Second-order nonadiabatic polarization potentials for positron-hydrogen and positron-helium elastic scattering

A self-consistent field theory for one-positron, many-electron systems derived by Schrader [Phys. Rev. A 1, 1070 (1970)] is solved in first order for positron-hydrogen and -helium elastic scattering below the positronium formation threshold. The perturbation is the positron-electron interaction, and the zeroth-order target, in the case of helium, is the Hartree-Fock atom. The first-order target equation is solved variationally. Unlike the polarized-orbital method of Callaway-Temkin, no further simplifying approximations are made here for s-wave phase shifts except the use of a finite basis in which to expand the perturbed electronic orbitals, and in the case of helium the neglect of a certain small and intractable integral at an intermediate stage of the calculation. In addition, for non-s-wave phase shifts we follow much previous work and neglect contributions of an excited target. Our approach is fully nonadiabatic, and is contrived in such a way that the zeroth-order target equation contains the positronic Laplacian operator. The nonadiabatic polarization potential thus generated by the positron differs significantly from that in other more approximate nonadiabatic theories. The positronic scattering phase shifts of the s, p, and d waves are calculated and compared to the best available results from other methods. Our calculated phase shifts for the hydrogen target lie midway between those for a static target and the exact results, thus revealing a serious lack of accuracy for any first-order method, at least when applied to systems similar to hydrogen. Our results for helium are more accurate but still not satisfactory. Our results thus cast doubt on the validity of many earlier calculations in which the polarization potential obtained from a first-order perturbation calculation is adjusted to improve the resulting phase shifts.