Head and Tail Deformations, Torsional Coriolis Coupling, and E1d-E2d Vibrational Mixing in Ethane-Like Molecules

Abstract

The mechanism of torsional Coriolis interaction of E1d and E2d vibrational modes in ethane-like molecules is investigated, and it is shown that this coupling can drastically affect the torsional splitting in the degenerate vibrational states. A basic point of our treatment is that the sets of coordinates of head and tail which combine with the + sign to generate E1d normal coordinates are in general different from those which combine with the − sign to generate E2d normal coordinates. It is shown that the ζγ torsional Coriolis coefficients calculated by the usual methods of normal mode analysis are related to the vibrational angular momenta within head and tail referred to the internal rotor axis systems. With knowledge of the L and L−1 matrices it is possible to transform these coefficients for reference to the molecule-fixed frame. It is peculiar that torsional Coriolis matrix elements occur between E1d and E2d vibrational components with the same x or y orientation in the molecule-fixed frame. The matrix elements of the torsional Coriolis operator and other operators responsible for the end-to-end coupling are determined, and a method for calculating vibration–torsion energies, and then torsional splittings, in degenerate vibrational states is outlined. Detailed calculations require a global model, involving all the degenerate vibrational basis states in a complex mechanism of interactions, but it is shown that useful information can be obtained by means of simplified models. Our semiempirical rule that degenerate vibrational states with a large negative value of the diagonal vibration–rotation Coriolis coefficient are likely to deviate much from the behavior of E1d or E2d vibrational states, with a sensible decrease of the torsional splittings, is confirmed.