Semiclassical theory of molecular collisions : three nearly coincident classical trajectories

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Abstract
The uniform asymptotic evaluation of integrals in semiclassical collision theory is considered. A uniform asymptotic approximation for a one-dimensional integral with three nearly coincident classical trajectories is derived by applying the method of Ursell. These integrals arise in the theory of rainbows in the elastic scattering of chemically reactive systems and in the theory of the collinear electronically adiabatic H + H2 chemical reaction. The uniform approximation is expressed in terms of a canonical integral and its derivatives. An exact series representation is obtained for the canonical integral. It is shown how the one-dimensional uniform approximation can be generalized to n-dimensional integrals. This is achieved by evaluating n-1 of the integrals by the ordinary saddle-point method and then applying the uniform method of Ursell to the remaining one.