On sum rules and Stillinger–Lovett conditions for inhomogeneous Coulomb systems

We show that, for the case of a Coulomb fluid bounded by a planar surface, the inhomogeneous Stillinger–Lovett condition is a consequence of the dipole moment sum rule of Blum et al. [J. Chem. Phys. 75, 5974 (1981)]. We also give a new derivation of the Stillinger–Lovett condition, valid for both homogeneous and inhomogeneous systems, which is entirely analogous to the derivation of the dipole moment sum rule. Our new derivation enables us to clarify certain subtleties that arise when image forces are present. We comment briefly on the generalization of these sum rules to systems of particles with higher multipole moments (e.g., ion–dipole mixtures).