Hohenberg–Kohn, Kohn–Sham, and Runge-Gross Density Functional Theories

Abstract

The two nondegenerate ground-state theorems of Hohenberg-Kohn (HK) are described with an emphasis on new understandings of the first theorem (HK1) and of its proof. Via HK1, the concept of a basic variable of quantum mechanics, a gauge invariant property knowledge of which uniquely determines the Hamiltonian to within a constant, and hence the wave functions of the system, is developed. HK1 proves that the basic variable is the nondegenerate ground state density. HK1 is generalized via a density preserving unitary transformation to prove the wave function must be a functional of the density and a gauge function of the coordinates in order for the wave function written as a functional to be gauge variant. A corollary proves that degenerate Hamiltonians representing different physical systems but yet possessing the same density cannot be distinguished on the basis of HK1. (This does not constitute a violation of HK1 as the Hamiltonians differ by a constant.) The primacy of the electron number N in the proof of the HK theorems is stressed. The Percus-Levy-Lieb (PLL) constrained-search path from the density to the wave functions is described. It is noted that the HK path is more fundamental, as knowledge of the property that constitutes the basic variable, as gleaned from HK1, is essential for the constrained-search proof of PLL. The Gunnarsson-Lundqvist theorems, the extension of the HK theorems to the lowest excited state of symmetry different from that of the ground state are described. The Runge-Gross (RG) theorems for time-dependent theory, with an emphasis on the first theorem (RG1), are explained. RG1 proves the basic variables to be the density and the current density. A density preserving unitary transformation generalizes RG1 to prove the wave function must be a functional of the density and a gauge function of the coordinates and time. A hierarchy based on gauge functions thereby exists for the fundamental first theorems of density functional theory. A corollary to RG1 similar to that for the time-independent case is proved. Kohn-Sham theory, a ground state theory, which constitutes the mapping from the interacting system to one of noninteracting fermions of the same density, is formulated. As this mapping is based on the HK theorems, the description of the model system is mathematical in that the energy is in terms of functionals of the density, and the local potentials defined as the corresponding functional derivatives.