Semiclassical theory of nonadiabatic transitions in a two-state exponential model

A general two-state exponential potential model is solved with use of the Bessel transformation and the WKB (Wentzel-Kramers-Brillouin) type semiclassical approximation. Accurate expressions are obtained for the nonadiabatic transition probability for one passage of the transition point and for the two dynamical phases. Functionalities of these quantities in terms of two basic parameters are the same as those obtained before by Nikitin. The two basic parameters are, however, expressed in more general and accurate forms. Accuracies of these expressions are numerically confirmed. The three quantities, the nonadiabatic transition probability and the two dynamical phases, constitute the nonadiabatic transition matrix and can be used to describe various (spectroscopic as well as scattering) processes not only for a two-state but also for a multichannel system. A possible generalization of the present theory is also briefly discussed to formulate a unified theory that can cover both Landau-Zener-Stueckelberg and Rosen-Zener-Demkov cases within the adiabatic state representation.