Exchange and correlation in the quasi-one-dimensional electron gas: The local-field correction

The local-field correction for the quasi-one-dimensional electron gas in cylindrical quantum wires with wire radius ${\mathit{R}}_0$ is calculated within the sum-rule approach of the self-consistent theory of Singwi, Tosi, Land, and Sjölander. The local-field correction is expressed by a generalized Hubbard form with two coefficients, which are determined self-consistently. Numerical results for the exchange energy and the correlation energy are presented for 0<${\mathit{r}}_{\mathit{s}}$<20, where ${\mathit{r}}_{\mathit{s}}$ is the random-phase-approximation parameter. We find that the exchange energy in the low-density regime is strongly enhanced compared to two and three dimensions: ${\mathrm{\varepsilon }}_{\mathrm{ex}}$(${\mathit{r}}_{\mathit{s}}$→∞)∝-ln(${\mathit{r}}_{\mathit{s}}$)/${\mathit{r}}_{\mathit{s}}$. For high density we find ${\mathrm{\varepsilon }}_{\mathrm{ex}}$(${\mathit{r}}_{\mathit{s}}$→0)∝-${\mathit{a}}^{\mathrm{*}}$/${\mathit{R}}_0$, where ${\mathit{a}}^{\mathrm{*}}$ is the Bohr radius. For the correlation energy we get ${\mathrm{\varepsilon }}_{\mathrm{cor}}$(${\mathit{r}}_{\mathit{s}}$→0)∝-${\mathit{r}}_{\mathit{s}}^2$/${\mathit{R}}_0^2$. The local-field correction strongly reduces the correlation energy for small carrier density if compared with the random-phase approximation. We study the pair-correlation function, the plasmon dispersion, and the compressibility, and we describe the effects of exchange and correlation on these quantities. The parameter for weak coupling in wire systems is described by ${\mathit{R}}_{\mathit{s}}$=4${\mathit{r}}_{\mathit{s}}$ ${\mathit{a}}^{\mathrm{*}}$/π${\mathit{R}}_0$1.