Quantal density-functional theory of excited states: The state arbitrariness of the model noninteracting system

Abstract

The quantal density-functional theory (Q-DFT) of nondegenerate excited-states maps the pure state of the Schr\"odinger equation to one of noninteracting fermions such that the equivalent excited state density, energy, and ionization potential are obtained. The state of the model S system is arbitrary in that it may be in a ground or excited state. The potential energy of the model fermions differs as a function of this state. The contribution of correlations due to the Pauli exclusion principle and Coulomb repulsion to the potential and total energy of these fermions is independent of the state of the S system. The differences are solely a consequence of correlation-kinetic effects. Irrespective of the state of the S system, the highest occupied eigenvalue of the model fermions is the negative of the ionization potential. In this paper we demonstrate the state arbitrariness of the model system by application of Q-DFT to the first excited singlet state of the exactly solvable Hookean atom. We construct two model S systems: one in a singlet ground state ${(1s}^{2}),$ and the other in a singlet first excited state $(1s2s).$ In each case, the density and energy determined are equivalent to those of the excited state of the atom, with the highest occupied eigenvalues being the negative of the ionization potential. From these results we determine the corresponding Kohn-Sham density-functional theory (KS-DFT) ``exchange-correlation'' potential energy for the two S systems. Further, based on the results of the model calculations, suggestions for the KS-DFT of excited states are made.