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

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(r₁₂, r₁₃, r₂₃⋯) 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 (H^-, He, Li⁺, and Be²⁺) by choosing g=1+αr₁₂ 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+αr₁₂) and (u, v)(1+αr₁₂) are discussed, and the forms obtained by truncating the expansion of Ψ in principal orbitals are studied in greater detail.