Many-body effects in the interacting quasi-one-dimensional electron gas: Oscillator confinement

Abstract

Many-body effects described by the local-field correction are calculated for the quasi-one-dimensional electron gas with an oscillator confinement of width parameter $b.$ The self-consistent theory of Singwi, Tosi, Land, and Sj\"olander is used with an analytical form for the local-field correction. We use a three-sum-rule approach in order to parametrize the local-field correction by three coefficients. The coefficients are determined self-consistently and depend on the width parameter $b$ and on the Wigner-Seitz parameter ${r}_{s}.$ Numerical results for the exchange energy and the correlation energy for $0<{r}_{s}<1000$ are presented. The exchange energy and the correlation energy in the low-density regime are described by ${\ensuremath{\varepsilon}}_{\mathrm{ex}}{(r}_{s}\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\propto}\ensuremath{-}\mathrm{ln}{(r}_{s}{)/r}_{s}$ and ${\ensuremath{\varepsilon}}_{\mathrm{cor}}{(r}_{s}\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\propto}\ensuremath{-}\mathrm{ln}{(r}_{s}{)/r}_{s}$ with ${\ensuremath{\varepsilon}}_{\mathrm{cor}}{(r}_{s}\ensuremath{\rightarrow}\ensuremath{\infty})/{\ensuremath{\varepsilon}}_{\mathrm{ex}}{(r}_{s}\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\approx}0.8.$ We derive analytical and numerical results for the compressibility, the chemical potential, screening properties, and bound-state energies of positively and negatively charged impurities. The long-distance behavior of the pair-correlation function is calculated. The compressibility sum rule and the long-wavelength behavior of the dielectric function are discussed in detail. The Hartree energy is calculated.