Hohenberg-Kohn Theorem Research Articles

Solution of the υ-representability problem on a one-dimensional torus

Abstract We provide a solution to the v-representability problem for a non-relativistic quantum many-particle system on a one-dimensional torus domain in terms of Sobolev spaces and their duals. Any one-particle density that is square-integrable, has a square-integrable weak derivative, and is gapped away from zero can be realized from the solution of a many-particle Schrödinger equation, with or without interactions, by choosing a corresponding external potential. This potential can contain a distributional contribution but still gives rise to a self-adjoint Hamiltonian. Importantly, this allows for a well-defined Kohn–Sham procedure but, on the other hand, invalidates the usual proof of the Hohenberg–Kohn theorem.

Geometrical perspective on spin-lattice density-functional theory.

A recently developed viewpoint on the fundamentals of density-functional theory for finite interacting spin-lattice systems that centers around the notion of degeneracy regions is presented. It allows for an entirely geometrical description of the Hohenberg-Kohn theorem and v-representability. The phenomena receive exemplification by an Anderson impurity model and other small-lattice examples. The case of adiabatic change and the time-dependent setting are examined as well.

Density functional theory from spherically symmetric densities: Ground and excited states of Coulomb systems.

Recently, Theophilou [J. Chem. Phys. 149, 074104 (2018)] proposed a peculiar version of the density functional theory by showing that the set of spherical averages of the density around the nuclei determines uniquely the external potential in atoms, molecules, and solids. Here, this novel theory is extended to individual excited states. The generalization is based on the method developed in the series of papers by Ayers, Levy, and Nagy [Phys. Rev. A 85, 042518 (2012)]. Generalized Hohenberg-Kohn theorems are proved to the set of spherically symmetric densities using constrained search. A universal variational functional for the sum of the kinetic and electron-electron repulsion energies is constructed. The functional is appropriate for the ground state and all bound excited states. Euler equations and Kohn-Sham equations for the set are derived. The Euler equations can be rewritten as Schrödinger-like equations for the square root of the radial densities, and the effective potentials in them can be expressed in terms of wave function expectation values. The Hartree plus exchange-correlation potentials can be given by the difference of the interacting and the non-interacting effective potentials.

Energetic Information from Information-Theoretic Approach in Density Functional Theory as Quantitative Measures of Physicochemical Properties.

The Hohenberg-Kohn theorem of density functional theory (DFT) stipulates that energy is a universal functional of electron density in the ground state, so energy can be thought of having encoded essential information for the density. Based on this, we recently proposed to quantify energetic information within the framework of information-theoretic approach (ITA) of DFT (J. Chem. Phys. 2022, 157, 101103). In this study, we systematically apply energetic information to a variety of chemical phenomena to validate the use of energetic information as quantitative measures of physicochemical properties. To that end, we employed six ITA quantities such as Shannon entropy and Fisher information for five energetic densities, yielding twenty-six viable energetic information quantities. Then, they are applied to correlate with physicochemical properties of molecular systems, including chemical bonding, conformational stability, intermolecular interactions, acidity, aromaticity, cooperativity, electrophilicity, nucleophilicity, and reactivity. Our results show that different quantities of energetic information often behave differently for different properties but a few of them, such as Shannon entropy of the total kinetic energy density and information gain of the Pauli energy density, stand out and strongly correlate with several properties across different categories of molecular systems. These results suggest that they can be employed as quantitative measures of physicochemical properties. This work not only enriches the body of our knowledge about the relationship between energy and information, but also provides scores of newly introduced explicit density functionals to quantify physicochemical properties, which can serve as robust features for building machine learning models in future studies.

A polarizable valence electron density based force field for high-energy interactions between atoms and molecules.

High-accuracy molecular force field models suited for hot gases and plasmas are not as abundant as those geared toward ambient pressure and temperature conditions. Here, we present an improved version of our previous electron-density based force field model that can now account for polarization effects by adjusting the atomic valence electron contributions to match abinitio calculated Mulliken partial charges. Using a slightly modified version of the Hohenberg-Kohn theorem, we also include an improved theoretical formulation of our model when applied to systems with degenerate ground states. We present two variants of our polarizable model, fitted from abinitio reference data calculated at CCSD(T)/cc-pVTZ and CCSD(T)/CEP-31G levels of theory, that both accurately model water dimer interaction energies. Further improvements include the additional interaction components with fictitious non-spherically symmetric, yet atom-centered, electron densities and fitting the exchange and correlation coefficients against analytical expressions. The latter removes all unphysical oscillations that are observed in the previous non-polarizable variant of our force field.

Synthesis of graded Cu-NiO composites and their performance in photocatalytic CO2 reduction of carbon dioxide

Abstract In this paper, firstly, the basic principle and efficiency of photocatalytic CO2 reduction are analyzed, and the thermodynamic and kinetic principles of photoreduction of CO2 are explained. Secondly, the graded Cu-NiO composites were prepared using the solution combustion method, and the atomic structure density properties of the graded Cu-NiO composites were calculated using the Hohenberg-Kohn theorem of density generalization theory. Finally, an empirical analysis was conducted to verify the effectiveness of the graded Cu-NiO composites for photocatalytic CO2 reduction. The results showed that the overall capacitance of Cu-NiO composites was between [−3.5*107F−2·cm2,−1.5*107F−2·cm2], and the methanol yield of 4% Cu-NiO composites reached 729 μmol/g cat and the ethanol yield of 4% Cu-NiO composites decreased to 171 μmol/g cat after 5 h of reaction.

Exposing minimal composition of Kohn–Sham theory and its extendability

Reducing the many-fermion problem to a set of single-particle (s.p.) equations, the Kohn–Sham (KS) theory has provided a practical tool to implement ab initio calculations of ground-state energies and densities in many-electron systems. There have been attempts to extend the KS theory so that it could describe other physical quantities, or it could be applied to other many-fermion systems. By generalizing and reformulating the KS theory in terms of the 1-body density matrix, we expose the minimal composition of the theory that enables the reduction of the many-fermion problem to the s.p. equations. Based on the reformulation, several basic issues are reconsidered. The v- and N-representabilities for the KS theory are distinguished from those for the Hohenberg-Kohn theorem. Criteria for the extendability of the KS theory are addressed.

The Structure of the Density-Potential Mapping. Part II: Including Magnetic Fields.

The Hohenberg-Kohn theorem of density-functional theory (DFT) is broadly considered the conceptual basis for a full characterization of an electronic system in its ground state by just one-body particle density. In this Part II of a series of two articles, we aim at clarifying the status of this theorem within different extensions of DFT including magnetic fields. We will in particular discuss current-density-functional theory (CDFT) and review the different formulations known in the literature, including the conventional paramagnetic CDFT and some nonstandard alternatives. For the former, it is known that the Hohenberg-Kohn theorem is no longer valid due to counterexamples. Nonetheless, paramagnetic CDFT has the mathematical framework closest to standard DFT and, just like in standard DFT, nondifferentiability of the density functional can be mitigated through Moreau-Yosida regularization. Interesting insights can be drawn from both Maxwell-Schrödinger DFT and quantum-electrodynamic DFT, which are also discussed here.

Extension of Bright Wilson's justification of the first Hohenberg Kohn theorem to non-nuclear maxima (NNM)

Extension of Bright Wilson's justification of the first Hohenberg Kohn theorem to non-nuclear maxima (NNM)

The Structure of Density-Potential Mapping. Part I: Standard Density-Functional Theory.

The Hohenberg-Kohn theorem of density-functional theory (DFT) is broadly considered the conceptual basis for a full characterization of an electronic system in its ground state by just the one-body particle density. Part I of this review aims at clarifying the status of the Hohenberg-Kohn theorem within DFT and Part II at different extensions of the theory that include magnetic fields. We collect evidence that the Hohenberg-Kohn theorem does not so much form the basis of DFT, but is rather the consequence of a more comprehensive mathematical framework. Such results are especially useful when it comes to the construction of generalized DFTs.

Geometry of Degeneracy in Potential and Density Space

In a previous work [J. Chem. Phys. 155, 244111 (2021)], we found counterexamples to the fundamental Hohenberg-Kohn theorem from density-functional theory in finite-lattice systems represented by graphs. Here, we demonstrate that this only occurs at very peculiar and rare densities, those where density sets arising from degenerate ground states, called degeneracy regions, touch each other or the boundary of the whole density domain. Degeneracy regions are shown to generally be in the shape of the convex hull of an algebraic variety, even in the continuum setting. The geometry arising between density regions and the potentials that create them is analyzed and explained with examples that, among other shapes, feature the Roman surface.

An exact one-particle theory of bosonic excitations: from a generalized Hohenberg–Kohn theorem to convexified N-representability

Motivated by the Penrose–Onsager criterion for Bose–Einstein condensation we propose a functional theory for targeting low-lying excitation energies of bosonic quantum systems through the one-particle picture. For this, we employ an extension of the Rayleigh–Ritz variational principle to ensemble states with spectrum w and prove a corresponding generalization of the Hohenberg–Kohn theorem: the underlying one-particle reduced density matrix determines all properties of systems of N identical particles in their w -ensemble states. Then, to circumvent the v-representability problem common to functional theories, and to deal with energetic degeneracies, we resort to the Levy–Lieb constrained search formalism in combination with an exact convex relaxation. The corresponding bosonic one-body w -ensemble N-representability problem is solved comprehensively. Remarkably, this reveals a complete hierarchy of bosonic exclusion principle constraints in conceptual analogy to Pauli’s exclusion principle for fermions and recently discovered generalizations thereof.

Machine learning the Hohenberg-Kohn map for molecular excited states

The Hohenberg-Kohn theorem of density-functional theory establishes the existence of a bijection between the ground-state electron density and the external potential of a many-body system. This guarantees a one-to-one map from the electron density to all observables of interest including electronic excited-state energies. Time-Dependent Density-Functional Theory (TDDFT) provides one framework to resolve this map; however, the approximations inherent in practical TDDFT calculations, together with their computational expense, motivate finding a cheaper, more direct map for electronic excitations. Here, we show that determining density and energy functionals via machine learning allows the equations of TDDFT to be bypassed. The framework we introduce is used to perform the first excited-state molecular dynamics simulations with a machine-learned functional on malonaldehyde and correctly capture the kinetics of its excited-state intramolecular proton transfer, allowing insight into how mechanical constraints can be used to control the proton transfer reaction in this molecule. This development opens the door to using machine-learned functionals for highly efficient excited-state dynamics simulations.

Extensibility of Hohenberg–Kohn Theorem to General Quantum Systems

AbstractThe Hohenberg–Kohn (HK) theorem for interacting electrons is a cornerstone of modern electronic structure calculations. For a general quantum system, a HK‐type Hamiltonian in the form of can always be defined, by grouping those terms with fixed or preknown coefficients into an internal part of the Hamiltonian , and factorizing the remaining as the superposition of a set of Hermitian operators . It is asked whether the HK theorem can be extended to such a general setting, so that the ground‐state expectation values of as the generalized density can in principle be used as the fundamental variables determining all the properties of the system. It is shown that the question can be addressed by the invertibility of generalized density correlation matrix (GDCM) defined with respect to the operators. This criterion is applied to several representative examples, including the quantum Ising dimer, frustration‐free systems, N‐level quantum systems and a fermionic Hubbard chain. It is suggested that for a finite‐size system, finding an invertible GDCM under one single configuration is typically sufficient to establish the generic extensibility of the HK theorem in the entire parameter space.

One-to-one correspondence between thermal structure factors and coupling constants of general bilinear Hamiltonians.

A theorem that establishes a one-to-one relation between zero-temperature static spin-spin correlators and coupling constants for a general class of quantum spin Hamiltonians bilinear in the spin operators has been recently established by Quintanilla, using an argument in the spirit of the Hohenberg-Kohn theorem in density functional theory. Quintanilla's theorem gives a firm theoretical foundation to quantum spin Hamiltonian learning using spin structure factors as input data. Here we extend the validity of the theorem in two directions. First, following the same approach as Mermin, the proof is extended to the case of finite-temperature spin structure factors, thus ensuring that the application of this theorem to experimental data is sound. Second, we note that this theorem applies to all types of Hamiltonians expressed as sums of bilinear operators, so that it can also relate the density-density correlators to the Coulomb matrix elements for interacting electrons in the lowest Landau level.

A closed local-orbital unified description of DFT and many-body effects

Density functional theory (DFT) is usually formulated in terms of the electron density as a function of position n(r). Here we discuss an alternative formulation of DFT in terms of the orbital occupation numbers {n α } associated with a local-orbital orthonormal basis set {ϕ α }. First, we discuss how the building blocks of DFT, namely the Hohenberg–Kohn theorems, the Levy–Lieb approach and the Kohn–Sham method, can be adapted for a description in terms of {n α }. In particular, the total energy is now a function of {n α }, E[{n α }], and a Kohn–Sham-like Hamiltonian is derived introducing the effects of the electron–electron interactions via effective potentials, . In a second step we consider the Hartree and exchange energies and discuss how to describe them, in the spirit of a DFT approach, in terms of the orbital occupation numbers. In this contribution special attention is paid to the description of the (intra-atomic) correlation energy and corresponding correlation potentials {V corr,α }. For this purpose, a model system is analyzed in detail, whereby an atomic Hamiltonian interacts with the environment via a simplified model; the use of this model allows us to obtain the correlation energy and potentials (in terms of {n α }) for different cases corresponding to low, intermediate and high electron correlations.

Adiabatic projection: Bridging ab initio, density functional, semiempirical, and embedding approximations.

Modern electronic structure approximations routinely employ reference systems described by approximate Hamiltonians. This work introduces the adiabatic projection formalism for building formally exact corrections to such reference systems. Starting from the real Hamiltonian of a many-electron system, one constructs a reference system Hamiltonian by projecting the kinetic and electron-electron interaction operators onto "interesting" states. The reference system is corrected by density functionals for the difference between the projected and unprojected kinetic and electron-electron energies. These density functionals are constructed from adiabatic connections between the reference and real systems. The Hohenberg-Kohn theorems imply the existence of exact functionals, which can ensure that the reference system's ground-state energy and density match the real system. Adiabatic projection further generalizes Kohn-Sham density functional theory (DFT) and the generalized adiabatic connection [W. Yang, J. Chem. Phys. 109, 10107 (1998)] and recovers these methods for certain choices of projection operators. Other choices of projection operators offer new opportunities, including formally exact and systematically improvable analogues to wavefunction-in-DFT embedding, DFT+U, and semiempirical theories. Numerical results are presented for two representative choices: a projected exchange-correlation correction to small-basis-set coupled cluster theory and a projected kinetic energy density functional correcting basis set errors in DFT. The latter offers performance for dimerization energies approaching the Boys-Bernardi counterpoise correction while also correcting intramolecular basis set superposition errors.

Density-functional theory on graphs.

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.

Excited-state theorem in both case of a nuclear potential and a more general external potential in correspondence to the Hohenberg-Kohn theorem

This paper is mainly aimed at presenting an excited-state theorem in both case of a nuclear potential and a more general external potential, in correspondence to the Hohenberg-Kohn (HK) theorem for the static ground state. The author uses the second-quantized scheme employed by HK. The electron field operator is expanded in terms of complete functions. State vectors are represented by electron occupation numbers of each basis state described by the complete function. The main idea is that for the electronic density as a function of spatial coordinates, the author found a one-to-one correspondence between the electronic density and a state vector via occupation numbers. It is found that in an atom and a more general system, there exists a one-to-one correspondence between the electronic density and external potential when the eigenenergy is specified. Thus, the theorem corresponding to the Hohenberg-Kohn theorem is presented for all states at the same level. The advantages and secondary works are as follows: (1) the Kohn-Sham formalism is extended to excited states with the electronic relaxation; (2) in the Schrödinger picture all time-dependent state vectors are derived within DFT; and (3) for simple cases numerical calculations were performed.

Many-body energy density functional

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