Dynamical local-field factors and effective interactions in the three-dimensional electron liquid

We present a study of the local-field factors for the homogeneous, isotropic, three-dimensional interacting electron liquid as a function of momentum, imaginary frequency (iw), and density (${\mathit{r}}_{\mathit{s}}$). A variational approach is used to solve integral equations for the density-density and spin-spin response functions, and it provides approximations to the local-field factors which are exact in the high-density limit. We derive sum rules which show that for large iw, the local-field factors possess maxima (in both q and iw) whose magnitudes are related to the pair-distribution function evaluated at zero separation. We introduce a parametrization scheme which incorporates these sum rules with the known compressibility, susceptibility, and third-moment sum rules. The local-field factors are then used to calculate the effective electron-electron interaction using the Vignale-Singwi formalism [Phys. Rev. B 32, 2156 (1985)]. It is found to be significantly larger than that predicted by Hubbard-type local-field factors for small to intermediate values of the wave vector and imaginary frequency.