Combined Use of the Methods of Superposition of Configurations and Correlation Factor on the Ground States of the Helium-Like Ions

Abstract

The accurate solution of the Schr\"odinger equation by means of a combination of the method of correlation factor and the method using superposition of configurations is discussed. For a many-electron system, the total wave function divided by the nodeless function $g=g({r}_{12}, {r}_{13}, {r}_{23}\ensuremath{\cdots})$ may be expanded in a series of Slater determinants built up from a complete one-electron set. For a two-electron system, this expansion becomes very simple and, by going over to principal orbitals, it may be "diagonalized" and brought to a particularly rapidly convergent form.The method is applied to the He-like ions (${\mathrm{H}}^{\ensuremath{-}}$, He, ${\mathrm{Li}}^{+}$, and ${\mathrm{Be}}^{2+}$) by choosing $g=1+\ensuremath{\alpha}{r}_{12}$ and by using a basis consisting of only three $s$-orbitals. The energies obtained differ from the exact values only by about 0.001-0.002 atomic units. Even the simpler wave functions ${(u)}^{2}(1+\ensuremath{\alpha}{r}_{12})$ and $(u, v)(1+\ensuremath{\alpha}{r}_{12})$ are discussed, and the forms obtained by truncating the expansion of $\ensuremath{\Psi}$ in principal orbitals are studied in greater detail.