Impact parameter methods with effective potentials for inelastic molecular collisions

Previously employed exact quantum impact parameter methods for elastic scattering are extended to inelastic scattering. This is done within the framework of a recently developed effective potential model for the interaction of two molecules. The effective potential V^eff and the transition matrix T are represented as integrals over a Bessel function times corresponding impact parameter amplitudes. Three particular impact parameter representations of the effective potential are discussed. It is shown that the form of the equation of motion for the T-matrix amplitude depends on the representation of the potential. This presents no difficulty once a particular choice of potential representation is made. A convenient characteristic wave number κ is introduced into the formulation, and it is shown that the resulting T-matrix is invariant with respect to its magnitude. The choice of κ is then argued on physical grounds. Several approximate equations of motion and their solutions are developed for the T-matrix amplitude. Some specific computational considerations are also discussed in conjunction with this development.