Impact-parameter method for proton-hydrogen-atom scattering variational bounds on the1scharge-exchange amplitudes

Abstract

A trial wave function that is a linear combination of two traveling hydrogenlike basis functions which contain two variable-charge parameters and a polarization parameter has been used to obtain bounds on the second-order error term in the variational principles of Demkov and Storm that are comparable to the magnitudes of approximate $1s$ charge-exchange amplitudes. It is demonstrated that the error function ${\ensuremath{\Delta}}_{1}({X}_{i})$ can be employed to judge various calculations and as an aid in adjusting parameters in the trial wave function to obtain better bounds. Based upon the use of ${\ensuremath{\Delta}}_{1}$ as a measure of error in an approximate trial wave function, we conclude that the Euler-Lagrange variational method is not the optimal approach. Suggestions are made for future work which might improve the calculated bounds.