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 $\ensuremath{\omega}(q)$, varying between the plasma frequency for small $q$ and $\frac{\ensuremath{\hbar}{q}^{2}}{2m}$ for large $q$. The coupling strength between the electron and the plasma modes is found by imposing the $f$ sum rule. $\ensuremath{\omega}(q)$ is determined by requiring the model to have a correct dielectric response. The exchange and correlation contributions to $E(k)$ 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 $k=0$ and ${k}_{F}$) is greater than unity. A specific local approximation to the exchange and correlation potential ${A}_{\mathrm{xc}}=\ensuremath{-}2.07{(n{a}_{0}^{3})}^{0.3}\mathrm{Ry}$, analogous to Slater's ${n}^{\frac{1}{3}}$ exchange potential, is accurate over 3 orders of magnitude in density. The (bare) momentum distribution $n(k)$, and the fraction $\ensuremath{\zeta}$ of electrons excited above ${k}_{F}$, are calculated as a function of density. For Li and Na, excluding band-structure effects, $\ensuremath{\zeta}=0.11 \mathrm{and} 0.14$, respectively.