Hypervirial theorems and scaling relations in density functional theory

Abstract

Within the framework of density functional theory, a hypervirial theorem for an operator <img src='http://www.niscair.res.in/jinfo/acaret.gif' border=0> is shown to be given by ∫d<b style="mso-bidi-font-weight:normal">r</b> <img src='http://www.niscair.res.in/jinfo/rho.gif' border=0>(<b style="mso-bidi-font-weight:normal">r</b>)^1/2 [µ, <img src='http://www.niscair.res.in/jinfo/acaret.gif' border=0>] <img src='http://www.niscair.res.in/jinfo/rho.gif' border=0>(<b style="mso-bidi-font-weight:normal">r</b>)^1/2= 0, where <img src='http://www.niscair.res.in/jinfo/rho.gif' border=0>(<b style="mso-bidi-font-weight:normal">r</b>) is the single particle electron density of a many-electron system. The chemical potential operator µ expressed as either a multiplicative or a differential operator, depending on the variant of DFT, is shown to yield equivalent results. The consequence of using approximate forms of the energy density functionals are discussed and for special cases of the operator <img src='http://www.niscair.res.in/jinfo/acaret.gif' border=0>, the virial theorem in DFT is recovered. Generalised scaling relations for the energy density functionals are obtained and the implications towards tensor virial theorem are discussed.

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