A comparative analysis of variational methods for inelastic and reactive scattering

In this paper we discuss recent developments in algebraic variational calculations of quantum mechanical amplitudes for rearrangement scattering in terms of a scattered wave variational principle (SWVP) equivalent to that of Schlessinger. We demonstrate that the variational functional of the generalized Newton variational principle (GNVP) involving the L² amplitude density can be interpreted in terms of the scattered wave, and we show that the functional is variationally correct to second order both in the error in the amplitude density and in the error in the scattered wave. This provides a way to unify recent treatments of rearrangement scattering and it suggests new choices for the basis functions in both the GNVP and the SWVP. The success of the GNVP suggests that an efficient basis set for the SWVP should include the half integrated Green's functions (HIGF) introduced earlier in the context of the GNVP. By using the formal relation between the HIGFs and the L² basis functions used in the GNVP, we may incorporate various physical effects either into the L² basis functions used to expand the L² amplitude density or into the HIGF used to expand the non-L² scattered wave. We show how these ideas can be used to make many of the matrix elements independent of energy, while at the same time retaining the convenience that most basis functions are L² and also retaining excellent convergence properties found in our earlier GNVP studies.