Impact-Parameter Method for Proton—Hydrogen - Atom Collisions. II. The Coupled-State Calculation, and Convergence of the Two - Centered Traveling Hydrogenic Expansion

Abstract

Several calculational procedures for generating approximate solutions to the time-dependent Schr\"odinger equation of the impact-parameter model are discussed in this work. It is emphasized that in these calculational procedures a trial wave function is selected from a definite class of functions the class of functions which have the asymptotic form of the exact wave function and then some property satisfied by the exact wave function is used to fix the trial wave function. The customary set of coupled equations can be obtained by requiring that the Schr\"odinger equation be satisfied in the subspace defined by the basis set used to express the trial wave function. Recently, it has been popular to show that these coupled equations follow by making certain functionals stationary. It is shown, however, that these functionals are not stationary about the exact solution for the variations represented by available trial wave functions which, as they must, reflect our ignorance of the exact nature of the true wave function. In this work, a functional related to the norm of the exact wave function is given, and this functional is shown to be stationary about the exact wave function for the variations represented by available trial wave functions. A calculational procedure based upon the stationary property of this functional is given by which variational parameters in a trial wave function can be determined. If these parameters are taken to be time-varying nuclear charges in the basis functions, one obtains the equations given by Cheshire. Finally, the question of convergence of the two-centered traveling hydrogenic orbital expansion is considered. We conclude that the two-centered expansion does not provide a scheme by which a sequence of approximate wave functions can be generated that converges to the exact wave function. However, it appears from the comparison of the approximate cross sections obtained in two-, four-, eight-, and 14-state calculations that the cross sections for the $1s$ exchange reaction and possibly for the $2s$ transitions may be tending to limiting values.