Rep:Mod:osef-module3

The Cope Rearrangement: A Computational Study

Introduction

The Cope Rearrangement is a classic example of a [3,3] sigmatropic reaction, involving a 1,5 diene. In this case we use the simplest 1,5 diene possible, 1,5 hexadiene. We will attempt to find the lowest energy conformations of the reactant (and consequently the product), as well as propose the reaction mechanism, based on the lowest energy transition state.

File:Pierre Cope Rearrangement.png File:Pierre Transition States.png

Optimising and Analysis of the Reactants/Products

Optimisation

The first structures to be optimised, were those in which the central C-C bond is arranged in an anti-periplanar fashion (i.e. dihedral angle approximately equal to 180º). These are presumed to be the lowest energy conformations, due to the fact that they exhibit the lowest level of steric hindrance. In this conformation, all bonds along the central C-C bond are aligned to undergo σ-conjugation. The so called "gauche" conformation also exhibits σ-conjugation, given that the dihedral angle along the central C-C bond is approximately 60º. Assuming that σ_CC and σ_CH have the same donor/acceptor capacity, we can assume that the electronic stabilisation resulting from σ-conjugation is the same in the gauche and anti conformations. Sterics will therefore play the main role in determining the energies of the conformations. From this statement, it is clear that, all other things being equal, the conformations in which the central C-C bond is in the anti conformation will be the lower in energy than its gauche analogue. Syn-periplanar (dihedral angle of 0º) and eclipsed (dihedral angle of 120º) conformations will all be higher in energy, due sterics, and the fact that they do not show σ-conjugation. We therefore only optimised the major conformers containing an anti or gauche central C-C bond. The optimisation for each of the conformers was performed using the Hartree-Fock method (HF), and a 3-21g basis set. The structures and their energies are shown in the table below, along with their jmol files and important dihedral angles.

Results of Optimisations

	File:Pierre react gauche 1.jpg	File:Pierre react gauche 2 (2).jpg
Point Group	C2	C2	C1	C2	C1	C1
Energy (Hartrees)	-231.6877	-231.6915	-231.6927	-231.6917	-231.6896	-231.6892
Relative Energy (Hartrees)	0.0049	0.0011	0.0000	0.0010	0.0030	0.0035
Energy (kcal/mol)	-145386.22	-145388.61	-145389.32	-145388.70	-145387.41	-145387.12
Relative Energy (kcal/mol)	3.10	0.71	0.00	0.62	1.91	2.20
Central C-C Dihedral Angle	75.8	67.7	64.2	63.7	71.1	70.0

File:Pierre Anti conformers.png

	File:Pierre react anti (1).jpg	File:Pierre react anti 2 (2).jpg
Point Group	C2	Ci	C2h	C1
Energy (Hartrees)	-231.6926	-231.6925	-231.6891	-231.6910
Relative Energy (Hartrees)	0.0001	0.0001	0.0036	0.0017
Energy (kcal/mol)	-145389.28	-145389.24	-145387.07	-145388.26
Relative Energy (kcal/mol)	0.04	0.08	2.25	1.06
Central C-C Dihedral Angle	176.9	180.0	180.0	178.4

Conformers Ordered in Terms of Energy

	Energy (Hartrees)	Relative Energy (Hartrees)
Gauche 3	-231.6927	0.00000
Anti 1	-231.6926	0.00006
Anti 2	-231.6925	0.00013
Gauche 4	-231.6917	0.00099
Gauche 2	-231.6915	0.00113
Anti 4	-231.6910	0.00169
Gauche 5	-231.6896	0.00305
Gauche 6	-231.6892	0.00350
Anti 3	-231.6891	0.00359
Gauche 1	-231.6877	0.00495

From the HF/3-21g optimisations made, it can be seen that a conformer with a gauche arrangement of its 4 central carbon atoms is the lowest in energy, contrary to what was predicted. This suggests that the central C-C bond is not the only major factor in determining the energy of the conformer. Indeed, such results are seen in the literature, and are rationalised by B.W. Gung et al. as being due to an attractive interaction between the vinyl proton and the π-orbital. The anti 2 conformer was further optimised using the DFT method and the B3LYP/6-31g(d) basis set. Although the conformation the conformation changes only very slightly in terms of dihedral angles (see table below), the energy of the 6-31g(d) optimised structure is considerably reduced (ca. -1832 kcal/mol). This is mainly due to the different methods used, and comparing energies obtained using two different methods is irrelevant. The structure optimised to 6-31g(d) shows the vinyl groups slightly more eclipsed with the hydrogen syn-related to them, as shown by the 1→4 and 3→6 dihedral angles.

Anti 2 Conformation
Energy (Hartree)	-231.69254	-234.61170
Central C-C Dihedral angle	-179.99	180.00
1→4 Dihedral angle	114.67	118.53
3→6 Dihedral angle	-114.67	-118.53

Vibrational analysis of Anti 2 conformer

A vibrational analysis of the B3LYP/6-31g(d) optimised anti 2 conformer was performed, giving the spectrum shown here. Other than the C-H related vibrations, it is worth noting that the only C=C stretch which appears in the spectrum is the asymmetric stretch. This confirms the Ci symmetry of the molecule, as vibrations with a center of inversion are not IR active.

C=C stretches in the Anti 2 conformer

	Symmetric	Asymmetric
Frequency	1731 cm-1	1734 cm-1
Intensity	0	18

File:Pierre react anti 2 6-31g(d) spectrum upside down.JPG	File:Pierre 1,5 hexadiene spectrum.gif
IR spectrum of Anti 2 Conformer	IR spectrum of 1,5 hexadiene

When comparing the spectrum obtained (Note: the calculated spectrum is turned upside-down for easier comparaison with the experimental spectrum) using the anti 2 conformation to the experimental IR spectrum of 1,5 hexadiene, we see that the C=C stretches do not coincide. This may be attributed to the fact that the anti 2 conformation is not the major conformation is solution, since it is only the 3rd most stable conformation of 1,5 hexadiene. In order to verify this conclusion, an optimisation and vibrational analysis at the DFT/B3LYP/6-31g(d) level of the gauche 3 conformer was run. The results showed C=C stretches at 1732 and 1733 cm^-1, similar to the 1731 and 1734 cm^-1 frequencies obtained for the anti 2 conformation, which are still far from the observed frequencies of 1850 and 1642 cm^-1. Considering that the computed frequencies generally have a 10% error on them, the experimental frequencies are within the error margin, leading to the conclusion that, although far from perfect, the results are acceptable.

Components of Energy of Anti 2 conformer

The results from the vibrational analysis also provide us with a thermochemical analysis. From this we can find the sum of the electronic component of the energy with various thermodynamic properties, such as the zero point energy (ZPE). The data obtained from the anti 2 conformer is shown in the table below.

Thermodynamic Data

Property	Energy (Hartrees)	Energy from lit. (Hartrees)
Eelec + ZPE	-234.469212	-234.469203
Eelec + Ethem	-234.461856	-234.461856
Eelec + Hthem	-234.460912	-
Eelec + Gtherm	-234.500821	-

We can see from the table above that the calculated numbers coincide quite well with those provided in the experimental script. We also see that the energies computed here are all higher than the purely electronic energy of the molecule (-234.61170280 Hartrees), suggesting that each of these thermodynamic quantities go towards a destabilisation of the molecule.

Optimising Transition States

Chair Transition State - Regular Optimisation

Allyl Fragment optimised to the HF/3-21g level

In order to optimise a transition state, an initial guess must be made. In the case of a [3,3] sigmatropic rearrangement, an allyl fragment best describes one half of the chair transition state, so such a fragment was drawn in Gaussview and optimised to a HF/3-21g level (Energy = -115.82304 Hartrees). Two of these optimised fragments were pasted together, with the two terminal C-C of each fragment directly above/below each other, with a distance of 2.2 Å separating the two fragments, as shown in the picture. The chair transition state was optimised to a TS(Berny) and the frequency analysis calculated using the HF method and a 3-21g basis set. The force constants were calculated only once, and Opt=NoEigen was added as additional keywords. This changed the structure slightly, decreasing the distance between the terminal carbons of the allyl fragments to 2.02 Å, and increasing the angle between these carbons and the central carbon of the ally fragments from 90.8º to 101.9º.

Guess	HF/3-21g
File:Pierre chair ts guess.jpg

The optimised structure clearly resembles a chair conformation a lot more than the guess structure. Furthermore, the imaginary frequency obtained from the calculation (-818 cm^-1) shows atomic motion which are very similar to what one expects from the Cope rearrangement.

Chair Transition State - Redundant Coordinate Editor Optimisation

Using the same guess structure as before, the coordinates of the terminal carbon atoms were frozen, with an interatomic distance of 2.2 Å, and optimised to a TS(Berny) at a HF/3-21g level. In a similar way to the previous optimisation without freezing coordinates, the angle between two adjacent terminal carbons and a central carbon was increased from 90.8º to 99.62º. In this case the increase is smaller, but still considerable, and the resulting structure is close to the chair conformation as well. The transition state was further optimised, but this time the reaction coordinate was defined as being the bond between the terminal carbons, the extra keyword Opt=NoEigen was added to the input file, and the force constants were not calculated. The angle evoked earlier was further increased to 101.85º (i.e. the same as for the method used without the RCE option), and the distance between the adjacent terminal carbon atoms was reduced from 2.2 Å to 2.02 Å, which again corresponds to the same bond length as in the non RCE optimisation. The energy of the transition states obtained in the two methods also correspond (-231.61932246 Hartrees for the non-RCE method and -231.61932239 Hartrees for the RCE method). Even though the RCE method may seem like it is an extra hassle, optimisation of individual bonds may be necessary for more complex structures.

Chair Transition State - Following the Intrinsic Reaction Coordinate (IRC)

The intrinsic reaction coordinate method allows us to compute the strucutre of the molecule as we advance along the reaction coordinate. In this case, we used the chair transition state optimised using the RCE method, and performed the IRC calculation in the forward direction, calculating the force constants once only, and limiting the calculation to 50 steps. The calculation did not converge, so the number of steps were increased to 70, and again the calculation did not converge. Repeating the calculation by computing the force constants at every step led to a successful convergence. The resulting file shows a reaction coordinate with 47 steps, with the structure gradually evolving from the transition state down to what appears to be the gauche 2 conformation.

Boat Transition State - QST2 Method

The boat transition state was optimised by using a different method to the TS(Berny). In the QST2 method, the reactant and product are specified in different molgroup windows, and the calculation finds the most suitable transition state. In the case of this experiment, the product and reactant are identical, so care must be taken in the numbering of the atoms, since this is the only difference between the two. A first attempt at finding the transition state was made by performing an opt+freq QST2 type calculation at an HF/3-21g level on the anti 2 conformer, after having optimised the latter to an HF/3-21g level as well. However, in this calculation, the displacement components did not converge. The transition state resulting from this calculation resembles the chair transition state, with the exception that the distance between the allyl fragments is 3.51/3.45 Å. The conclusion drawn from this is that the structures submitted to Gaussian for the calculations are too far from the desired transition state. Therefore, the dihedral angles of the central C-C bond was modified to 110º, and the internal angles between C2-C3-C4 and C3-C4-C5 were changed to 100º. The same procedure was performed on the product molecule, and the calculation was submitted to Gaussian again. Once again, the calculation did not converge, but the outputted structure for the transition state is close to the twist boat conformation. In an effort to make the calculation converge (to get a frequency analysis), the anti 2 fragment was optimised to the B3LYP/6-31g(d) level, its angles modified as before, and the QST2 calculation was run. This time, the calculation converged, giving a transition state with a boat conformation, and an imaginary vibration at -840 cm-1 which, as seen below, illustrates the bond forming/breaking nature of the Cope rearrangement. The distance between the two allyl-like fragments is approximately 2.14 Å, and its energy is -231.60280 Hartrees.

	File:Pierre Boat ts (twist).jpg

Imaginary Frequency at -840 cm^-1

File:Pierre -840 vibration (boat).gif

Boat Transition State - Following the Intrinsic Reaction Coordinate (IRC)

As for the chair transition state, the IRC for the boat transition state (previously optimised using the QST2 method) was computed. As for the chair TS, the reaction coordinate was followed only in the forward direction, over 50 steps, calculating the force constants at every point. The result is shown below. We can see that as the reaction proceeds, two ends of the allyl fragments come closer together, and the other two move further apart. We also see the 1,5 related unsaturations being formed quite early in the reaction, and the latter stages involve optimisation of the energy, via the rotation of the terminal unsaturations. The final strucuture shown in the IRC is a syn-periplanar conformer, which we know from our previous studies is not the lowest in energy. We can therefore predict that the conformer will most likely rotate so as to form lower energy conformers (eg. gauche or anti conformers).

Summary

In cyclohexane, the boat conformation is higher in energy than the chair conformation because all its groups are eclipsing each other. Furthermore, the hydrogens in the so-called ‘flagstaff’ position clash sterically, making the structure approximately 6 kcal/mol higher in energy than the chair conformation. Due to the similarities in the structure of the transition states and cyclohexane, it can safely be said that the boat TS will be higher in energy than the chair TS. This is indeed what is observed, since at 0 K, the difference in energy between the chair and the boat TS is: 7.899 kcal/mol (B3LYP/6-31g(d)). This is relatively close to the experimental difference, and suggests that the B3LYP/6-31g(d) model is quite accurate. A further comparaison with experimental data is shown in the table below.

Summary of Energies (kcal/mol)
	HF/3-21g			B3LYP/6-31g(d)
	Eelec (Hartrees)	Eelec+ZPE (Hartrees)	Eelec+Etherm (Hartrees)	Eelec (Hartrees)	Eelec+ZPE (Hartrees)	Eelec+Etherm (Hartrees)
Chair TS	-231.619322	-231.466700	-231.461341	-234.556984	-234.414929	-234.409008
Boat TS	-231.602802	-231.450928	-231.445298	-234.593093	-234.402341	-234.396006
Reactant Anti 2	-231.692535	-231.539540	-231.532566	-234.611703	-234.469212	-234.461856

1 Hartee = 627.509 kcal/mol

Summary of Activation Energies (kcal/mol)
	HF/3-21g		B3LYP/6-31g(d)		Experimental
	at 0 K	at 298.15 K	at 0 K	at 298.15 K	at 0 K
Ea Chair	45.71	49.07	34.06	33.16	33.5 ± 0.5
Ea Boat	55.60	59.14	41.96	41.32	44.7 ± 2.0

Overall, the use of computational methods to predict the transition states and corresponding activation energies is justified, seeing as the results are in good agreement with the experimental values. This is especially true for the B3LYP/6-31g optimised structures, which, taking into account the experimental error, come within 1 kcal/mol of the experimental values. Optimisation at a higher level (B3LYP/6-31g instead of HF/2-21g) shows a clear improvement in the accuracy of the activation energies.