Unitary approximations to transition probabilities: Generalized impact parameter theory and the time dependent projection operator method

A general unitary approximation scheme is presented for transition probabilities based upon the application of the time dependent version of the Zwanzig–Feshbach projection operator method to the generalized impact parameter approximation. The overall approximation scheme is based upon the time dependent picture of the binary collision process. To begin with, the translational and internal motions are decoupled with the translational motion being treated in one of three ways, namely, as a straight line trajectory, as a single curved reference classical trajectory, or as a collection of such trajectories, each of which is associated with a pair of internal states. The internal dynamics is then parameterized by the set of given trajectories which are functions of the impact parameter. This motion is then formally solved using the Zwanzig–Feshbach projection operator method. A set of unitary approximations to these solutions are then presented, based upon a time disordered approximation. These latter approximations are applicable to each of the three treatments of the translational motion. As well, the basis used to describe the internal degrees of freedom may be either atomic or molecular (diabatic or adiabatic). Thus this combined approximation scheme provides a systematic theory for testing various effects associated with either the treatment of the translational motion or the internal motion.