An analytical preaveraging approximation for the evaluation of matrix elements

Abstract

A method is developed for the rapid numerical evaluation of matrix elements which may be averaged in order to approximately account for explicit uncertainties in the integrand caused by, for example, uncertainties in the interaction potential. The exact matrix element average is approximated by an analytically preaveraged integrand which selectively filters out high-frequency, long-range components of the original integrand. Results of this approximation are compared with the original matrix elements and their exact averages over these uncertainties. The analytical preaveraging approximation (APA) is applied first to a simple, analytically soluble integral and is then extended to treat inelastic matrix elements which utilize more complicated (and realistic) uniformized JWKB wavefunctions. APA matrix elements are computed for vibrationally inelastic transitions in Ar-N 2 scattering and are compared with exactly averaged results.