Low-density limit of the correlation energy in the random-phase approximation for charged particles of arbitrary statistics

Within the random-phase approximation (RPA) or ring sum, the ground-state correlation energy for a uniform gas of charged particles with density parameter ${\mathit{r}}_{\mathit{s}}$ tends as ${\mathit{r}}_{\mathit{s}}$\ensuremath{\rightarrow}\ensuremath{\infty} to (-0.803 Ry) ${\mathit{r}}_{\mathit{s}}^{\mathrm{\ensuremath{-}}3/4}$. This limit holds for fermions, as for bosons and distinguishable particles. For electrons, the next term in the low-density expansion (of order ${\mathit{r}}_{\mathit{s}}^{\mathrm{\ensuremath{-}}1}$) cancels the exchange energy. Corrections to RPA must cancel the ${\mathit{r}}_{\mathit{s}}^{\mathrm{\ensuremath{-}}3/4}$ term, and can modify the ${\mathit{r}}_{\mathit{s}}^{\mathrm{\ensuremath{-}}1}$ term.