) calculation compared to the HF/3-21G calculation. The central C-C bond length (1.55 Å) and C-C-C-C dihedral angle (180 °) is the same for both calculations. The C=C bond length increased slightly from 1.32 Å to 1.33 Å. The energy however is significantly lower for the second optimisation at -234.61171 Hartree compared to -231.69254 Hartree for the first. Vibrational analysis confirmed that the structure was minimised. Some selected vibrations are shown below.
| Vibration | Frequency | Intensity |
|---|---|---|
| 1734.3 | 18 | |
| 3031.5 | 54 | |
| 3080.2 | 36 | |
| 3136.9 | 56 | |
| 3155.5 | 15 |
The frequency analysis can also be used to obtain thermodynamic information contained within the log file. The calculation was carried out at 298.15 K. This provides the following data:
- Sum of Electronic and Zero-point Energies – the energy at 0 K this includes the zero-point vibrational energy
- Sum of Electronic and Thermal Energies – the energy at 298.15 K and 1 atm this includes contributions from the translational, rotational and vibrational energy
- Sum of Electronic and Thermal Enthalpies – this contains an additional correction for RT which is important when analysing dissociation reactions
- Sum of Electronic and Thermal Free Energies – this includes the entropic contribution to the free energy
| Energy / au | |
|---|---|
| Sum of Electronic and Zero-point Energies | -234.469203 |
| Sum of Electronic and Thermal Energies | -234.461857 |
| Sum of Electronic and Thermal Enthalpies | -234.460913 |
| Sum of Electronic and Thermal Free Energies | -234.500777 |
Optimisation of Transition States
The two transition states were constructed from two C3H5 allyl fragments which had been optimised using the HF/3-21G method. For each transition state structures the ends of the two fragments were positioned 2.2 Å apart.
Chair Transition State Structure
TS(Berny) Method
The chair transition state was optimised using the "Optimise to a TS(Berny) option". The structure used as an input needs to be accurate for this method. If the stucture is far from the transition state then it will probably not optimise correctly. This calculation scans the potential energy surface by determining the force constant (Hessian) matrix for the initial Structure and then slowly altering the structure until a transition state is reached. The additional keywords "Opt=NoEigen" were included to stop the calculation failing if more than one imaginary frequency is detected. The HF/3-21G method and basis set was used for this optimisation. This produced the structure shown below.
The energy of this optimised structure is -231.61932 Hartree and the partially formed/broken C-C bonds are 2.02 Å long. Frequency analysis confirmed that a maxima was achieved as an imaginary frequency at -817.9 cm-1 was predicted. This imaginary frequency corresponds to the Cope rearrangement. The vibration shows the concerted dissociation of one C-C bond and formation of another.
Frozen Coordinate Method
The chair transition state structure was then optimised via a different method. The frozen coordinate method fixes the terminal carbons in place and optimises the rest of the structure. The constraints on the geometry are then relaxed and the program optimises the geometry by differentiating along the two co-ordinates until a maxima is achieved. The same method, basis set and additional keywords were used in these optimisations as for the TS(Berny) method. The advantage of this Method is that the Hessian matrix does not need to be calculated and the default guess can be used instead. This reduces the demand on computational resources.
The energy of this optimised structure and C-C bond lengths are the same as for the previous calculation method. Frequency analysis confirmed that a maxima was found. The imaginary frequency was predicted at -818.1 cm-1 again corresponding to the Cope rearrangement. This suggests that both methods produced the same transition state structures.
Boat Transition State Structure
The boat transition state structure was also calculated. This was done by using the QST2 method, this interpolates between the reactant and product geometries to find the energy maximum (transition state). The anti 2 conformer was used as the product and reactant molecule. In order for the calculation to find the transition state structure the numbering of the atoms in the reactant and product must be consistent. The atoms were labelled according to the diagram on the right and then subjected to optimisation using the QST2 technique and B3LYP/6-31G(d) method and basis set. This produced a distorted chair transition state because the calculation interpolated between the 2 structures by translation of the allyl fragment rather than considering rotations. The structures were therefore adjusted so that the allyl fragments were eclipsed and the internal bond angles were reduced to 100°. The calculation was then run again with the same settings as before.
| Structure | Energy / Hartree | Vibration | Frequency / cm-1 | ||||
|---|---|---|---|---|---|---|---|
| -234.54309 | -530.6 |
Intrinsic Reaction Coordinate (IRC) Analysis
In order to find which conformer leads to the rearrangement an IRC calculation was run. This follows the minimum energy pathway from the transition state. The optimised transition state structure is used as an input, the coordinates are then adjusted to move in the direction visualised above as the imaginary frequency. This essentially moves across the potential energy surface in the direction of the steepest negative gradient causing the structure to move incrementally towards the product structure. The product structure can then be analysed and compared to the previous results to determine which conformer the reaction path will lead to.
The IRC was only calculated in the forward direction as the reaction is symmetrical, this reduces the time for the calculation. To improve the accuracy of the calculation the force constants were calculated for every cycle in the calculation. The number of points on the IRC was initially set to 50. However, in the case of the boat transition state this did not optimise the product structure sufficiently and so was increased to 100.
| Final Structure | Energy / Hartree | Conformer |
|---|---|---|
| -231.61932 | Gauche 2 | |
| -231.69266 | Gauche 3 |
| Reaction Coordinate Through Chair Transition State | Reaction Coordinate Through Chair Transition State |
|---|---|
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|
|
|
The IRC suggests that the two transition states lead to slightly different conformers. The structure resulting from the chair transition state has an energy higher than any of the conformers of 1,5-Hexadiene, This is because the relatively low number of points on the IRC means that the structure has not fully optimised. The structure appears to be converging on that of the gauche 2 conformer. Comparison of the energy and structure of the product resulting from the boat transition state indicates that the gauche 3 conformer is formed.
Thermochemistry
The activation energy for a reaction is simply the energy difference between the reactants and the transition state. Both these energies are calculated as part of the optimisation process. The thermochemical data which results can be useful in predicting the rates of reaction for example. The Transition states were reoptimised to the B3LYP/6-31G(d) level of theory. This should provide a more accurate result.
| Chair Transition State | Boat Transition State | |
|---|---|---|
| Sum of Electronic and Zero-point Energies / Hartree | -234.414925 | -234.402339 |
| Sum of Electronic and Thermal Energies / Hartree | -234.409006 | -234.396005 |
| Sum of Electronic and Thermal Enthalpies / Hartree | -234.408061 | -234.395061 |
| Sum of Electronic and Thermal Free Energies / Hartree | -234.443808 | -234.431747 |
| 0.00 K | Expt. (0 K) | 298.15 K | |
|---|---|---|---|
| ΔEChair / kcal mol-1 | 34.4 | 33.5 ± 0.5 | 33.1 |
| ΔEBoat / kcal mol-1 | 43.1 | 44.7 ± 2.0 | 41.3 |
The calculated activation energies are calculated relative to the anti 2 conformer. This gives activation energies which are very close to the experimental values.
Diels-Alder Cycloaddition
The Diels-Alder reaction is a pericyclic reaction involving a conjugated diene with an alkene forming two new C-C bonds in a concerted manner to give cyclohexene products. This reaction will be studied using molecular orbital analysis and transition state analysis. The reaction between ethene and butadiene provides a simple case study for this reaction.
Optimisation of Reactants
The structures of both reactants were optimised. The Diels-Alder reaction requires that the diene is in the s-cis conformation, therefore the dihedral angle between the four C atoms was set to 0° prior to optimisation. Both reactants were first optimised using the low level HF/3-21G calculation to achieve approximate structures. These were then optimised further using the B3LYP/6-31G(d) level of theory. Optimisation was verified in each case by vibrational analysis.
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|
| |
| Energy / Hartree | -78.58746 | -155.98595 |
|---|
Molecular Orbital Analysis
The frontier orbitals of both reactants are shown below. The symmetry of these molecules determines whether the reaction occurs as the orbitals can only interact constructively if they are of the same symmetry and similar energy.
| Frontier Orbital | Energy / Hartree | Symmetry | ||
|---|---|---|---|---|
| HOMO | 
|
|
-0.26664 | Symmetric |
| LUMO | 
|
|
0.01880 | Antisymmetric |
| Frontier Orbital | Energy / Hartree | Symmetry | ||
|---|---|---|---|---|
| HOMO - 1 | 
|
|
-0.31708 | Symmetric |
| HOMO | 
|
|
-0.22736 | Antisymmetric |
| LUMO | 
|
|
-0.03013 | Symmetric |
| LUMO + 1 | 
|
|
0.09644 | Antisymmetric |
As can be seen above there are many combinations of orbitals which could potentially lead to a bonding interaction. The most important of these is likely to be the HOMO-LUMO interactions as the energies are most similar. This π-π* interaction allows for the concerted formation of two C-C bonds.
Optimisation of the Transition State
The approximate transition state was created using the optimised structures of the reactants. The Transition state has an envelope structure which maximises the orbital overlap and gives the greatest interaction. Therefore the ethene molecule was positioned approaching from above the plane of the butadiene. The distance between the terminal C atoms was set to 2.00 Å. This structure was then optimised to a transition state using The TS(Berny) technique, calculating the force constants once, using the B3LYP/6-31G method and basis set and with the additional keywords "opt=NoEigen".
| Structure | Energy / Hartree | Vibration | Frequency / cm-1 | ||||
|---|---|---|---|---|---|---|---|
| -234.54390 | -525.1 |
Vibrational analysis confirmed that a transition state was reached. An imaginary frequency at -525.1 cm-1 indicates that the structure is a maxima on the PES. Visualisation of this frequency shows the formation of the two C-C bonds corresponding to the diels-alder reaction. Bond lengths also suggest that this is a transition state. The Terminal C atoms are 2.27 Å apart this is much longer than a typical C-C bond (1.53 Å [2]) suggesting that it is only partially formed. Similarly, the C-C bond in the butadiene fragment has a length of 1.41 Å, This is intermediate between a typical single and double bond (1.34 Å [2]). The C=C bonds are also slightly lengthened suggesting that the π bonds are weakening.
Molecular Orbital Analysis
| HOMO | LUMO | |
|---|---|---|
|
| |
| Energy / Hartree | -0.21896 | 0.00860 |
| Symmetry | Symmetric | Symmetric |
Both the HOMO and LUMO are symmetrical They must therefore result from overlap of symmetrical orbitals. The HOMO and LUMO are formed from overlap of the ethene HOMO with the butadiene LUMO. There is a large amount of electron density between the two fragments indicating that bonds are beginning to form. As expected the bonds appear to be forming suprafacially.
Regiochemistry in the Diels-Alder Reaction
The Diels-Alder reaction between maleic anhydride and cyclohexadiene can give two different products depending on the relative orientation of dieneophile as it approaches the diene. The major isomer formed in this reaction is the endo product as it forms via a lower energy transition state. secondary orbital interactions have been invoked to account for the low energy of this structure.
Optimisation of Reactants
Cyclohexadiene and Maleic anhydride were first optimised using the HF/3-21g method and then further optimised to the B3LYP/6-31G(d)level. The resulting structures are shown below. Optimisation was verified in each case by vibrational analysis.
|
|
| |
| Energy / Hartree | -233.41891 | -379.28954 |
|---|
Optimisation of the Transition State
The approximate transition states were created using the optimised structures of the reactants. The distance between the diene and alkene was set to 2.30 Å. The transition state structures were then optimised to the B3LYP/6-31G(d) level of theory using TS(Berny) method. These optimisations failed and the structures produced were unrealistic. This is most likely because the input structures were not close enough to the transition state structure. The transition states were then optimised using the frozen coordinates method and the same settings as before. Vibrational analysis confirmed a transition state structure
| Structure | Energy / Hartree | Vibration | Frequency / cm-1 | |||||
|---|---|---|---|---|---|---|---|---|
| Exo Transition State | -612.67931 | -448.0 | ||||||
| Endo Transition State | -612.68340 | -446.9 |
The bond lengths for both transition states are all indicative of partially formed C-C and C=C bonds. The endo transition state is 2.56 kcal mol-1 lower in energy than the exo transition state. Since both are formed from the same reactants then the endo will have a lower activation energy. Under kinetic control this would lead to the observed endo selectivity. The exo form is likely more strained as the (C=O)-O-(C=O) fragment is brought close to the CH2-CH2 group. This would induce some steric strain destabilising the structure. The steric clash in the endo transition state is less as the (C=O)-O-(C=O)group is brought close to the planar alkene rather than tetrahedral carbons. This also allows for the possibility of a secondary orbital interaction. The molecular orbitals of the endo transition state have the appropriate symmetry and nodal properties to allow a stabilising interaction between the C=O π* (LUMO of maleic anhydride) with the HOMO of cyclohexadiene. This stabilising interaction further favours the endo transition state.
| Orbital | Exo TS | Endo TS |
|---|---|---|
| HOMO | 
|
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| LUMO | 
|
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The HOMO of both transition states shows electron density between the two halves of the molecule, this corresponds to the bond being formed during the reaction. The endo LUMO could potentially be involved in the secondary orbital overlap. This is not possible with the exo transition state.
Conclusion
This module has modelled a series of reactions computationally. The transition state structures have been calculated and the imaginary frequencies corresponding to bond forming processes have been visualised. Molecular orbitals and thermochemical data have been determined. This has allowed the outcome of the reactions to be rationalised. The calculations are all a compromise between accuracy and calculation time but the methods and basis sets used appear to give reasonable results which compare well with the literature.