Forward and inverse functional variations in rotationally inelastic scattering

Abstract

This paper considers the response of various rotational energy transfer processes to functional variations about an assumed model intermolecular potential. Attention is focused on the scattering of an atom and a linear rigid rotor. The collision dynamics are approximated by employing both the infinite order sudden (IOS) and exponential distorted wave (EDW) methods to describe Ar–N2 and He–H2, respectively. The following cross sections are considered: state-to-state differential and integral, final state summed differential and integral, and effective diffusion and viscosity cross sections. Attention is first given to the forward sensitivity densities δ0/δV(R,r) where 0 denotes any of the aforementioned cross sections, R is the intermolecular distance, and r is the internal coordinates. These forward sensitivity densities (functional derivatives) offer a quantitative measure of the importance of different regions of the potential surface to a chosen cross section. Via knowledge of the forward sensitivities and a particular variation δV(R,r) the concomitant response δ0 is generated. It was found that locally a variation in the potential can give rise to a large response in the cross sections as measured by these forward densities. In contrast, a unit percent change in the overall potential produced a 1%–10% change in the cross sections studied indicating that the large + and − responses to local variations tend to cancel. In addition, inverse sensitivity densities δV(R,r)/δ0 are obtained. These inverse densities are of interest since they are the exact solution to the infinitesimal inverse scattering problem. Although the inverse sensitivity densities do not in themselves form an inversion algorithm, they do offer a quantitative measure of the importance of performing particular measurements for the ultimate purpose of inversion. Using a set of state-to-state integral cross sections we found that the resultant responses from the infinitesimal inversion were typically small such that ‖δV(R,r)‖≪‖V(R,r)‖. From the viewpoint of an actual inversion, these results indicate that only through an extensive effort will significant knowledge of the potential be gained from the cross sections. All of these calculations serve to illustrate the methodology, and other observables as well as dynamical schemes could be explored as desired.