Violation of sum rules by random-phase-approximation surface response calculations based on a density-functional ground-state description

It is shown that the use of the random-phase approximation (RPA) to determine the response of a metal surface to a static uniform external field in conjunction with a local-density description of exchange and correlation in the ground state violates the exact force sum rule. As a result, the static image plane is located considerably closer to the surface and the external field is less effectively screened than in a consistent treatment of exchange and correlation in both ground state and response. In addition, this implementation of the RPA violates the so-called surface f-sum rule, giving a too low number of electrons available for excitation near a metal surface. As a consequence of these violations, the first and first-inverse frequency moment of the imaginary part of the dynamic induced charge as well as its small-frequency behavior are incorrectly described.