Variational formulation of the coupled-state impact-parameter method

In the coupled-state impact-parameter method, the exact impact-parameter wave function is approximated by a trial impact-parameter wave function which is a finite linear combination of $N$ basis vectors with time-dependent coefficients. In the standard approach the coefficients are determined by solving the equations obtained by projecting the impact-parameter Schrödinger equation with the exact wave function replaced by the trial wave function onto the $N$ basis vectors which define the trial wave function. It is well known that this method gives variational estimates of the transition amplitudes if the basis vectors represent physical states. However, if in proton-hydrogen-atom scattering Sturmian basis vectors, which are for the most part nonphysical basis vectors, are used, the method is not variational. In this paper it is shown that if Sturmian basis vectors are used the method can be made variational by projecting the Schrödinger equation with the exact wave function replaced by the trial wave function onto the physical basis vectors of interest rather than onto the Sturmian basis vectors; the resultant equations are no more difficult to solve than the standard equations. It is also shown that useful variational bounds on the error of the estimate of a transition amplitude exist only if the trial wave function is expanded in terms of a physical basis set.