Dielectric Response of the Electron Liquid in Generalized Random-Phase Approximation: A Critical Analysis

We critically examine various approximate theories which have been put forward within the spirit of generalized random-phase approximation (GRPA) for dielectric response of the degenerate electron liquid at metallic densities. The exchange-correlation contribution to the effective field acting on an electron is expressed in terms of a frequency-independent function $G\left(\overrightarrow{\mathrm{q}}\right)$ in GRPA. There are requirements of certain sum rules, e.g., the compressibility sum rule, the fluctuation-dissipation theorem, and the third-frequency-moment sum rule, which impose restrictions on $G\left(\overrightarrow{\mathrm{q}}\right)$. The theory of Vashishta and Singwi for $G\left(\overrightarrow{\mathrm{q}}\right)$ satisfies the compressibility sum rule and the fluctuation-dissipation theorem, while another recent theory by Pathak and Singwi satisfies the third-moment sum rule and the fluctuation-dissipation theorem. The second work neglects the correlation contribution to the kinetic energy, while the first one takes it into account through the introduction of an ad hoc parameter. In this paper, we show that if the correlation kinetic energy part is correctly taken into account then $G\left(\overrightarrow{\mathrm{q}}\right)$ cannot be made to satisfy the compressibility sum rule and the third-moment sum rule simultaneously, in the sense that doing this would violate the ground-state-energy theorem of Ferrell.