Variational Lower Bounds on Electron Hydrogens- Wave Phase Shifts

The variational-bound (or minimum-principle) formulation for single-channel scattering was applied to the calculation of rigorous lower bounds on the $s$-wave phase shifts for the scattering of electrons by hydrogen atoms ($e^{-}\mathrm{H}$) for energies below the excitation threshold. This method was previously applied to positron-hydrogen ($e^{+}\mathrm{H}$) elastic scattering. Because of the exchange effect in the $e^{-}\mathrm{H}$ case, a projection operator is employed which projects out of the wave function those states in which either or both of the electrons are in the ground state. The method requires the exact solution of the static equation, and of the corresponding static Green's function $G^P$, which is expressed in terms of the solution of an integral equation. Trial functions are chosen which are spatially symmetrized (or antisymmetrized) and correspond to states of the hydrogen atom of given angular momentum quantum number $l$; the associated radial functions contain nonlinear variational parameters. The $e^{-}\mathrm{H}$ calculations are found to give much more accurate results for a given number of parameters than the $e^{+}\mathrm{H}$ calculations.