Impact-Parameter Method for Proton-Hydrogen-Atom Collisions. IV. Variational Methods for the Time-Dependent Impact-Parameter Model

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Abstract
A stationary functional and two variational principles are given in this work by which approximate transition amplitudes for the charge-exchange and electronic excitation processes occurring in proton — hydrogen-atom scattering can be calculated. The calculated transition amplitudes will be "second-order accurate." It is demonstrated how the stationary property of the functional can be used for a general variation to determine optimal approximate wave functions. The equations for these wave functions should be no more difficult to solve than the usual set of coupled equations, and the solution will automatically satisfy the unitary condition and the condition imposed by the time-reversal symmetry of the Hamiltonian. An upper bound on the second-order error term is derived which only depends upon the interaction and the trial wave function. The upper bound approaches zero as the error between the exact and trial wave functions vanishes, and thus the magnitude of the upper bound is a measure of the quality of a trial wave function. In addition, this upper bound can be used in conjunction with the variational principles to obtain upper and lower bounds which bracket the absolute values of the exact transition amplitudes; thus, one can in principle obtain exact values for the transition amplitudes to any desired accuracy. One of the variational principles offers the advantage that the trial wave function need not satisfy any asymptotic conditions. With this variational principle, one can therefore circumvent the difficulties associated with the use of Sturmian or pseudostate basis functions.

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