Quantum number and energy scaling for nonreactive collisions

Two new theoretical developments are presented in this article. First an energy corrected sudden (ECS) approximation is derived by explicitly incorporating both the internal energy level spacing and the finite collision duration into the sudden S-matrix. An application of this ECS approximation to the calculation of rotationally inelastic cross sections is shown to yield accurate results for the H⁺-CN system. Second, a quantum number and energy scaling relationship for nonreactive S-matrix elements is derived based on the ECS method. A few detailed illustrations are presented and scaling predictions are compared to exact results for R-T, V-T, and V-R, T processes in various atom-molecule systems. The agreement is uniformly very good -even when the sudden approximation is inaccurate. An important result occurs in the analysis of V-T processes: the effects of anharmonic wavefunctions (coupling) and decreasing vibrational energy gaps (energetics) are separated. Each factor makes significant contributions to the deviation of the anharmonic from the harmonic scaling relationship.