The locality hypothesis in density-functional theory: An exact theorem

Abstract

The locality hypothesis in density-functional theory (DFT) states that the functional derivative of the Hohenberg-Kohn universal functional can be expressed as a local multiplicative potential function, and this is the basis of DFT and of the successful Kohn-Sham model. Nesbet has in several papers [Phys. Rev. A \bf{58}, R12 (1998); \it{ibid.} A \bf{65}, 010502 (2001); Adv. Quant. Chem, \bf{43}, 1 (2003)] claimed that this hypothesis is in conflict with fundamental quantum physics, and as a consequence that the Hohenberg-Kohn theory cannot be generally valid. We have in a Comment to the Physical Review [Phys. Rev. A \bf{67}, 056501 (2003)] commented upon these works and recently extended the arguments [Adv. Quant. Chem. \bf{43}, 95 (2003)]. We have shown that there is no such conflict and that the locality hypothesis is inherently exact. In the present work we have furthermore verified this numerically by constructing a local Kohn-Sham potential for the $1s2s ^3S$ state of helium that generates the many-body electron density and shown that the corresponding $2s$ Kohn-Sham orbital eigenvalue agrees with the ionization energy to nine digits. Similar result is obtained with the Hartree-Fock density. In addition to verifying the locality hypothesis, this confirms the theorem regarding the Kohn-Sham eigenvalue of the highest occupied orbital.