Semiclassical theory of molecular collisions: Many nearly coincident classical trajectories

Abstract

The uniform asymptotic evaluation of integral representations for S-matrix elements and scattering amplitudes in semiclassical collision theory is considered. A uniform approximation for a one-dimensional integral with many coalescing saddle-points is derived by applying the asymptotic techniques of Chester et al. and Ursell. Physically, each saddle-point is associated with a real or complex-valued classical trajectory. A concrete example of an integral with four coalescing classical trajectories whose positions depend on two parameters is discussed qualitatively to provide motivation for the calculations. The uniform approximation is expressed in terms of a canonical integral and its derivatives. The topological structure of the classical trajectories determines the canonical integral. A series representation is derived for the canonical integral. The series repesentation can be used to evaluate the canonical integral for small to moderate values of its arguments. The extension of the one-dimensional uniform approximation to multidimensional integrals is also considered. The results in this paper generalize and unify previous calculations on two and three nearly coincident classical trajectories.