Equation satisfied by the energy-density functional for electron-electron mutual Coulomb repulsion

It is shown that the electron-electron mutual Coulomb repulsion energy-density functional Veeγ[ρ] satisfies the equationVeeγ[ρN1]-Veeγ[ρN-1γ]=∫d3r(δVeeγ[ρN1]/δρN1(r))[ρN1(r)-ρN-1γ(r)], where ρN1(r) and ρN-1γ(r) are N-electron and (N-1)-electron densities determined from the same adiabatic scaled external potential of the N-electron system at coupling strength γ.