Microscopic reversibility and complex-valued trajectories in a semiclassical theory of molecular collisions

The role of microscopic reversibility and time reversal in a semiclassical theory of molecular collisions is discussed. The boundary conditions for time-reversed classical trajectories are written in terms of those for forward trajectories in a manner consistent with microscopic reversibility. Special care must be taken in calculating time-reversed complex-valued trajectories when branch points exist in the complex time plane. Such points can arise from an explicit branch point structure of the potential energy function, such as in a semiclassical treatment of electronic transitions, and/or from the branch point structure of the numerically computed equations of motion. If a forward trajectory crosses a branch cut in the upper (lower) half time plane, the time-reversed trajectory must cross the corresponding branch cut in the lower (upper) half time-plane.