Bounds for Some Equilibrium Properties of an Electron Gas

Upper and lower bounds are derived for the average potential energy and Helmholtz free energy of an electron gas with uniform positive background. In the ground-state limit, upper and lower bounds are given for the average kinetic energy, average potential energy, and total ground-state energy. Inequalities are derived for the static form factor $S\left(k\right)\mathrm{}$ and wave-number-dependent dielectric function $\epsilon (k, 0)$, making use of exact sum rules for the Fourier-transformed density-density commutator and of the assumption that $S\left(k\right)\mathrm{}\le 1$. Comparison is made with the exact behavior of these quantities for small $k$. The sum rules are used to construct an approximate nonlinear integral equation for the ground-state static form factor of the electron gas.