A direct method for determining moments of final state distributions in molecular collisions

Abstract

A method is developed for directly determining the energy moments of final state distributions in molecular collisions. The method involves the solution of a coupled master-like equation for moments (of a chosen order) corresponding to different possible initial conditions (states) of the collision system. This direct moment equation is derived from exact quantum mechanics through the application of the generalized cumulant expansion approach previously developed. Two levels of treatment are considered : one in which all molecular degrees of freedom are allowed to be fully correlated, with the only approximation on the dynamics being the truncated cumulant expansion ; and a second where only a subset of the complete set of degrees of freedom is treated by the first level of treatment, and effects due to motions of the remaining degrees of freedom are averaged over. A classical mechanical direct moment equation is also derived.

Applications of the direct moment (DM) method using a classical impulsive approximation for translational motion are made to rotationally inelastic scattering in He + H₂ and He + CO₂. For He + H₂, good agreement (errors less than 20 per cent) between DM and close coupling (CC) energy transfers and standard deviations is found at translational energies of 0·4 and 0·9 eV and all partial waves J. In addition, the direct moment equation results are found to agree well with those obtained using transition probabilities from a master equation also derived using truncated cumulant expansions. For He + CO₂, the DM energy transfers are less quantitative (errors of 40–50 per cent), apparently because of differences between the manner in which CC and DM results approach statistical limits (the first oscillates and the second varies smoothly).

The computational efficiency of the direct moment method is assessed, and a major advantage of this approach seems to be its ability to provide a few low-order moments with usually much less computational effort than is needed to obtain the corresponding moments by summing transition probabilities from master equation, coupled states or coupled channel methods.