Exact integral constraint requiring only the ground-state electron density as input on the exchange-correlation force −∂Vxc(r)/∂r for spherical atoms

Abstract

Following some studies of ∫n(r)∇V(r)dr by earlier workers for the density functional theory (DFT) one-body potential V(r) generating the exact ground-state density, we consider here the special case of spherical atoms. The starting point is the differential virial theorem, which is used, as well as the Hiller–Sucher–Feinberg [Phys. Rev. A 18, 2399 (1978)] identity to show that the scalar quantity paralleling the above vector integral, namely, ∫n(r)∂V(r)/∂rdr, is determined solely by the electron density n(0) at the nucleus for the s-like atoms He and Be. The force −∂V/∂r is then related to the derivative of the exchange-correlation potential Vxc(r) by terms involving only the external potential in addition to n(r). The resulting integral constraint should allow some test of the quality of currently used forms of Vxc(r). The article concludes with results from the differential virial theorem and the Hiller–Sucher–Feinberg identity for the exact many-electron theory of spherical atoms, as well as for the DFT for atoms such as Ne with a closed p shell.