Mod:kwlittle2

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Cope Rearrangement

Optimising reactants and products

In Gaussview, a molecule of 1,5-hexadiene in the 'anti' conformation was drawn by placing an ethyl fragment and replacing anti-periplanar hydrogen atoms with vinyl fragments, then optimised using the Hartree-Fock method, 3-21G basis set[1]. A 'gauche' molecule was drawn by replacing a benzene ring fragment's atoms with methyl and vinyl fragmets, then optimised[2].

I predicted that the lowest-energy conformer would be similar to the gauche conformer but with the central dihedral angle at 180 ('anti3' in the appendix table); both this and 'anti2' were drawn and optimised[3][4]. The energies and symmetries of all these molecules are:

Conformation Point Group Energy / au
gauche C2 -231.68771613
anti C2h -231.69253525
anti2 Ci -231.69253529
anti3 C2h -231.68907062

The anti2 molecule was re-optimised[5] using the B3LYP method and 6-31G* basis set, to give an energy of -234.61171035 au and slightly larger C-C-C angles. This optimised molecule was submitted for frequency analysis at STP[6] and at absolute zero[7] (actually at 0.01K, as Gaussian would not calculate 0K) to give specific energy terms:

Energies of 'anti2' 1,5-hexadiene in Hartrees
Energy type 298.15K 0K
Sum of electronic and zero-point Energies -234.469204 -234.469204
Sum of electronic and thermal Energies -234.461857 -234.469203
Sum of electronic and thermal Enthalpies -234.460913 -234.469203
Sum of electronic and thermal Free Energies -234.500777 -234.469203

Gaussian output files

  1. ↑ File:ANTIHF.LOG
  2. ↑ File:GAUCHEHF.LOG
  3. ↑ File:ANTI3HF.LOG
  4. ↑ File:ANTI2HF.LOG
  5. ↑ File:ANTI26-31G.LOG
  6. ↑ File:ANTI2FREQ.LOG
  7. ↑ File:ANTI2FREQ0K.LOG

Optimising the chair and boat transition structures

The chair transition state.
The boat transition state.

Chair

An allyl fragment was drawn in GaussView and optimised with the HF/3-21G method[1]. This was duplicated, and one of the fragments rotated to make an approximation of the chair transition state. The true transition structure was found from this via two different methods, each using the HF/3-21G method:

First, the approximate structure was simply optimised to a Berny Transition State with the HF/3-21G method[2], calculating force constants once, to give a molecule with a single imaginary vibration a -818cm-1.

The approximate structure was also optimised to a minimum, but with modified 'redundant coordinates'. Two coordinate systems were set: each with two carbon atoms at the end of either allyl fragment as a 'Frozen Bond' coordinate, and 'opt=ModRedundant' was added to the keywords. This failed initially as only a single frozen coordinate was set, making an asymmetrical molecule. This was corrected to give a true optimised molecule[3]. This was then optimised to a transition state[4] without calculating force constants, but with the redundant coordinates changed to 'Bond Add' type.

These two methods produce equivalent transition structures, both with forming/breaking bond lengths of 2.02A and C-C bond lengths of 1.39A.

Boat

The boat structure was found by the QST2 method, which interpolates between a reactant and product. In this case, the two are the same but with different atom numbering representing the changed bond structure. The optimised 'anti2' conformer was opened, then copied and pasted as a second molecule in the same group. The atom labelling of the second molecule was changed, and the group submitted for optimisation to a QST2 transition state with HF/3-21G method[5].

This first optimisation failed, giving a structure similar to the chair transition state. The original molecule group was opened, then the central dihedral angle of each molecule changed to 0 and the central C-C-C angles changed to 100, making them similar to the 'gauche2' conformer. This was optimised but failed, giving a twist-boat-like structure[6]. The optimisation was repeated, this time calculating force constants once, which went to completion to produce the boat transition state[7].

Intrinsic reaction coordinate

By inspection, the chair transition structure looks as if it will form the gauche conformer of 1,5-hexadiene. This was tested by the Intrinsic Reaction Coordinate method, which follows the reaction from the transition state to its products or reactants. The chair structure as calculated by the TS(Berny) force constants method was opened and submitted for IRC, calculating force constants once and setting the maximum cycles to 100[8]. This calculation failed, stating "Maximum number of corrector steps exceded"; however, the final molecule produced looked close to the gauche2 conformer.

This was confirmed in two ways: by optimising this outputted molecule to a minimum[9] at the B3LYP/6-31G* level, and also by repeating the IRC method, calculating force constants at each step[10] and then optimising the final molecule as above[11]. Both methods produced equivalent gauche2 conformers.

Activation energies

The activation energies of the reaction for the anti2 conformer were calculated by taking the difference between its energy and that of each transition state at both the HF/3-21G and B3LYP/6-31G* levels of theory.

All energies in Hartrees
Theory anti2 chair Ea(chair) boat Ea(boat)
HF/3-21G -231.69253529 -231.61932232 0.073213 -231.60280203 0.089733
HF/3-21G (electronic and thermal free energies) -231.570913 -231.495206 0.075707 -231.479773 0.09114
B3LYP/6-31G* -234.61171035 -234.55698303 0.054727 -234.54309307 0.068617
B3LYP/6-31G* (electronic and thermal free energies) -234.500777 -234.443814 0.056963 -234.431752 0.069025

These activation energies in kcal/mol are:

Theory Ea(chair) Ea(boat)
HF/3-21G 0K 45.94183193 56.30847033
HF/3-21G 298.15K 45.94183193 57.19121300
B3LYP/6-31G* 0K 34.34191151 43.05799294
B3LYP/6-31G* 298.15K 35.74482188 43.31384110
Experimental 33.5 ± 0.5 44.7 ± 2.0

Thus, the computationally-calculated energies closest to experiment are those using the B3LYP method, 6-31G* basis set and with thermal corrections.

Gaussian output files

  1. ↑ File:ALLYL.LOG
  2. ↑ File:CHAIRFORCE.LOG
  3. ↑ File:CHAIRFROZEN3.LOG
  4. ↑ File:CHAIRFROZENTS.LOG
  5. ↑ File:CHAIRQST2.LOG
  6. ↑ File:CHAIRQST2MOD.LOG
  7. ↑ File:CHAIRQST2MOD3.LOG
  8. ↑ File:CHAIRIRCONCE.LOG
  9. ↑ File:CHAIRIRCOPT.LOG
  10. ↑ File:CHAIRIRCALWAYS.LOG
  11. ↑ File:CHAIRIRCALWAYSOPT.LOG

The Diels-Alder cycloaddition

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