Catastrophes and molecular collisions

Abstract

Catastrophe theory is applied to the asymptotic evaluation of integrals that occur in semiclassical collision theory with special reference to Thom's seven elementary catastrophes. It is shown how the topological structure of a family of classical trajectories can determine the caustics and canonical integrals of semiclassical collision theory. For the elementary catastrophes, the canonical integrals are of two types: one-dimensional cuspoids, and the two-dimensional umbilics which are non-separable in general. With the help of Morse's Lemma, the elementary catastrophes can also occur in the asymptotic evaluation of higher dimensional integrals. The uniform approximations are used to calculate the index of a catastrophe. The index is a measure of the order in ħ of a uniform approximation near a caustic. The hyperbolic umbilic is also used to discuss the validity of some semi-classical formulae currently used in the theory of atom-surface and atom-molecule collisions.