On the susceptibility sum rule in the electron liquid

Abstract

Various attempts have been made (1.~) to develop a theory of the magnetic response of an electron liquid at metallic density, in which self-consistency is achieved between the response and the static spin correlation functions. Most recently YASUItARA and WATAB~ (5) have considered the problem with part icular at tention on the susceptibility sum rule, which requires that the long-wavelength l imit of the static magnetic response in the paramagnetic state coincide with the thermodynamic value of the spin susceptibil i ty as obtained by a second derivative of the energy with respect to the magnetization. They have consequently found it necessary to develop a self-consistent theory which involves second (functional) derivatives of the spin correlation functions with respect to the spin densities. Their approach is based on the Kohn-Sham theory (6) and parallels the approach developed by SH~M (7) for the dielectric response of the paramagnetic state. In this note we show that (as is the case of the dielectric problem (s)) the paramagnetic susceptibility can be obtained from the sole knowledge of the ]irst derivatives of the spin correlation functions with respect to the spin densities. This result hinges on the existence of the relation