Approximations for the interparticle interaction energy in an exactly solvable two-electron model atom

The capability of different ansatz kernels, denoted as $K(\mathbf{r},{\mathbf{r}}^{\text{'}})$, in the calculation of the electron-electron interaction energy is investigated here for an exactly solvable two-electron model atom proposed by Moshinsky. The model atom is in the spin-compensated, paramagnetic ground state. The exact expression for the interaction energy in this state, derived by the diagonal of the second-order density matrix, is used as a rigorous background for comparison. It is found that the form of $K_M(\mathbf{r},{\mathbf{r}}^{\text{'}})=2\rho (\mathbf{r})\rho ({\mathbf{r}}^{\text{'}})-{\gamma }^p(\mathbf{r},{\mathbf{r}}^{\text{'}}){\gamma }^q({\mathbf{r}}^{\text{'}},\mathbf{r})$, expressed via the $\rho (\mathbf{r})$ density distributions and operator powers of the one-body density matrix $\gamma (\mathbf{r},{\mathbf{r}}^{\text{'}})$, results in the exact value for the interparticle interaction energy of the two-electron model atom if and only if $p=q=1/2$. Approximate forms with $p=q\ne 1/2$ and with $p\ne q$ at $(p+q)=1$ give deviations from the exact expression.

Figure 1

The $F_{p,q}(\xi )$ function, defined by Eq. (17), as a function of $\sqrt{\xi }$ at different parametrizations. Dashed curve refers to $p=q=0.65$; dotted curve to $p=0.65$ and $q=0.35$. The solid line at unity corresponds to $p=q=1/2$, i.e., the exact representation.