Studies of inelastic molecular collisions using impact parameter methods. I. Model calculations

In this paper we explore the behavior of an exponential approximation to the impact parameter amplitudes for model systems with potential matrices of simple form. In these matrices the magnitude of the coupling is governed by the difference in the quantum numbers of the coupled states. In the limit of large numbers of states the amplitude functions may be expressed as single integrals, which in many cases can be evaluated in terms of known mathematical functions. The simple structure of this result allows us to analyze the general features of the amplitude functions and the resulting cross section integrands versus impact parameter. Specific examples for potentials with Gaussian and exponential radial dependence are used for illustration. An exploration of the effect of varying the potential parameters for the Gaussian potential is made. In addition, the behavior of functions related to the generalized cross sections for spectral line broadening are considered. Finally, we explore the generalization of the simple model to more elaborate coupling schemes.