Path integral approach to methyl group rotation

Abstract

Methyl group rotation is a simple one-dimensional example of the phenomenon of molecular rotation tunnelling, which is in turn a special case of the more general physical problem of barrier penetration. A real-time path integral formulation of a methyl group interacting with a lattice ('the environment') is presented. The generalised Langevin equation (Haken, 1975) and influence functional theory (Feynman and Vernon, 1966) are used to distinguish between the zero- and finite-temperature effects of the interaction. The finite-temperature effects are treated by eliminating the environment coordinates to obtain the reduced Green function. The resulting path integrals are evaluated in the long-time limit using a classical-path approximation to obtain results for the shifting and broadening of the ground torsional energy levels and hence the ground tunnel splitting. In the low-temperature limit the results obtained are the same as those obtained by Hewson (1982). In the high-temperature limit the results obtained agree with the predictions of a classical hopping model. The use of the same theoretical framework in both regimes provides an understanding of the transition from quantum mechanical to classical behaviour as the temperature is increased.