Ring-diagram summations and the self-energy of the homogeneous electron gas at its weak-correlation limit

Abstract

The small-${r}_{s}$ asymptotics of the self-energy $\ensuremath{\Sigma}(k,\ensuremath{\omega})$ of the homogeneous electron gas (HEG) is studied in terms of the Feynman diagrams involving the noninteracting one-body Green's function ${G}_{0}$ and the static bare Coulomb repulsion ${v}_{0}$. The lowest-order approximation to $\ensuremath{\Sigma}(k,\ensuremath{\omega})$ is given by the product of ${G}_{0}$ and ${v}_{0}$. The nature of the proper ring-diagram summation (equivalent to the random-phase approximation) for $\ensuremath{\Sigma}(k,\ensuremath{\omega})$ that affords the correct small-${r}_{s}$ single behavior of ${r}_{s}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{r}_{s}$ is investigated. Reexamination of ring-diagram summations for several properties of the HEG proves in a rigorous manner that the product ${G}_{0}{v}_{\mathrm{r}}$, where ${v}_{\mathrm{r}}$ is the ring-diagram-summed dynamically screened repulsion, yields the correct lowest-order asymptotics, whereas ${G}_{\mathrm{r}}{v}_{0}$, where ${G}_{\mathrm{r}}$ is the ring-diagram-summed Green's function, contributes only to higher-order terms.