Theorem For Density Functional Theory Research Articles

Tilted-Plane Structure of the Energy of Finite Quantum Systems.

The piecewise linearity condition on the total energy with respect to the total magnetization of finite quantum systems is derived using the infinite-separation-limit technique. This generalizes the well-known constancy condition, related to static correlation error, in approximate density functional theory. The magnetic analog of Koopmans' theorem in density functional theory is also derived. Moving to fractional electron count, the tilted-plane condition is derived, lifting certain assumptions in previous works. This generalization of the flat-plane condition characterizes the total energy surface of a finite system for all values of electron count N and magnetization M. This result is used in combination with tabulated spectroscopic data to show the flat-plane structure of the oxygen atom, among others. We find that derivative discontinuities with respect to electron count sometimes occur at noninteger values. A diverse set of tilted-plane structures is shown to occur in d-orbital subspaces, depending on chemical coordination. General occupancy-based total-energy expressions are demonstrated thereby to be necessarily dependent on the symmetry-imposed degeneracies.

Machine learning electronic structure methods based on the one-electron reduced density matrix

The theorems of density functional theory (DFT) establish bijective maps between the local external potential of a many-body system and its electron density, wavefunction and, therefore, one-particle reduced density matrix. Building on this foundation, we show that machine learning models based on the one-electron reduced density matrix can be used to generate surrogate electronic structure methods. We generate surrogates of local and hybrid DFT, Hartree-Fock and full configuration interaction theories for systems ranging from small molecules such as water to more complex compounds like benzene and propanol. The surrogate models use the one-electron reduced density matrix as the central quantity to be learned. From the predicted density matrices, we show that either standard quantum chemistry or a second machine-learning model can be used to compute molecular observables, energies, and atomic forces. The surrogate models can generate essentially anything that a standard electronic structure method can, ranging from band gaps and Kohn-Sham orbitals to energy-conserving ab-initio molecular dynamics simulations and infrared spectra, which account for anharmonicity and thermal effects, without the need to employ computationally expensive algorithms such as self-consistent field theory. The algorithms are packaged in an efficient and easy to use Python code, QMLearn, accessible on popular platforms.

An interpretation of quantum foundations based on density functional theory and polymer self-consistent field theory

The Feynman quantum-classical isomorphism between classical statistical mechanics in 3+1 dimensions and quantum statistical mechanics in 3 dimensions is used to connect classical polymer self-consistent field theory with quantum time-dependent density functional theory. This allows the theorems of density functional theory to relate non-relativistic quantum mechanics to a classical statistical mechanical derivation of polymer self-consistent field theory for ring polymers in a 4 dimensional thermal-space. One dynamic postulate is added to two static postulates which allows for a description of quantum physics from a 5 dimensional thermal-space-time ensemble perspective. A connection with aspects of classical field theory can be made in the classical limit.

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.

Core-Level Excitation Energies of Nucleic Acid Bases Expressed as Orbital Energies of the Kohn-Sham Density Functional Theory with Long-Range Corrected Functionals.

The core electron binding energies (CEBEs) and core-level excitation energies of thymine, adenine, cytosine, and uracil are studied by the Kohn-Sham (KS) method with long-range corrected (LC) functionals. The CEBEs are estimated according to the Koopmans-type theorem for density functional theory. The excitation energies from the core to the valence π* and Rydberg states are calculated as the orbital energy differences between core-level orbitals of a neutral parent/cation and unoccupied π* or Rydberg orbitals of its cation. The model is intuitive, and the spectra can easily be assigned. Core excitation energies from oxygen 1s, nitrogen 1s, and carbon 1s to π* and Rydberg states, and the chemical shifts, agree well with previously reported theoretical and experimental data. The straightforward use of KS orbitals in this scheme carries the advantage that it can be applied efficiently to large systems such as biomolecules and nanomaterials.

Information‐theoretic approach in density functional theory and its recent applications to chemical problems

AbstractUsing simple functionals of the electron density to appreciate and quantify molecular structure and chemical reactivity properties is a recent endeavor in density functional theory (DFT) toward the development of a new chemical reactivity theory. According to the first Hohenberg–Kohn theorem in DFT, the electron density alone should be able to determine any property in the ground state. Exchange and correlation energies are such properties, so are molecular structure and chemical reactivity, and hence they all should accurately be determined by electron density functionals. Quantities such as Shannon entropy, Fisher information, Ghosh–Berkowitz–Parr entropy, information gain, Onicescu information energy, etc., from the information‐theoretic approach are simple electron density functionals, whose analytical forms are exactly known. In this article, we demonstrate their usefulness and validity to quantify regioselectivity, stereoselectivity, and other structure and reactivity properties. We will outline the current understanding of its theoretical framework at first, and then highlight recent applications to chemical problems including isomeric and conformational stability, electrophilicity and nucleophilicity, strong covalent and weak noncovalent interactions, acidity and basicity, aromaticity and antiaromaticity, and numerous other properties. The effort of employing electron density functionals to quantify chemical concepts should open up a new door for us to ultimately develop a chemical reactivity theory using the DFT language.This article is characterized under:Structure and Mechanism > Reaction Mechanisms and CatalysisElectronic Structure Theory > Density Functional TheoryStructure and Mechanism > Molecular StructuresMolecular and Statistical Mechanics > Molecular Interactions

An alternative derivation of orbital-free density functional theory.

Polymer self-consistent field theory techniques are used to derive quantum density functional theory without the use of the theorems of density functional theory. Instead, a free energy is obtained from a partition function that is constructed directly from a Hamiltonian so that the results are, in principle, valid at finite temperatures. The main governing equations are found to be a set of modified diffusion equations, and the set of self-consistent equations are essentially identical to those of a ring polymer system. The equations are shown to be equivalent to Kohn-Sham density functional theory and to reduce to classical density functional theory, each under appropriate conditions. The obtained noninteracting kinetic energy functional is, in principle, exact but suffers from the usual orbital-free approximation of the Pauli exclusion principle in addition to the exchange-correlation approximation. The equations are solved using the spectral method of polymer self-consistent field theory, which allows the set of modified diffusion equations to be evaluated for the same computational cost as solving a single diffusion equation. A simple exchange-correlation functional is chosen, together with a shell-structure-based Pauli potential, in order to compare the ensemble average electron densities of several isolated atom systems to known literature results. The agreement is excellent, justifying the alternative formalism and numerical method. Some speculation is provided on considering the timelike parameter in the diffusion equations, which is related to temperature, as having dimensional significance, and thus picturing pointlike quantum particles instead as nonlocal, polymerlike, threads in a higher dimensional thermal-space. A consideration of the double-slit experiment from this point of view is speculated to provide results equivalent to the Copenhagen interpretation. Thus, the present formalism may be considered as a type of "pilot-wave," realist, perspective on density functional theory.

About the relation of electron–electron interaction potentials with exchange and correlation functionals

We investigate, numerically, the possibility of associating to each approximation to the exchange-and-correlation functional in density-functional theory (DFT), an optimal electron–electron interaction potential for which it performs best. The fundamental theorems of density-functional theory (DFT) make no assumption about the precise form of the electron–electron interaction: to each possible electron–electron interaction corresponds an exchange-and-correlation functional. This fact suggests the opposite question: given some functional of the density, is there any electron–electron interaction for which it is the exact exchange-and-correlation functional? And, if not, what is the interaction for which the functional produces the best results? Within the context of lattice DFT, we study these questions by working on the one-dimensional Hubbard chain. The idea of associating an optimal interaction potential to each approximation to the exchange and correlation functionals suggests, finally, a procedure to optimise parameterised families of functionals: find that one whose associated interaction most closely resembles the real one.

Metric-space approach to potentials and its relevance to density-functional theory

External potentials play a crucial role in modelling quantum systems, since, for a given inter- particle interaction, they define the system Hamiltonian. We use the metric space approach to quantum mechanics to derive, from the energy conservation law, two natural metrics for potentials. We show that these metrics are well defined for physical potentials, regardless of whether the system is in an eigenstate or if the potential is bounded. In addition, we discuss the gauge freedom of potentials and how to ensure that the metrics preserve physical relevance. Our metrics for potentials, together with the metrics for wavefunctions and densities from [I. D'Amico, J. P. Coe, V. V. Franca, and K. Capelle, Phys. Rev. Lett. 106, 050401 (2011)] paves the way for a comprehensive study of the two fundamental theorems of Density Functional Theory. We explore these by analysing two many- body systems for which the related exact Kohn-Sham systems can be derived. First we consider the information provided by each of the metrics, and we find that the density metric performs best in distinguishing two many-body systems. Next we study for the systems at hand the one-to-one relationships among potentials, ground state wavefunctions, and ground state densities defined by the Hohenberg-Kohn theorem as relationships in metric spaces. We find that, in metric space, these relationships are monotonic and incorporate regions of linearity, at least for the systems considered. Finally, we use the metrics for wavefunctions and potentials in order to assess quantitatively how close the many-body and Kohn-Sham systems are: We show that, at least for the systems analysed, both metrics provide a consistent picture, and for large regions of the parameter space the error in approximating the many-body wavefunction with the Kohn-Sham wavefunction lies under a threshold of 10%.

Exact maps in density functional theory for lattice models

In the present work, we employ exact diagonalization for model systems on a real-space lattice to explicitly construct the exact density-to-potential and graphically illustrate the complete exact density-to-wavefunction map that underly the Hohenberg–Kohn theorem in density functional theory. Having the explicit wavefunction-to-density map at hand, we are able to construct arbitrary observables as functionals of the ground-state density. We analyze the density-to-potential map as the distance between the fragments of a system increases and the correlation in the system grows. We observe a feature that gradually develops in the density-to-potential map as well as in the density-to-wavefunction map. This feature is inherited by arbitrary expectation values as functional of the ground-state density. We explicitly show the excited-state energies, the excited-state densities, and the correlation entropy as functionals of the ground-state density. All of them show this exact feature that sharpens as the coupling of the fragments decreases and the correlation grows. We denominate this feature as intra-system steepening and discuss how it relates to the well-known inter-system derivative discontinuity. The inter-system derivative discontinuity is an exact concept for coupled subsystems with degenerate ground state. However, the coupling between subsystems as in charge transfer processes can lift the degeneracy. An important conclusion is that for such systems with a near-degenerate ground state, the corresponding cut along the particle number N of the exact density functionals is differentiable with a well-defined gradient near integer particle number.

Application of Koopmans’ theorem for density functional theory to full valence-band photoemission spectroscopy modeling

In this work, Koopmans’ theorem for Kohn-Sham density functional theory (KS-DFT) is applied to the photoemission spectra (PES) modeling over the entire valence-band. To examine the validity of this application, a PES modeling scheme is developed to facilitate a full valence-band comparison of theoretical PES spectra with experiments. The PES model incorporates the variations of electron ionization cross-sections over atomic orbitals and a linear dispersion of spectral broadening widths. KS-DFT simulations of pristine rubrene (5,6,11,12-tetraphenyltetracene) and potassium–rubrene complex are performed, and the simulation results are used as the input to the PES models. Two conclusions are reached. First, decompositions of the theoretical total spectra show that the dissociated electron of the potassium mainly remains on the backbone and has little effect on the electronic structures of phenyl side groups. This and other electronic-structure results deduced from the spectral decompositions have been qualitatively obtained with the anionic approximation to potassium–rubrene complexes. The qualitative validity of the anionic approximation is thus verified. Second, comparison of the theoretical PES with the experiments shows that the full-scale simulations combined with the PES modeling methods greatly enhance the agreement on spectral shapes over the anionic approximation. This agreement of the theoretical PES spectra with the experiments over the full valence-band can be regarded, to some extent, as a collective validation of the application of Koopmans’ theorem for KS-DFT to valence-band PES, at least, for this hydrocarbon and its alkali-adsorbed complex.

On Koopmans’ theorem in density functional theory

This paper clarifies why long-range corrected (LC) density functional theory gives orbital energies quantitatively. First, the highest occupied molecular orbital and the lowest unoccupied molecular orbital energies of typical molecules are compared with the minus vertical ionization potentials (IPs) and electron affinities (EAs), respectively. Consequently, only LC exchange functionals are found to give the orbital energies close to the minus IPs and EAs, while other functionals considerably underestimate them. The reproducibility of orbital energies is hardly affected by the difference in the short-range part of LC functionals. Fractional occupation calculations are then carried out to clarify the reason for the accurate orbital energies of LC functionals. As a result, only LC functionals are found to keep the orbital energies almost constant for fractional occupied orbitals. The direct orbital energy dependence on the fractional occupation is expressed by the exchange self-interaction (SI) energy through the potential derivative of the exchange functional plus the Coulomb SI energy. On the basis of this, the exchange SI energies through the potential derivatives are compared with the minus Coulomb SI energy. Consequently, these are revealed to be cancelled out only by LC functionals except for H, He, and Ne atoms.

Correlation effects in the valence ionization spectra of large conjugated molecules: p-Benzoquinone, anthracenequinone and pentacenequinone

A review of an extensive series of theoretical studies of the valence one-electron and shake-up ionization spectra of polycyclic aromatic hydrocarbons is presented, along with new results for three planar quinone derivatives, obtained using one-particle Green's function (1p-GF) theory along with the so-called third-order algebraic diagrammatic construction [ADC(3)] scheme and the outer-valence Green's function (OVGF) approximation. These results confirm both for the π- and σ-band systems the rapid spreading, upon increasing system size, of many shake-up lines with significant intensities at outer-valence energies. Linear regressions demonstrate that with large conjugated molecules the location of the shake-up onset in the π-band system is merely determined by the energy of the frontier (HOMO, LUMO) orbitals. Electron pair removal effects are found to almost compensate the electron relaxation effects induced by ionization of π-levels, whereas the latter effects strongly dominate the ionization of more localized lone-pair ( n) levels, and may lead to inversions of the energy order of Hartree–Fock (HF) orbitals. Therefore, although it increases upon a lowering of the HF band gap, and thus upon an increase of system size, the dependence of the one-electron ionization energies onto the quality of the basis set is lesser for π-levels than for σ-levels relating to electron lone pairs ( n). Basis sets of triple- and quadruple-zeta quality are therefore required for treatments of the outermost π- and n-ionization energies approaching chemical accuracy [1 kcal/mol, i.e. 0.04 eV]. When 1p-GF theory invalidates Koopmans’ theorem and the energy order of HF orbitals, a comparison with Kohn–Sham orbital energies confirms the validity of the meta-Koopmans’ theorem for density functional theory.

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.

Differential virial theorem in density-functional theory in terms of the Pauli potential for spherically symmetric electron densities: Illustrative example for the family of Be-like atomic ions

The differential virial theorem relates the force $\ensuremath{-}\ensuremath{\partial}V∕\ensuremath{\partial}\mathbf{r}$ associated with the one-body potential $V(\mathbf{r})$ of density-functional theory to the Laplacian ${\ensuremath{\nabla}}^{2}n$ of the ground-state density $n(\mathbf{r})$ and to a quantity ${\mathbf{z}}_{s}(\mathbf{r})$ involving the kinetic energy density tensor ${t}_{\ensuremath{\alpha}\ensuremath{\beta}}(\mathbf{r})$. Having the concept of the Pauli potential ${V}_{P}(\mathbf{r})$, ${z}_{s}$ is derived for spherically symmetric ground-state densities $n(r)$ in terms of the von Weizs\"acker kinetic energy density and the first derivative of ${V}_{P}(r)$. ${\mathbf{z}}_{s}$ is related solely to the gradient kinetic energy density ${t}_{G}(r)$ for Be-like atomic ions.

Frustrated magnetism in superconducting hexagonal Fe: Calculation of inter-atomic pair exchange interactions

We calculate magnetic exchange interaction parameters in hcp Fe using magnetic force theorem in density functional theory in a range of volumes, which covers the region of stability of the superconducting state. The calculations are done for a paramagnetic state using disordered local moment model within a local fixed spin-moment constraint. It is found that exchange interaction parameters strongly favor antiferromagnetic inter-plane stacking of the basal hcp planes. At the same time, the nearest neighbor in-plane exchange interaction parameters are also strongly antiferromagnetic, which should lead to magnetic frustrations in the planar triangular lattice. Eventual consequences of the magnetic frustration effects on the interplay between magnetism and superconductivity in hcp Fe under pressure are discussed.

Simple implementation of complex functionals: Scaled self-consistency

We explore and compare three approximate schemes allowing simple implementation of complex density functionals by making use of self-consistent implementation of simpler functionals: (i) post-local-density approximation (LDA) evaluation of complex functionals at the LDA densities (or those of other simple functionals) (ii) application of a global scaling factor to the potential of the simple functional, and (iii) application of a local scaling factor to that potential. Option (i) is a common choice in density-functional calculations. Option (ii) was recently proposed by Cafiero and Gonzalez [Phys. Rev. A 71, 042505 (2005)]. We here put their proposal on a more rigorous basis, by deriving it, and explaining why it works, directly from the theorems of density-functional theory. Option (iii) is proposed here for the first time. We provide detailed comparisons of the three approaches among each other and with fully self-consistent implementations for Hartree, local-density, generalized-gradient, self-interaction corrected, and meta-generalized-gradient approximations, for atoms, ions, quantum wells, and model Hamiltonians. Scaled approaches turn out to be, on average, better than post approaches, and unlike these also provide corrections to eigenvalues and orbitals. Scaled self-consistency thus opens the possibility of efficient and reliable implementation of density functionals of hitherto unprecedented complexity.

Differentiability in density-functional theory: Further study of the locality theorem

The locality theorem in density-functional theory (DFT) states that the functional derivative of the Hohenberg-Kohn universal functional can be expressed as a local multiplicative potential function, and this is the basis of DFT and of the successful Kohn-Sham model. Nesbet has in several papers [Phys. Rev. A 58, R12 (1998); Phys. Rev. A65, 010502 (2001); Adv. Quant. Chem, 43, 1 (2003)] claimed that this theorem is in conflict with fundamental quantum physics, and as a consequence that the Hohenberg-Kohn theory cannot be generally valid. We have commented upon these works [Comment, Phys. Rev. A 67, 056501 (2003)] and recently extended the arguments [Adv. Quantum Chem. 43, 95 (2003)]. We have shown that there is no such conflict and that the locality theorem is inherently exact. In the present work we have furthermore verified this numerically by constructing a local Kohn-Sham potential for the $1s2s^{3}S$ state of helium that generates the many-body electron density and shown that the corresponding $2s$ Kohn-Sham orbital eigenvalue agrees with the ionization energy to nine digits. Similar result is obtained with the Hartree-Fock density. Therefore, in addition to verifying the locality theorem, this result also confirms the so-called ionization-potential theorem.

Hund?s multiplicity rule: a unified interpretation

Hund’s multiplicity rule, stating that a higher spin state has a lower energy within the same electron configuration, is empirical but has shown to be valid for both atoms and molecules. Several theoretical interpretations for its validity, including explanations in terms of the lower interelectron repulsion and the greater electron–nuclear attraction in the higher spin state, are available. None of them, however, are satisfactory. Here we show that Hund’s rule can be explained by the Janak theorem in density functional theory, extended to excited states and multiplets. In the exact density functional theory theory, it leads to ΔE ST=E S−E T=ΔeHOMO, with E S and E T the singlet and triplet state energies and eHOMO the highest occupied molecular orbital energies of the spin states. This relationship was previously obtained by M. Levy [(1995) Physical Review A 52:R4313]. In this paper, numerical results within the Hartree–Fock framework for both atoms and molecules confirm the previously mentioned justification. Good results of the Hartree–Fock method come from the accurate description of the exchange effect from where Hund’s multiplicity rule originated.

Physical interpretation and evaluation of the Kohn–Sham and Dyson components of the ε–I relations between the Kohn–Sham orbital energies and the ionization potentials

Theoretical and numerical insight is gained into the ε–I relations between the Kohn–Sham orbital energies εi and relaxed vertical ionization potentials (VIPs) Ij, which provide an analog of Koopmans’ theorem for density functional theory. The Kohn–Sham orbital energy εi has as leading term −niIi−∑j∈Ωs(i)njIj, where Ii is the primary VIP for ionization (φi)−1 with spectroscopic factor (proportional to the intensity in the photoelectron spectrum) ni close to 1, and the set Ωs(i) contains the VIPs Ij that are satellites to the (φi)−1 ionization, with small but non-negligible nj. In addition to this “average spectroscopic structure” of the εi there is an electron-shell step structure in εi from the contribution of the response potential vresp. Accurate KS calculations for prototype second- and third-row closed-shell molecules yield valence orbital energies −εi, which correspond closely to the experimental VIPs, with an average deviation of 0.08 eV. The theoretical relations are numerically investigated in calculations of the components of the ε–I relations for the H2 molecule, and for the molecules CO, HF, H2O, HCN. The derivation of the ε–I relations employs the Dyson orbitals (the ni are their norms). A connection is made between the KS and Dyson orbital theories, allowing the spin-unrestricted KS xc potential to be expressed with a statistical average of individual xc potentials for the Dyson spin–orbitals as leading term. Additional terms are the correction vc,kin,σ due to the correlation kinetic effect, and the “response” vresp,σ, related to the correction to the energy of (N−1) electrons due to the correlation with the reference electron.