Generalisation of the virial theorem and related theorems for jellia with arbitrary background densities and application to special geometriees and profiles. I. General theory and application to the jellium sphere and the semi-infinite electron gas

Abstract

Virial and Budd-Vannimenus theorems (1973) are generalised and derived unitarily and simplified: for a jellium with an arbitrary background density rho (r) four different but equivalent expressions of the pressure term in the virial theorem are found, containing the total energy Erho , the electrostatic potential phi rho (r), the electric field Erho (r) and the electron density nrho (r), respectively. The expression with phi rho (r) generalises the Budd-Vannimenus theorem for arbitrary geometries and background density profiles and the other expressions with Erho (r) and nrho (r) give electrostatically equivalent formulations of the Budd-Vannimenus theorem. The jellium sphere is studied especially for the limiting case of infinite radius R to infinity , yielding bulk and surface theorems. The latter involves a correction of a theorem given by Vannimenus and Budd, arising from the necessity to start with a finite system for a consistent treatment of the pressure. Also an expression given by Heinrichs (1979) is corrected.