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

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 $b.\mathrm{}$ The self-consistent theory of Singwi, Tosi, Land, and Sjölander 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 $0<\mathrm{}{r}_s<1000\mathrm{}$ are presented. The exchange energy and the correlation energy in the low-density regime are described by ${\varepsilon }_{\mathrm{ex}}{(r}_s\to \infty )\propto -\ln {(r}_s{)/r}_s$ and ${\varepsilon }_{\mathrm{cor}}{(r}_s\to \infty )\propto -\ln {(r}_s{)/r}_s$ with ${\varepsilon }_{\mathrm{cor}}{(r}_s\to \infty )/{\varepsilon }_{\mathrm{ex}}{(r}_s\to \infty )\approx \mathrm{0.8.}$ 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.