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.
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ACKNOWLEDGEMENT The authors acknowledge William H. Miller for helpful correspondence and David W. Schwenke for helpful conversations. This work was supported in part by the National Science Foundation under grants nos. CHE86-00363 and CHE86-17063 and by the R. A. Welch Foundation under Grant E-608.