Variational method for a three-body problem of two-electron atoms

The ground-state wave function of a heliumlike atom is assumed to be $\Psi =N(1+\mu \lambda r_{12})\times \exp [-\mu (\beta r_{>}+r_{<}\left)\right]$, where $r_{<},r_{>}=\mathrm{m}\mathrm{i}\mathrm{n},max(r_1,r_2)$, $r_1$ and $r_2$ are the distances of the electrons from the nucleus, and $r_{12}$ is the distance between the electrons. The parameters $\mu$, $\beta$, and $\lambda$ are determined by minimizing the energy of the system and it can be reduced to a two-parameter problem. Numerical results are presented to show the improvement for the groundstate energy from the two- or three-parameter calculations.