Semiclassical perturbation theory of electron-molecule collisions

Abstract

The theory of the classical or semiclassical $S$ matrix is combined with the use of perturbation dynamics to derive an approximately unitary expression for the scattering matrix for a very general class of potential interactions. The $S$ matrix takes the form of a sum over products of Bessel functions whose orders are related to the changes in quantum numbers occurring in the transition, and whose arguments depend on the dynamical variables of the problem, including the unperturbed quantum numbers. In the general case, these arguments can be expressed as simple integrals over the unperturbed trajectory, and for well-behaved potentials they can be explicitly evaluated in terms of the modified Bessel functions $K_0$ and $K_1$. The connection between the semiclassical perturbation scattering theory and other approximations, such as the Born and eikonal approximations, is demonstrated. The general theory is illustrated by applications to electron- (or ion-) polar-molecule scattering, including quadrupole as well as dipole interactions and including coupling to vibrations in both harmonic and anharmonic approximations. The more complicated interactions involve lengthier products of Bessel functions in the sum-and-product representation, but these are easily and systematically evaluated, and they reduce smoothly to the appropriate simpler expressions when the coupling coefficients of the higher-order terms become small. More complicated potentials, including interactions between polyatomic molecules, can be handled by a simple systematic extension of the same principles. For electron-molecule scattering, these expressions can be used in their present form since the sums are dominated by Bessel functions of comparatively low order which can be evaluated directly; extensions to molecule-molecule scattering and ion-molecule scattering are equally valid formally, but their practical application will often require the use of asymptotic approximations to the Bessel functions.