Superconductivity in the homogeneous electron gas: Exchange and correlation effects

We examine the influence of exchange and correlation effects on normal-state and superconducting properties of the homogeneous electron gas. Our investigations extend an earlier paper by Rietschel and Sham (RS) on exact solutions of Eliashberg’s equations for the random-phase-approximation (RPA)-screened Coulomb interaction between the electrons. We do this in two steps: In a first step, the dielectric function is modified by including local-field corrections, while in a second step, the spin dependence of the electron-electron interaction is taken into account explicitly within a phenomenological model first proposed by Kukkonen and Overhauser. This latter step is synonymous with consideration of vertex corrections. Our main result is the following: Local-field corrections alone are not able to remove the artifact of plasmon-induced superconductivity in the RPA-screened electron gas as reflected by negative values for the Coulomb pseudopotential ${\mathrm{\mu }}^{\mathrm{*}}$(RS). Only additional consideration of spin-dependent vertex corrections leads to values for ${\mathrm{\mu }}^{\mathrm{*}}$ which are positive throughout and of order 0.05–0.10 for electronic densities 1≤${\mathit{r}}_{\mathit{s}}$≤5.