Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1404.3297 · Physical Review A

Nonexistence of a Hohenberg-Kohn variational principle in total current-density-functional theory

Andre Laestadius, Michael Benedicks

Published Mar 23, 2015• 18 citations
arXivHTMLPDF

Abstract

For a many-electron system, whether the particle density $\ensuremath{\rho}(\mathbf{r})$ and the total current density $\mathbf{j}(\mathbf{r})$ are sufficent to determine the one-body potential $V(\mathbf{r})$ and vector potential $A(\mathbf{r})$ 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 ${V}_{0}$ and ${\mathbf{A}}_{0}$. Furthermore, for an arbitrary number of electrons and under the assumption that a Hohenberg-Kohn theorem exists formulated with $\ensuremath{\rho}$ and $\mathbf{j}$, we discuss the possibility of a variational principle in total current-density-functional theory such as that of Hohenberg-Kohn.

DOI
10.1103/physreva.91.032508

Related papers

The Hohenberg–Kohn theorem

Carsten A. Ullrich, Adam Wasserman

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.

•Aug 20, 2026•0 citations

Hohenberg-Kohn theorem for nonlocal external potentials

Thomas L. Gilbert

The Hohenberg-Kohn theorem is extended to the case that the external potential is nonlocal. It is shown that, in this more general case, a nondegenerate ground-state wave function is a universal functional of the one-particle density kernel $\ensuremath{\mu}(x,{x}^{\ensuremath{'}})$, but probably not of the particle density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})=\ensuremath{\Sigma}{s}^{}\ensuremath{\mu}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s,\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s)$. The variational equations for the local and nonlocal cases are compared. The former must be replaced by a variational equation for an equivalent system of noninteracting particles, following a prescription of Kohn and Sham, in order to obtain a Schr\"odinger-like form, and contains only local potentials. The latter may be obtained directly in Schr\"odinger-like form, but the exchange-correlation potential is nonlocal. If the nonlocal pseudo-Hamiltonian exists [i.e., if the functional derivative $\frac{\ensuremath{\delta}E}{\ensuremath{\delta}\ensuremath{\mu}(x,{x}^{\ensuremath{'}})}$ exists for a nondegenerate ground-state density kernel], then the eigenfunctions of the pseudo-Hamiltonian are natural spin orbitals, and all partially occupied orbitals ($0<〈{\ensuremath{\varphi}}_{i}|\ensuremath{\mu}|{\ensuremath{\varphi}}_{i}〉<1$) belong to the same degenerate eigenvalue of the pseudo-Hamiltonian. Finally, it is shown, as a corollary of Coleman's theorem for $N$-representable density kernels, that any finite non-negative differentiable function is an $N$-representable particle density.

Physical review. B, Solid state•Sep 15, 1975•819 citations

Many-body energy density functional

Alejandro Kievsky, Giuseppina Orlandini, M. Gattobigio

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 $^{4}\mathrm{He}$ atomic clusters, to put in evidence the advantages of this formulation in terms of physical insights.

arXiv:2106.13582•Physical Review A•Sep 20, 2021•2 citations

Ab initioapproach for many-electron systems without invoking orbitals: An integral formulation of density-functional theory

Weitao Yang

A new approach for the calculation of ground states of many-electron systems is proposed via an integral formulation of the Hohenberg-Kohn-Sham density-functional theory. Only equations for the total electron density are involved; orbitals are not employed. Exchange and correlation effects are incorporated. In place of the set of single-electron equations, the total electron density is explicitly expressed in terms of the Kohn-Sham effective local potential through multidimensional integrations. The development is based on the first-order density matrix as obtained from the one-body Green's function in polygonal and Fourier path-integral representations. This might open up the possibility of ab initio calculations for molecules with very many electrons. It also provides explicit solutions to two long-standing problems: electron kinetic energy and momentum density as functionals of the total electron density. The formulation can also be used in calculations for general fermions.

Physical Review A•Dec 1, 1988•19 citations

Variational principle for the ground-state energy as a functional of the one-particle density matrix: Beyond Hartree-Fock theory

Abraham Klein, Reiner M. Dreizler

In parallel with standard density-functional theory, we study the energy of the ground state of a finite many-body system as a functional of the one-particle density matrix. We show that the formulation of a variational principle that is valid beyond the Hartree-Fock limit requires that two-body correlations be included not only in the ground-state energy but also in the constraints. As an illustration, we apply a linear-response argument to derive formulas for first-order corrections to the Hartree-Fock density matrix. Further analysis suggests an approach in terms of the density matrix of an independent-particle system, which can be introduced by the application of an alternative variational principle. This approach is reminiscent of Kohn-Sham theory, but the effective external potential is not required to be local. This variational method can be implemented in a systematic fashion by means of the linked-cluster expansion. In an appendix we study a variant of the Hohenberg-Kohn theorem for nonlocal potentials.

Physical Review A•Apr 1, 1998•11 citations

The problem of the universal density functional and the density matrix functional theory

Viktor Borisovich Bobrov, S. A. Trigger

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.

Journal of Experimental and Theoretical Physics•Apr 1, 2013•18 citations
For a many-electron system, whether the particle density \ensuremathρ(r)\ensuremath{\rho}(\mathbf{r})\ensuremathρ(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 EV0,A0(\ensuremathρ,j)=\ensuremath⟨\ensuremathψ(\ensuremathρ,j),H(V0,A0)\ensuremathψ(\ensuremathρ,j)\ensuremath⟩{\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}EV0​,A0​​(\ensuremathρ,j)=\ensuremath⟨\ensuremathψ(\ensuremathρ,j),H(V0​,A0​)\ensuremathψ(\ensuremathρ,j)\ensuremath⟩ 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 \ensuremathρ\ensuremath{\rho}\ensuremathρ 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.
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 theorem is extended to the case that the external potential is nonlocal. It is shown that, in this more general case, a nondegenerate ground-state wave function is a universal functional of the one-particle density kernel $\ensuremath{\mu}(x,{x}^{\ensuremath{'}})$, but probably not of the particle density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})=\ensuremath{\Sigma}{s}^{}\ensuremath{\mu}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s,\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s)$. The variational equations for the local and nonlocal cases are compared. The former must be replaced by a variational equation for an equivalent system of noninteracting particles, following a prescription of Kohn and Sham, in order to obtain a Schr\"odinger-like form, and contains only local potentials. The latter may be obtained directly in Schr\"odinger-like form, but the exchange-correlation potential is nonlocal. If the nonlocal pseudo-Hamiltonian exists [i.e., if the functional derivative $\frac{\ensuremath{\delta}E}{\ensuremath{\delta}\ensuremath{\mu}(x,{x}^{\ensuremath{'}})}$ exists for a nondegenerate ground-state density kernel], then the eigenfunctions of the pseudo-Hamiltonian are natural spin orbitals, and all partially occupied orbitals ($0<〈{\ensuremath{\varphi}}_{i}|\ensuremath{\mu}|{\ensuremath{\varphi}}_{i}〉<1$) belong to the same degenerate eigenvalue of the pseudo-Hamiltonian. Finally, it is shown, as a corollary of Coleman's theorem for NNN-representable density kernels, that any finite non-negative differentiable function is an NNN-representable particle density.
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.
A new approach for the calculation of ground states of many-electron systems is proposed via an integral formulation of the Hohenberg-Kohn-Sham density-functional theory. Only equations for the total electron density are involved; orbitals are not employed. Exchange and correlation effects are incorporated. In place of the set of single-electron equations, the total electron density is explicitly expressed in terms of the Kohn-Sham effective local potential through multidimensional integrations. The development is based on the first-order density matrix as obtained from the one-body Green's function in polygonal and Fourier path-integral representations. This might open up the possibility of ab initio calculations for molecules with very many electrons. It also provides explicit solutions to two long-standing problems: electron kinetic energy and momentum density as functionals of the total electron density. The formulation can also be used in calculations for general fermions.
In parallel with standard density-functional theory, we study the energy of the ground state of a finite many-body system as a functional of the one-particle density matrix. We show that the formulation of a variational principle that is valid beyond the Hartree-Fock limit requires that two-body correlations be included not only in the ground-state energy but also in the constraints. As an illustration, we apply a linear-response argument to derive formulas for first-order corrections to the Hartree-Fock density matrix. Further analysis suggests an approach in terms of the density matrix of an independent-particle system, which can be introduced by the application of an alternative variational principle. This approach is reminiscent of Kohn-Sham theory, but the effective external potential is not required to be local. This variational method can be implemented in a systematic fashion by means of the linked-cluster expansion. In an appendix we study a variant of the Hohenberg-Kohn theorem for nonlocal potentials.
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.