Three-body problem of two-electron atoms

The ground-state wave function of a heliumlike atom is assumed to be $\psi =\phi \left(r_3\right)\exp (-\alpha r_{>}-\beta r_{<})$, where $r_{<}, r_{>}=\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{ }\max (r_1, r_2)$, $r_1$ and $r_2$ being the distances of the electrons from the nucleus. An equation which determines the electron-electron correlation function $\phi \left(r_3\right)$ is derived and numerically solved. The parameters $\alpha$ and $\beta$ 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 ($\mathrm{ee}{e}^{+}$) 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 $\mathrm{ee}{e}^{+}$ 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.