Ritz-Hylleraas Solutions of the Ground State of Two-Electron Atoms Involving Fractional Powers

Abstract

Earlier work on the inclusion of half-integral powers in the Ritz-Hylleraas ground-state solutions of the nonrelativistic wave equation for the helium atom is extended through functions involving 18 parameters. Energies that are lower than those found with other published comparable functions are obtained in all cases. Preliminary results are also given of calculations involving more general fractional powers, $Z$ values different from 2, and half-integral-power solutions for which the expectation value of the square of the Hamiltonian is finite. With the latter type of expansions one obtains, at least at an early stage, an additional improvement in the approximation. Thus, with 11 parameters one finds the energy -2.903704 atomic units, which differs by only 0.0007% from the 80-parameter solution of Kinoshita. The initial results found for $Z=8$ indicate an improvement in convergence over that obtained for $Z=2$; the energy -59.156560 atomic units, which was obtained with a 12-parameter function, was only 0.00006% larger than the energy obtained by Pekeris with his 210-term function. The computed mass polarization corrections give also satisfactory results as judged by the similar results obtained with the most extensive solutions available.