Static Density Response Function of a Degenerate Electron Gas Calculated by Quantal Hyper-Netted Chain Equation

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Abstract
The density distribution n(r∣v) around a fixed electron in an electron gas with density n0 is calculated by the two quantal versions (QHNC and QHNC') of the hyper-netted chain (HNC) equation, which gives the radial distribution funcion g(r) in the best fir with the computer simulation of the classical electron liquid. The one (QHNC) is derived from the Hohenberg-Kohn-Sham formalism by treating the kinetic energy in an exact manner and the other (QHNC') is obtained by approximating it as in the Thomas-Fermi theory. The result is compared with those of the Hartree and Sjölander-Stott equations. In the high density region (rs=1) where the exchange-correlation effect is small, the QHNC n(r∣v) is virtually identical with the Hartree n(r∣v): this may be considered as providing a numerical evidence for the validity of the Ansatz that the Fourier transform of n(r∣v)-n0 is represented in terms of the static density-density response function χQ. With the aid of this Ansatz, χQ is calculated from the QHNC and QHNC' equations. The local-field factor G(Q) in χQ calculated by QHNC has a maximum and two types of singularities in its slope at Q=2QF (QF; the Fermi wave-vector) and approaches a constant value 1-n(r=0∣v)/n0 in the large wave-vector limit, while the QHNC' G(Q) increases monotonically with Q, which is similar to that of Singwi's group. This difference indicates that in the calculation of G(Q) it is important to treat the kinetic energy in an exact manner.