Moshinsky introduced an exactly soluble model of a two-electron atom consisting of two spin-$\frac{1}{2}$ particles interacting via harmonic forces and moving in a harmonic-oscillator potential. Here, the exact ground-state density $\rho \mathrm{}\left(r\right)\mathrm{}$ is related to the (also analytically known) Hartree-Fock density ${\rho }_{\mathrm{HF}}\mathrm{}\left(r\right).$ The generalization to the off-diagonal matrix $\gamma (\mathbf{r},{\mathbf{r}}^{\prime })$ is then effected, this being related to the idempotent ${\gamma }_{\mathrm{HF}}(\mathbf{r},{\mathbf{r}}^{\prime })/2.\mathrm{}$ This exact information on this “model atom” prompts us to propose an approximate form of $\gamma (\mathbf{r},{\mathbf{r}}^{\prime })$ for the He-like ions, the ${\mathrm{H}}_2$ molecule and, in general, all two-electron systems. $\gamma (\mathbf{r},{\mathbf{r}}^{\prime })$ is constructed solely from the exact $\rho \left(\mathbf{r}\right)$ and its Hartree-Fock counterpart. Some detailed treatment of the two-electron Hookean atom with spring constant $k=1/4\mathrm{}$ (atomic units) is also presented.

• Received 20 June 2002

DOI:https://doi.org/10.1103/PhysRevA.67.022509