Three-body problem of two-electron atoms

The ground-state wave function of a heliumlike atom is assumed to be ψ=φ(r₃)exp(-αr_>-βr_<), where r_<, r_>=min, max(r₁, r₂), r₁ and r₂ being the distances of the electrons from the nucleus. An equation which determines the electron-electron correlation function φ(r₃) is derived and numerically solved. The parameters α and β are determined by minimizing the energy of the system. This is an extension of the Feshbach-Rubinow-type approximation which has recently been applied to the same problem. The positronium negative ion (eee⁺) is also examined by the same method. It is found for the binding energies that the improvement over the Feshbach-Rubinow approximation is substantial for the eee⁺ system, but less significant for heliumlike atoms. It is claimed that the proposed method is particularly useful in calculating quantities which are sensitive to the short-range correlation between the electrons.