Phase-index problem in the semiclassical description of molecular collisions

Semiclassical approaches of molecular collisions as proposed by Miller and Marcus involve phase indices, related to focal and turning points along trajectories contributing to S matrix elements. The main purpose of the work is to revisit the previous approaches in the case of two degrees of freedom rotationally inelastic collisions so as to make phase indices explicitly appear from first principles. Classical S matrix theory (CSMT) and three semiclassical initial value representation (SCIVR) treatments, respectively involving simple, double, and triple integrals, are considered. The phase index is either the Maslov index of the classical configuration space Green function (CSMT and the first two SCIVR methods), or the Maslov index of the Van Vleck-Gutzwiller space-time propagator (third SCIVR method). In order to assess the validity of the four previous approaches, their predictions are compared with exact quantum scattering results for interaction potentials leading to strong quantum interferences. The Gaussian weighting procedure, recently introduced in the quasiclassical trajectory method, is used here for practical CSMT calculations. We finally discuss the standard application of CSMT in the light of the previous developments and results.