Scaling relations, virial theorem and energy densities for long-range and short-range density functionals

We analyze a decomposition of the Coulomb electron-electron interaction into a long-range and a short-range part in the framework of density functional theory, deriving some scaling relations and the corresponding virial theorem. We study the behavior of the local density approximation in the high-density limit for the long-range and the short-range functionals by carrying out a detailed analysis of the correlation energy of a uniform electron gas interacting via a long-range only electron-electron repulsion. Possible definitions of exchange and correlation energy densities are discussed and clarified with some examples.