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The Hohenberg–Kohn theorem

Carsten A. Ullrich, Adam Wasserman

Published Aug 20, 2026• 0 citations
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Abstract

Abstract In this chapter, the Hohenberg-Kohn theorem of DFT is formulated and proved. This theorem says that there is a unique one-to-one correspondence between potentials and ground-state densities in electronic many-body systems. The original proof of Hohenberg and Kohn is presented and discussed, which proceeds in two steps: first, the mapping from potentials to wave functions is shown to be unique, followed by the mapping from wave functions to densities. Two corollaries of the theorem are then discussed: the variational principle, whereby the exact ground-state energy can be found by variation of the energy functional, and then the universality of parts of this energy functional. The so-called constrained search formalism is also briefly discussed.

DOI
10.1093/oso/9780198910664.003.0008

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Abstract In this chapter, the Hohenberg-Kohn theorem of DFT is formulated and proved. This theorem says that there is a unique one-to-one correspondence between potentials and ground-state densities in electronic many-body systems. The original proof of Hohenberg and Kohn is presented and discussed, which proceeds in two steps: first, the mapping from potentials to wave functions is shown to be unique, followed by the mapping from wave functions to densities. Two corollaries of the theorem are then discussed: the variational principle, whereby the exact ground-state energy can be found by variation of the energy functional, and then the universality of parts of this energy functional. The so-called constrained search formalism is also briefly discussed.
The Hohenberg-Kohn (HK) theorem is one of the most fundamental theorems of quantum mechanics, and constitutes the basis for the very successful density-functional approach to inhomogeneous interacting many-particle systems. Here we show that in formulations of density-functional theory (DFT) that employ more than one density variable, applied to systems with a degenerate ground state, there is a subtle loophole in the HK theorem, as all mappings between densities, wave functions, and potentials can break down. Two weaker theorems which we prove here, the joint-degeneracy theorem and the internal-energy theorem, restore the internal, total, and exchange-correlation energy functionals to the extent needed in applications of DFT to atoms, molecules, and solids. The joint-degeneracy theorem constrains the nature of possible degeneracies in general many-body systems.
The Hohenberg-Kohn theorem and the Kohn-Sham equations, which are at the basis of the density-functional theory, are reformulated in terms of a particular many-body density, which is translational and Galilean invariant and therefore is relevant for self-bound systems. In a similar way that there is a unique relation between the one-body density and the external potential that gives rise to it, we demonstrate that there is a unique relation between that particular many-body density and a definite many-body potential. The energy is then a functional of this density, and its minimization leads to the ground-state energy of the system. As a proof of principle, the analogous of the Kohn-Sham equation is solved in the specific case of 4He^{4}\mathrm{He}4He atomic clusters, to put in evidence the advantages of this formulation in terms of physical insights.
For a many-electron system, whether the particle density $\ensuremath{\rho}(\mathbf{r})$ and the total current density j(r)\mathbf{j}(\mathbf{r})j(r) are sufficent to determine the one-body potential V(r)V(\mathbf{r})V(r) and vector potential A(r)A(\mathbf{r})A(r) is still an open question. For the one-electron case, a Hohenberg-Kohn theorem exists formulated with the total current density. Here we show that the generalized Hohenberg-Kohn energy functional ${\mathcal{E}}_{{V}_{0},{\mathbf{A}}_{0}}(\ensuremath{\rho},\mathbf{j})=\ensuremath{\langle}\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j}),H({V}_{0},{\mathbf{A}}_{0})\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j})\ensuremath{\rangle}$ can be minimal for densities that are not the ground-state densities of the fixed potentials V0{V}_{0}V0​ and A0{\mathbf{A}}_{0}A0​. Furthermore, for an arbitrary number of electrons and under the assumption that a Hohenberg-Kohn theorem exists formulated with $\ensuremath{\rho}$ and j\mathbf{j}j, we discuss the possibility of a variational principle in total current-density-functional theory such as that of Hohenberg-Kohn.
It is shown that the Hohenberg-Kohn lemma and theorem are direct consequences of the statement that the ground state energy (or free energy) of a system of interacting particles in an external field is a unique functional of the potential of this field. This means that, if the Hohenberg-Kohn theorem is valid, the nonuniform density in the equilibrium system and the external field potential are biunique functionals. In this case, the nonuniform density is intimately related to the inverse response function. On this basis, a regular procedure can be constructed for determining the density functional for the free energy or ground state energy.
The analysis in this paper shows that the Hohenberg-Kohn theorem is the constellation of two statements: (i) the mathematically rigorous Hohenberg-Kohn lemma, which demonstrates that the same ground-state density cannot correspond to two different potentials of an external field, and (ii) the hypothesis of the existence of the universal density functional. Based on the obtained explicit expression for the nonrel-ativistic particle energy in a local external field, we prove that the energy of the system of more than two non-interacting electrons cannot be a functional of the inhomogeneous density. This result is generalized to the system of interacting electrons. It means that the Hohenberg-Kohn lemma cannot provide justification of the universal density functional for fermions. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect. In the framework of the density matrix functional theory, the hypothesis of the existence of the universal density matrix functional corresponds to the cases of noninteracting particles and to interaction in the Hartree-Fock approximation.
The principles of density-functional theory are studied for finite lattice systems represented by graphs. Surprisingly, the fundamental Hohenberg-Kohn theorem is found void, in general, while many insights into the topological structure of the density-potential mapping can be won. We give precise conditions for a ground state to be uniquely v-representable and are able to prove that this property holds for almost all densities. A set of examples illustrates the theory and demonstrates the non-convexity of the pure-state constrained-search functional.