Hohenberg-Kohn Theorem Research Articles

Hohenberg–Kohn theorems in the presence of magnetic field

In this article, we examine Hohenberg–Kohn theorems for Current Density Functional Theory, that is, generalizations of the classical Hohenberg–Kohn theorem that includes both electric and magnetic fields. In the Vignale and Rasolt formulation (Vignale and Rasolt, Phys. Rev. Lett. 1987, 59, 2360), which uses the paramagnetic current density, we address the issue of degenerate ground states and prove that the ensemble‐representable particle and paramagnetic current density determine the degenerate ground states. For the formulation that uses the total current density, we note that the proof suggested by Diener (Diener, J. Phys.: Condens. Matter. 1991, 3, 9417) is unfortunately not correct. Furthermore, we give a proof that the magnetic field and the ensemble‐representable particle density determine the scalar and vector potentials up to a gauge transformation. This generalizes the result of Grayce and Harris (Grayce and Harris, Phys. Rev. A 1994, 50, 3089) to the case of degenerate ground states. We moreover prove the existence of a positive wavefunction that is the ground state of infinitely many different Hamiltonians. © 2014 Wiley Periodicals, Inc.

Density functional theory across chemistry, physics and biology

The past decades have seen density functional theory (DFT) evolve from a rising star in computational quantum chemistry to one of its major players. This Theme Issue, which comes half a century after the publication of the Hohenberg-Kohn theorems that laid the foundations of modern DFT, reviews progress and challenges in present-day DFT research. Rather than trying to be comprehensive, this Theme Issue attempts to give a flavour of selected aspects of DFT.

Density-functional theory, finite-temperature classical maps, and their implications for foundational studies of quantum systems

The advent of the Hohenberg-Kohn theorem in 1964, its extension to finite-T, Kohn-Sham theory, and relativistic extensions provide the well-established formalism of density-functional theory (DFT). This theory enables the calculation of all static properties of quantum systems without the need for an n-body wavefunction ψ. DFT uses the one-body density distribution instead of ψ. The more recent time-dependent formulations of DFT attempt to describe the time evolution of quantum systems without using the time-dependent wavefunction. Although DFT has become the standard tool of condensed-matter computational quantum mechanics, its foundational implications have remained largely unexplored. While all systems require quantum mechanics (QM) at T=0, the pair-distribution functions (PDFs) of such quantum systems have been accurately mapped into classical models at effective finite-T, and using suitable non-local quantum potentials (e.g., to mimic Pauli exclusion effects). These approaches shed light on the quantum → hybrid → classical models, and provide a new way of looking at the existence of non- local correlations without appealing to Bell's theorem. They also provide insights regarding Bohmian mechanics. Furthermore, macroscopic systems even at 1 Kelvin have de Broglie wavelengths in the micro-femtometer range, thereby eliminating macroscopic cat states, and avoiding the need for ad hoc decoherence models.

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

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.

Variational principle, Hohenberg–Kohn theorem, and density function origin shifts

Origin shifts performed on the density functions (DF) permit to express the Hohenberg–Kohn theorem (HKT) as a consequence of the variational principle. Upon ordering the expectation values of Hermitian operators, an extended variational principle can be described using origin shifted DF. Under some restrictions, HKT can be extended for some specific Hermitian operators.

Hohenberg-Kohn theorem including electron spin

The Hohenberg-Kohn theorem is generalized to the case of a finite system of $N$ electrons in external electrostatic $\mathcal{E}(\mathbf{r})=\ensuremath{-}\ensuremath{\nabla}v(\mathbf{r})$ and magnetostatic $\mathbf{B}(\mathbf{r})=\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathbf{A}(\mathbf{r})$ fields in which the interaction of the latter with both the orbital and spin angular momentum is considered. For a nondegenerate ground state a bijective relationship is proved between the gauge invariant density $\ensuremath{\rho}(\mathbf{r})$ and physical current density $\mathbf{j}(\mathbf{r})$ and the potentials ${v(\mathbf{r}),\mathbf{A}(\mathbf{r})}$. The possible many-to-one relationship between the potentials ${v(\mathbf{r}),\mathbf{A}(\mathbf{r})}$ and the wave function is explicitly accounted for in the proof. With the knowledge that the basic variables are ${\ensuremath{\rho}(\mathbf{r}),\mathbf{j}(\mathbf{r})}$, and explicitly employing the bijectivity between ${\ensuremath{\rho}(\mathbf{r}),\mathbf{j}(\mathbf{r})}$ and ${v(\mathbf{r}),\mathbf{A}(\mathbf{r})}$, the further extension to $N$-representable densities and degenerate states is achieved via a Percus-Levy-Lieb constrained-search proof. A ${\ensuremath{\rho}(\mathbf{r}),\mathbf{j}(\mathbf{r})}$-functional theory is developed. Finally, a Slater determinant of equidensity orbitals which reproduces a given ${\ensuremath{\rho}(\mathbf{r}),\mathbf{j}(\mathbf{r})}$ is constructed.

Implications of the two nodal domains conjecture for ground state fermionic wave functions

The nodes of many-body wave functions are mathematical objects important in many different fields of physics. They are at the heart of the quantum Monte Carlo methods but outside this field their properties are neither widely known nor studied. In recent years a conjecture, already proven to be true in several important cases, has been put forward related to the nodes of the fermionic ground state of a many-body system, namely that there is a single nodal hypersurface that divides configuration space into only two connected domains. While this is obviously relevant to the fixed node diffusion Monte Carlo method, its repercussions have ramifications in various fields of physics as diverse as density functional theory or Feynman and Cohen's backflow wave function formulation. To illustrate this we explicitly show that, even if we knew the exact Kohn-Sham exchange correlation functional, there are systems for which we would obtain the exact ground state energy and density but a wave function quite different from the exact one. This paradox is only apparent since the Hohenberg-Kohn theorem relates the energy directly to the density and the wave function is not guaranteed to be close to the exact one. The aim of this paper is to stimulate the investigation of the properties of the nodes of many-body wave functions in different fields of physics. Furthermore, we explicitly show that this conjecture is related to the phenomenon of avoided nodal crossing but it is not necessarily caused by electron correlation, as sometimes has been suggested in the literature. We explicitly build a many-body uncorrelated example whose nodal structure shows the same phenomenon.

Monte Carlo implementation of density-functional theory

We propose a conceptually easy and relatively straigthforward numerical method for calculating the ground-state properties of many-particle systems based on the Hohenberg-Kohn theorems. In this ``density-functional Monte Carlo'' method a direct numerical minimization of the energy functional is performed by a Monte Carlo algorithm in which the density is simulated by a distribution of Bernoulli walkers. The total number of particles is conserved by construction, unlike for other implementations of density-functional theory. The feasibility of the method is illustrated by applying it to a nanoshell.

Hohenberg–Kohn theorem for Coulomb type systems and its generalization

Density functional theory (DFT) has become a basic tool for the study of electronic structure of matter, in which the Hohenberg–Kohn theorem plays a fundamental role in the development of DFT. In this paper, we present a simple, selfcontained and mathematically rigorous proof using the Fundamental Theorem of Algebra. We also show the Hohenberg–Kohn theorem for systems with some more general external potentials.

Demonstration of the Gunnarsson-Lundqvist theorem and the multiplicity of potentials for excited states

The Gunnarsson-Lundqvist (GL) theorem of density functional theory states that there is a one-to-one relationship between the density of the lowest nondegenerate excited state of a given symmetry and the external potential. As a consequence, knowledge of this excited state density determines the external potential uniquely. [The GL theorem is the equivalent for such excited states of the Hohenberg-Kohn (HK) theorem for nondegenerate ground states.] For other excited states, there is no equivalent of the GL or HK theorem. For these states, there thus exist multiple potentials that generate the excited-state density. We show, by example, the satisfaction that the GL theorem holds and the multiplicity of potentials for excited states.

Generalization of the Hohenberg–Kohn theorem to the presence of a magnetostatic field

The Hohenberg–Kohn theorem is generalized to the case of electrons in the presence of both an external electrostatic E(r)=−∇v(r) and magnetostatic B(r)=∇×A(r) field. For the non-degenerate ground state, it is established that the basic variables are the ground state density ρ(r) and physical current density j(r) by proving the relationship between the densities {ρ(r),j(r)} and the external potentials {v(r),A(r)} is one-to-one. A constrained-search proof is also provided. It is further explained why the basic variables cannot be the density ρ(r) and the paramagnetic current density jp(r) as presently thought to be the case.

Off-diagonal matrix elements: a modus tollens approach

A numerical application of the modus tollens rule of classical logic is described and is shown to permit the calculation of off-diagonal matrix elements by using a small basis set of implicit wavefunctions. The success of the calculation is discussed in terms of the most simple form of the Hohenberg–Kohn theorem.

Applications of density matrix in the fractional quantum mechanics: Thomas–Fermi model and Hohenberg–Kohn theorems revisited

The many-body space fractional quantum system is studied using the density matrix method. We give the new results of the Thomas–Fermi model, obtain the quantum pressure of the free electron gas. We also show the validity of the Hohenberg–Kohn theorems in the space fractional quantum mechanics and generalize the density functional theory to the fractional quantum mechanics.

Mathematical aspects of the LCAO MO first order density function (4): a discussion on the connection of Taylor series expansion of electronic density (TSED) function with the holographic electron density theorem (HEDT) and the Hohenberg-Kohn theorem (HKT)

Taylor series expansion of electronic density (TSED) functions are set up in order to propose them as an alternative description of the holographic electron density theorem (HEDT) functions. The manipulation of the obtained TSED general formulation leads to a connection between TSED, HEDT and Hohenberg-Kohn theorem (HKT).

On the simple constrained‐search reformulation of the Hohenberg–Kohn theorem to include degeneracies and more (1964–1979)

AbstractThe constrained‐search reformulation provides a transparent proof of the Hohenberg–Kohn theorem, while extending it to include degeneracies, solving the w‐representability problem, and more. This proof, as first presented at the 1979 Sanibel meeting, is reviewed within the context of a brief historical perspective and my interactions at that meeting. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

Kohn-Sham potentials for fullerenes and spherical molecules

We present a procedure for the construction of accurate Kohn-Sham potentials of quasispherical molecules starting from the first-principles valence densities. The method is demonstrated for the case of icosahedral ${\mathrm{C}}_{20}{}^{2+}$ and ${\mathrm{C}}_{60}$ molecules. Provided the density is $N$ representable the Hohenberg-Kohn theorem guarantees the uniqueness of the obtained potentials. The potential is iteratively built following the suggestion of R. van Leeuwen and E. J. Baerends [Phys. Rev. A 49, 2421 (1994)]. The high symmetry of the molecules allows a parametrization of the angular dependence of the densities and the potentials using a small number of symmetry-adapted spherical harmonics. The radial behavior of these quantities is represented on a grid and the density is reconstructed from the approximate potential by numerically solving the coupled-channel Kohn-Sham equations. Subsequently, the potential is updated and the procedure is continued until convergence is achieved.

Correlated fermionic densities for many harmonically trapped particles interacting with repulsive forces

This study is motivated by the very recent work on correlation energy as approximated by the Thomas–Fermi (TF) semiclassical limit [B.R. Landry, et al., Phys. Rev. Lett. 103 (2009) 066401]. In contrast, and motivated by the Hohenberg–Kohn theorem, our work is focussed primarily on the correlated TF ground-state density. We invoke directly the Holas et al. result that for two-fermion systems with harmonic trapping, the fermion–fermion interaction u simply adds to the trapping potential. We conclude this report with some results on correlation kinetic energy for two-fermion systems.

Analysis of multi-configuration density functional theory methods: theory and model application to bond-breaking

We consider the extension of the standard single-determinant Kohn–Sham method to the case of a multi-configuration auxiliary wave function. By applying the rigorous Kohn–Sham method to this case, we construct the proper interacting and auxiliary energy functionals. Following the Hohenberg–Kohn theorem for both energy functionals, we derive the corresponding multi-configuration Kohn–Sham equations, based on a local effective potential. At the end of the analysis we show that, at the ground state, the auxiliary wavefunction must collapse into a single-determinant wave function, equal to the regular KS wavefunction. We also discuss the stability of the wavefunction in multi-configuration density functional theory methods where the auxiliary system is partially interacting, and the remaining (residual) correlation is evaluated as a functional of the density. As an example showing both the challenges and the possibilities, we implement such a procedure for the perfect pairing wavefunction, using a residual correlation functional that is based on the Lee–Yang–Parr functional, and present results for an elementary bond-breaking process.

Density functional theory and quantum computation

This paper establishes the applicability of density functional theory methods to quantum computing systems. We show that ground-state and time-dependent density functional theory can be applied to quantum computing systems by proving the Hohenberg-Kohn and Runge-Gross theorems for a fermionic representation of an N qubit system. As a first demonstration of this approach, time-dependent density functional theory is used to determine the minimum energy gap Delta(N) arising when the quantum adiabatic evolution algorithm is used to solve instances of the NP-Complete problem MAXCUT. It is known that the computational efficiency of this algorithm is largely determined by the large-N scaling behavior of Delta(N), and so determining this behavior is of fundamental significance. As density functional theory has been used to study quantum systems with N ~ 1000 interacting degrees of freedom, the approach introduced in this paper raises the realistic prospect of evaluating the gap Delta(N) for systems with N ~ 1000 qubits. Although the calculation of Delta(N) serves to illustrate how density functional theory methods can be applied to problems in quantum computing, the approach has a much broader range and shows promise as a means for determining the properties of very large quantum computing systems.

Analysis of the stability of finite subspaces in density functional theory

We study the problem of the stability of finite subspaces with respect to the external potential in the formulation of the Hohenberg-Kohn theorem in density functional theory. We provide general procedures to construct potentials that make any finite dimensional subspace unstable, i.e., we construct potentials that acting over functions that belong to the subspace, generate functions that do not belong to that subspace. Explicit calculations of these instability generating potentials are carried out for the particle-in-a-box problem and for the hydrogen atom. We also discuss the consequences of these instabilities on the Kohn–Sham equations, as well as conditions for stability and the relation between instability and nonuniqueness of potentials.