The energy as a functional of the charge density and the charge-density susceptibility: A simple, exact, nonlocal expression for the electronic energy of a molecule

Abstract

Simple, new expressions relate the electronic potential energy 〈V〉 and the total electronic energy E of a molecule to its averaged electron density 〈ρe(r)〉, the nonlocal charge-density susceptibility χe(r,r′;iω), the nuclear positions {RN}, and the nuclear charges {ZN}. The expressions derived in this work are exact nonrelativistically, within the Born–Oppenheimer approximation. The results give a nonlocal form for the electronic energy in density functional theory. The virial theorem for a system with Coulomb forces is used to derive the expectation value of the kinetic energy in terms of the expectation values of the potential energy and the derivatives of the potential energy operator with respect to nuclear coordinates; gradient expansions of the kinetic energy functional are not needed. Exchange and correlation effects on 〈V〉 and E are determined by an integral of the charge-density susceptibility χe(r,r′;iω), over imaginary frequencies. The results for 〈V〉 and E are first derived by use of the fluctuation-dissipation theorem and the symmetry properties of the charge-density susceptibility with respect to a change in the sign of ω. Identical results are derived by integration of χe(r,r′;iω) over imaginary frequencies and use of the closure relation.