The classical path approximation in time-dependent quantum collision theory

Time-dependent treatments of molecular collisions have frequently employed the combination of quantum equations for the internal degrees of freedom (i.e., vibration, rotation) with a classical description of the translational motion. In this paper, it is shown how energy-conserving classical path equations may be derived from first principles, thereby obtaining quantum correction terms of arbitrary order. These expressions have interesting implications for the question of how the classical limit is approached. If the translational degree of freedom is assumed to be described by a very narrow wave packet, the correction term to first order in ℏ is purely imaginary and therefore non-Hermitian. These techniques can be generalized to a classical treatment of other degrees of freedom, and the case of rotation in an atom-vibrotor collision is explicitly considered. For both the translational and rotational examples, the full series of corrections takes on an interesting and suggestive exponential form. The failure of classical path methods to satisfy microscopic reversibility is ascribed to a difficulty with the boundary conditions for the translational motion.