An Approach to the Internal Rotation Problem

Abstract

A method is described for a direct calculation of the barrier to internal rotation in ethane and ethanelike molecules. The approach suggested should identify the physical origin of the barrier, provided a satisfactory account would be forthcoming from exact Hartree—Fock calculations of the energies of the staggered and eclipsed forms in the Born—Oppenheimer approximation. A simple model is considered in which the electronic distribution is taken to be cylindrically symmetric about the central bond. It is demonstrated that this model predicts energy differences between staggered and eclipsed forms, composed solely of nuclear—nuclear repulsion energy, that are consistently about 5/3 of the experimental energy barriers. A slight expansion of the wavefunction in the eclipsed configuration is shown to be required by the virial theorem, but this does not appreciably change the predicted magnitude of the barrier. It is proposed that the model can be refined by a calculation of the electronic energies of the actual staggered and eclipsed forms of ethane relative to that for the cylindrical distribution. By means of second-order Hartree—Fock perturbation theory, it is shown that the energy differences are a simple sum of contributions from the nine occupied molecular orbitals. The individual contributions can be calculated by the solution of certain differential equations or their variational equivalent. To simplify the treatment further, an approximate set of uncoupled differential equations is suggested for finding the perturbed functions from estimates for the charge densities of the cylindrical-model molecular orbitals.