Correlation in Fermi liquids: Analytical results for the local-field correction in two and three dimensions

The local-field correction for the two-dimensional and the three-dimensional electron gas is calculated within a sum-rule version of the self-consistent approach of Singwi, Tosi, Land, and Sjölander. Correlation effects are studied. Results for 0.001${\mathit{r}}_{\mathit{s}}$${\mathit{r}}_{\mathit{s}}$ is the random-phase-approximation parameter. An analytical expression for the static structure factor, representing a generalized Feynman-Bijl spectrum, is used in the calculation. We derive analytical expressions for the density dependence of the local-field correction and we compare the results for the ground-state energy for the interacting electron gas with Monte Carlo computations. The pair-correlation function and the compressibility are studied. Exchange and correlation effects for quantum wells and heterostructures are calculated: numerical and analytical results are derived. In two dimensions and at low density a roton structure in the plasmon dispersion is found. We discuss an instability in layered structures of two-dimensional electron gases.