Self-consistent Overhauser model for the pair distribution function of an electron gas in dimensionalities D=3 and D=2

We present self-consistent calculations of the spin-averaged pair distribution function g(r) 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(r) that are in excellent agreement with quantum Monte Carlo data at low and intermediate coupling strength r_s, extending up to r_s≈10 in dimensionality D=3. 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 in D=2, and has limited accuracy in regard to the spin-resolved components g_↑↑(r) and g_↑↓(r). We also report calculations of the electron-electron s-wave scattering length, to test an analytical expression proposed by Overhauser in D=3 and to present new results in D=2 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.