Some Aspects of the Quantal and Semiclassical Calculation of Phase Shifts and Cross Sections for Molecular Scattering and Transport

Abstract

A critical study is made of the JWKB approximation for phase shifts by comparison of the approximate phases with those calculated by numerical solution of the radial wave equation. A region of E—l—Λ* space is mapped out for the Lennard-Jones (12–6) potential in which the JWKB approximation is unsatisfactory. Reasons for the failure of the JWKB approximation are noted, and a simple but effective method of predicting the region of E—l—Λ* space where failure occurs is suggested. The study is continued by a comparison of the quantal and semiclassical transport and total cross sections. It is shown that the JWKB transport cross sections are of limited practical value and that the semiclassical total cross sections are useful only at high values of E and low values of Λ*. Particular attention is centered on features associated with classical orbiting and with semiclassical rainbow and glory scattering. An apparent sign change in the quantum corrections for the viscosity cross section found by de Boer and Bird is shown to be an artifact caused by the neglect of the attractive part of the interaction potential. Finally, small-angle quantal differential cross sections for a repulsive r—12 potential are compared with some recent semiclassical calculations. The behavior as exp (—cχ2) at small angles is found to be valid to larger angles than previously thought.