Simplified Theory of Electron Correlations in Metals

Abstract

A model is developed which allows one to easily calculate correlation effects of interacting electrons. Upon considering a particular electron one replaces the excitation spectrum of all other electrons by a single mode $\omega \mathrm{}\left(q\right)\mathrm{}$, varying between the plasma frequency for small $q$ and $\frac{\hslash q^2}{2m}$ for large $q$. The coupling strength between the electron and the plasma modes is found by imposing the $f$ sum rule. $\omega \mathrm{}\left(q\right)\mathrm{}$ is determined by requiring the model to have a correct dielectric response. The exchange and correlation contributions to $E\left(k\right)\mathrm{}$ have nearly opposite $k$ dependence. However, there is a residual oscillation near $k_F$ which causes the effective mass $m^{*}$ to be less than unity, even though the mean mass (between $k=0\mathrm{}$ and $k_F$) is greater than unity. A specific local approximation to the exchange and correlation potential $A_{\mathrm{xc}}=-2.\mathrm{}07{\left(n{a}_0^3\right)}^{0.3}\mathrm{Ry}$, analogous to Slater's $n^{\frac{1}{3}}$ exchange potential, is accurate over 3 orders of magnitude in density. The (bare) momentum distribution $n\left(k\right)\mathrm{}$, and the fraction $\zeta$ of electrons excited above $k_F$, are calculated as a function of density. For Li and Na, excluding band-structure effects, $\zeta =0.\mathrm{}11 \mathrm{and} 0.14$, respectively.