Calculation of the dielectric function for an electron liquid

Abstract

A theory for calculating the frequency- and wave-number-dependent dielectric function of an electron liquid is presented by solving the equations of motion for the double-time retarded commutator of the charge-density fluctuation operators. It is based on a decoupling of the higher-order Green's functions, which has been achieved by demanding certain proportionality between the higher-order Green's functions and the lower order. The proportionality coefficient is determined by conserving the various frequency moments. It is shown that by conserving the first frequency moment only, we reproduce the Toigo-Woodruff result for the dielectric function. However, the present theory has the advantage of conserving frequency moments to an infinite order. The dielectric function obtained in this paper is a functional of the function $G (\overrightarrow{\mathrm{k}},\omega )$ which is considered to account for the short-range correlations arising from both the exchange and Coulomb effects. This turns out to be the same as the one derived by Rajagopal who solved variationally the integral equation for the irreducible vertex function containing only linear exchange processes. Numerical evaluation of the function $G (\overrightarrow{\mathrm{k}},\omega )$ has been made for various values of $k$ in the limit $\omega =0\mathrm{}$, and the results are compared with those of the earlier theories. In contrast to all the earlier theories, we find a very sharp peak in the value of our $G\mathrm{}\left(k\right)\mathrm{}$ around $k=2\mathrm{}{k}_F$. It is further interesting to note that the present theory satisfies exactly the compressibility sum rule. An important result of this theory is that the value of $G\left(k\right)\mathrm{}$, obtained in this paper in the limit $k\to \infty$, happens to be $\frac{1}{3}$, which is in complete agreement with the value of $G \left(\infty \right)$ in the Hartree-Fock approximation. This we consider to be a great success over the other existing theories.