The virial theorem and the surface energy of an electron gas

The virial theorem is used to derive an exact relation between the kinetic and interaction parts of the surface energy of an electron gas. A similar result is derived within the framework of the density-functional formalism. These results are generalized to the surface energy of a d-dimensional electron gas. In four dimensions we find that the density dependence of the surface energy is of the form σ=Aρ5/4.