Nonuniqueness of the Effective Potential Energy and Wave Function in Quantal Density Functional Theory

Abstract

Since its advent, a key precept of Kohn–Sham density functional theory (KS-DFT) [1, 2] has been the uniqueness of the local effective potential energy function v s (r), or equivalently of the electron-interaction potential energy function vee(r) of the model S system of noninteracting fermions. Nondegenerate ground state KS-DFT maps an interacting system in its ground state to an S system that is also in its ground state. As the density ρ(r) of the interacting and noninteracting fermions is the same, it then follows from the first Hohenberg–Kohn theorem, as explained in Chap. 4, that the potential energy vee(r) is unique. In Kohn–Sham terms, there is only one such potential energy function because vee(r) is the functional derivative \(\delta E^{{{\rm KS}}}_{{{\rm ee}}}[\rho ]/ \delta \rho (r)\) taken at the ground state density, where \(E^{{{\rm KS}}}_{{ee}}\left[ \rho \right]\) is the unique KS ground state electron-interaction energy functional. It is evident, however, from quantal density functional theory (Q-DFT) [3], that one is not limited to constructing S systems only in a ground state. One can equally well construct S systems in an excited state such that the ground state density ρ(r), energy E, and ionization potential I of the interacting system are reproduced [3, 4]. The state of the model S system is arbitrary. This means that there exist, in principle, an infinite number of local effective potential energy functions that generate the density, energy and ionization potential of the interacting system in its ground state. Furthermore, it is solely the Correlation-Kinetic components of these potential energy functions that differ.