Theory of the local field correction in an electron gas

Abstract

The wave-vector- and frequency-dependent dielectric function ɛ(k,ω) of an electron gas can be expressed in terms of Lindhard's function and a complex local field correctionG(k,ω) which incorporates all the effects of dynamic exchange and correlation in the system. The general properties ofG(k,ω) are discussed, in particular the static and high-frequency limits. It is shown that for smallk, bothG(k, 0) andG(k, ∞) vary ask 2, with different coefficients, but both determined by the average kinetic and potential energies per particle. For largek,G(k, ∞) varies again ask 2 and it is argued that the same holds true forG(k, 0), with both coefficients (though different) determined by the average kinetic energy per particle. General formulas for the plasma dispersion relation and damping, involving, respectively, the real and imaginary parts ofG(k,ω), are given. The term in the plasma frequency which is proportional tok 2 is given directly in terms of the average kinetic and potential energies per particle, a result true at all temperatures. A calculation of the frequency dependence ofG(k,ω), starting from the exact equation of motion for the particle-hole operator and employing a decoupling approximation introduced previously by Toigo and Woodruff, is presented. Explicit results forG(k,ω) are obtained for smallk and allω. The complete expressions forG(k, 0) andG(k, ∞) in this approximation have been obtained and are plotted.