Dynamic local-field factor of an electron liquid in the quantum versions of the Singwi-Tosi-Land-Sjölander and Vashishta-Singwi theories

We have investigated in detail the complex, dynamic local field G(q,{ω}) of an electron liquid in the quantum versions of the Singwi-Tosi-Land-Sj\"olander (STLS) and Vashishta-Singwi (VS) theories. We have worked out the various analytical properties of G(q,{ω}) and shown that the STLS and VS theories are the high-frequency limits of the quantum cases. Variation of G(q,{ω}) with {ω} is found to be rather mild. Interestingly, the static local field G(q,0) exhibits, in the region of metallic densities, a peaked structure around q{}2.8kF, in contrast to the monotonically increasing local-field factors of the STLS and VS theories. The height of the peak reaches values greater than 1, which causes the effective particle-hole interaction to become attractive. This gives rise to the possibility of a charge-density-wave instability. We also give a parametric representation of G(q,0), which fulfills exactly the compressibility sum rule.