An investigation of the exponential atomic collision model: analytical results

Abstract

A simple, but versatile, atomic collision model for treating non-adiabatic phenomena associated with curve-crossings and non-crossings is the exponential model of Nikitin. This two-state model leads, within a semiclassical framework, to a collision S -matrix made up of classical trajectory transition probabilities and semiclassical phases. It can be used to discuss physical processes such as excitation and charge transfer. In applying the model expressions to real situations, phase-integral expressions for the S -matrix elements are needed. Such expressions have been obtained earlier by Crothers through a process of interpretation and abstraction. To shed new light on this interpretation we give a phase-integral derivation of the two-state semiclassical collision S -matrix within a general exponential model. Only analytical results are presented here; numerical ones will be reported in future publications. Representing the S -matrix as a product of two ‘half-way house’ matrices and assuming only one principal transition zone in each half we may parametrize the elements of S in a semiclassically consistent way. The parameters are computed in terms of certain Stokes constants and complex Coulomb phases arising from the classical trajectory equations describing the evolution of the diabatic (l. c. a. o.) electronic states. The Stokes constants and complex Coulomb phases associated with the canonical (pure exponential model) one-pole, two-transition point problem are parametrized via the comparison equation method, supplemented by strong-coupling asymptotics. The corresponding quantities for a perturbed canonical form (general exponential model) are suitably abstracted in terms of simple known functions and physically significant phase integrals. In this way the complete semiclassical collision S -matrix is derived. A short discussion of its applicability is given.