Semiclassical Theory of Vibrationally Inelastic Scattering in Three Dimensions

A hierarchy of semiclassical approximations is derived to treat vibrationally inelastic scattering in three dimensions. The radial wavefunctions of the coupled Schrödinger equations for the combined vibrationally and rotationally inelastic scattering are expressed in terms of the radial wavefunctions for elastic scattering from a spherically symmetric potential to ``remove'' the elastic scattering from the problem. The rotationally inelastic scattering is diagonalized using the sudden approximation. The orbital angular momentum is diagonalized using a classical action‐angle transformation. The result is a simple set of coupled differential equations—one for each vibrational state. For small changes in relative energy we obtain the coupled equations for a time‐dependent perturbation acting along a single classical trajectory. If the energy changes can be neglected, the sudden approximation for the vibrationally inelastic scattering is obtained. For large changes in energy a multitrajectory method is developed which uses a separate trajectory for each vibrational state. Using the sudden approximation, vibrationally inelastic transition probabilities are calculated for $\mathrm{He}+{\mathrm{H}}_2$ as a function of molecular orientations and impact parameters. Orientation‐averaged cross sections cannot be obtained from collinear trajectories by a simple steric factor independent of energy and the specific transition. It is also shown that, in the sudden limit, the orientation‐averaged differential cross section is approximately equal to the differential cross section for the orientation‐averaged potential.