Self-consistent Overhauser model for the pair distribution function of an electron gas in dimensionalitiesD=3andD=2

We present self-consistent calculations of the spin-averaged pair distribution function $g\left(r\right)\mathrm{}$ for a homogeneous electron gas in the paramagnetic state in both three and two dimensions, based on an extension of a model that was originally proposed by Overhauser [Can. J. Phys. 73, 683 (1995)] and further evaluated by Gori-Giorgi and Perdew [Phys. Rev. B 64, 155102 (2001)]. The model involves the solution of a two-electron scattering problem via an effective Coulombic potential, which we determine within a self-consistent Hartree approximation. We find numerical results for $g\left(r\right)\mathrm{}$ that are in excellent agreement with quantum Monte Carlo data at low and intermediate coupling strength $r_s,$ extending up to $r_s\approx 10$ in dimensionality $D=3.\mathrm{}$ However, the Hartree approximation does not properly account for the emergence of a first-neighbor peak at stronger coupling, such as at $r_s=5\mathrm{}$ in $D=2,$ and has limited accuracy in regard to the spin-resolved components $g_{\uparrow \uparrow }\mathrm{}\left(r\right)\mathrm{}$ and $g_{\uparrow \downarrow }\mathrm{}\left(r\right).$ We also report calculations of the electron-electron s-wave scattering length, to test an analytical expression proposed by Overhauser in $D=3\mathrm{}$ and to present new results in $D=2\mathrm{}$ at moderate coupling strength. Finally, we indicate how this approach can be extended to evaluate the pair distribution functions in inhomogeneous electron systems and hence to obtain improved exchange-correlation energy functionals.