Short-range correlation in the uniform electron gas: Extended Overhauser model

We use the two-electron wave functions (geminals) and the simple screened Coulomb potential proposed by Overhauser [Can. J. Phys. 73, 683 (1995)] to compute the pair-distribution function $g(r)$ for a uniform electron gas, finding the exact $g(0)$ for this model and extending the results from $g(0)$ to $g(r).$ We find that the short-range $(r<{r}_{s})$ part of this $g(r)$ is in excellent agreement with quantum Monte Carlo simulations for a wide range of electron densities. We are thus able to estimate the value of the second-order ${(r}^{2})$ coefficient of the small interelectronic-distance expansion of the pair-distribution function. The coefficients of the small-$r$ expansion of the spin-resolved ${g}_{\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}}}(r)$ have density or ${r}_{s}$ dependencies which we parametrize in a way that makes it easy to find their coupling-constant averages. Their spin-polarization or $\ensuremath{\zeta}$ dependencies are estimated from a proposed spin-scaling relation.