Dielectric function of the electron gas with dynamical-exchange decoupling. II. Discussion and results

Abstract

By making the dynamical-exchange decoupling in the equation of motion for the Wigner distribution function, exchange effects in the dielectric function of the homogeneous electron gas were, in an earlier derivation, described by a frequency-dependent local-field correction $G(q,\ensuremath{\omega})$. In paper I, details were provided how the sixfold integral for $G(q,\ensuremath{\omega})$ can be reduced analytically into a double integral, adapted for numerical purposes. In this paper, the consequences of dynamical-exchange effects are studied, and the theory is tested for its internal consistency. The evaluation of the pair correlation function $g(r)$ at the origin from both the static limit and the high-frequency limit of the frequency-dependent local-field correction $G(q,\ensuremath{\omega})$ leads to the same value $g(0)=\frac{1}{2}$, in contrast to other theories where, from both limits, different results are obtained. Also, the compressibility, calculated from the dielectric function including exchange, agrees with the Hartree-Fock result. Furthermore, it is shown that the high-frequency limit of $G(q,\ensuremath{\omega})$ satisfies the general properties implied by the third-frequency-moment sum rule, resulting again in $g(0)=\frac{1}{2}$ and leading to the Hartree-Fock ground-state energy. This consistency between the static and high-frequency behavior of $G(q,\ensuremath{\omega})$ cannot be fulfilled by any static approximation to $G(q,\ensuremath{\omega})$, because an adequate treatment of dynamical-exchange effects involves excited states that are consistent with the Hartree-Fock ground state. In the static limit, $G(q,\ensuremath{\omega})$ exhibits a relatively sharp peak near $q=2{k}_{F}$. This peak induces an instability of the homogeneous electron gas, quite similar to and in the same density range as the instability of the spin susceptibility discussed by Hamann and Overhauser. For ${r}_{s}\ensuremath{\ge}10.6$, a supplementary instability relative to charge-density deformations occurs. The frequency dependence of $G(q,\ensuremath{\omega})$ is examined, and numerical values for $\mathrm{Re}G(q,\ensuremath{\omega})$ and $\mathrm{Im}G(q,\ensuremath{\omega})$ are presented. The real part of the dielectric function and the imaginary part of the inverse dielectric function are plotted for several densities in the metallic range. Compared to the random-phase approximation (RPA), the frequencies of the maxima in the structure factor obtained in the present work (i.e., with dynamical-exchange decoupling) are shifted to lower frequencies. The plasmon dispersion is considerably closer to recent experimental data in aluminium than with RPA. Finally, it turns out that the inclusion of frequency-dependent exchange effects results in the natural occurrence of spin- and charge-density waves. These are the dynamical extension of the instability relative to magnetic perturbations, found in the static limit at low densities.