About the compatibility between ansatzes and constraints for a local formulation of orbital‐free density functional theory

Functional properties that are exact for the Hohenberg–Kohn functional may turn into mutually exclusive constraints at a given level of ansatz. This is exemplarily shown for the local density approximation. Nevertheless, it is possible to reach exactly the Kohn–Sham data from an orbital‐free density functional framework based on simple one‐point functionals by starting from the Levy–Perdew–Sahni formulation. The energy value is obtained from the density‐potential pair, and therefore does not refer to the functional dependence of the potential expression. Consequently, the potential expression can be obtained from any suitable model and is not required to follow proper scaling behavior.     

Exchange and correlation energy in density functional theory

Several different versions of density functional theory (DFT) that satisfy Hohenberg–Kohn theorems are characterized by different definitions of a reference or model state determined by an N‐electron ground state. A common formalism is developed in which exact Kohn–Sham equations are derived for standard Kohn–Sham theory, for reference‐state density functional theory, and for unrestricted Hartree–Fock (UHF) theory considered as an exactly soluble model Hohenberg–Kohn theory. A natural definition of exchange and correlation energy functionals is shown to be valid for all such theories. An easily computed necessary condition for the locality of exchange and correlation potentials is derived. While it is shown that in the UHF model of DFT the optimized effective potential (OEP) exchange satisfies this condition by construction, the derivation shows that this condition is not, in general, sufficient to define an exact local exchange potential. It serves as a test to eliminate proposed local potentials that are not exact for ground states. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 521–525, 2000     

A simple model for the Slater exchange potential and its performance for solids

A simple local model for the Slater exchange potential is determined by least square fit procedure from Hartree–Fock (HF) atomic data. Since the Slater potential is the exact exchange potential yielding HF electron density from Levy‐Perdew‐Sahni density functional formalism (Levy et al., Phys. Rev. A 1984, 30, 2745), the derived local potential is significantly more negative than the conventional local density approximation. On the set of 22 ionic, covalent and van der Waals solids including strongly correlated transition metal oxides, it has been demonstrated, that this simple model potential is capable of reproducing the band gaps nearly as good as popular meta GGA potentials in close agreement with experimental values.     

Energy‐surfaces from the upper bound of the Pauli kinetic energy

Based on the Kohn–Sham Pauli potential and the Kohn–Sham electron density, the upper bound of the Pauli kinetic energy is tested as a suitable replacement for the exact Pauli kinetic energy for application in orbital‐free density functional calculations. It is found that bond lengths for strong and moderately bound systems can be qualitatively predicted, but with a systematic shift toward larger bond distances with a relative error of 6% up to 30%. Angular dependence of the energy‐surface cannot be modeled with the proposed functional. Therefore, the upper bound model is the first parameter‐free functional expression for the kinetic energy that is able to qualitatively reproduce binding curves with respect to bond distortions. © 2016 Wiley Periodicals, Inc.     

Accurate Kohn–Sham ionization potentials from scaled‐opposite‐spin second‐order optimized effective potential methods

One important property of Kohn–Sham (KS) density functional theory is the exact equality of the energy of the highest occupied KS orbital (HOMO) with the negative ionization potential of the system. This exact feature is out of reach for standard density‐dependent semilocal functionals. Conversely, accurate results can be obtained using orbital‐dependent functionals in the optimized effective potential (OEP) approach. In this article, we investigate the performance, in this context, of some advanced OEP methods, with special emphasis on the recently proposed scaled‐opposite‐spin OEP functional. Moreover, we analyze the impact of the so‐called HOMO condition on the final quality of the HOMO energy. Results are compared to reference data obtained at the CCSD(T) level of theory. © 2016 Wiley Periodicals, Inc.     

Correlation potentials for a multiconfigurational-based density functional theory with exact exchange

A density functional theory based on a complete active space self consistent field (CASSCF) reference function with exact exchange is discussed. It is first shown that such a theory may be formulated with a correlation potential dependent on the density function and on the active space used. Auxiliary functions, such as the on-top two-particle density, are used to define uniquely the potential for different active spaces. The paper also analyses the correlation functional for some atomic and molecular cases. Large ab initio calculations are performed to obtain accurate density functions. A correlation potential is then fitted such that the reference CASSCF function gives the same density. The correlation potential values are saved in a data base for future analysis.     

Density functional theory on phase space

Forty‐five years after the point de départ [Hohenberg and Kohn, Phys Rev, 1964, 136, B864] of density functional theory, its applications in chemistry and the study of electronic structures keep steadily growing. However, the precise form of the energy functional in terms of the electron density still eludes us—and possibly will do so forever [Schuch and Verstraete, Nat Phys, 2009, 5, 732]. In what follows we examine a formulation in the same spirit with phase space variables. The validity of Hohenberg–Kohn–Levy‐type theorems on phase space is recalled. We study the representability problem for reduced Wigner functions, and proceed to analyze properties of the new functional. Along the way, new results on states in the phase space formalism of quantum mechanics are established. Natural Wigner orbital theory is developed in depth, with the final aim of constructing accurate correlation‐exchange functionals on phase space. A new proof of the overbinding property of the Müller functional is given. This exact theory supplies its home at long last to that illustrious ancestor, the Thomas–Fermi model. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

About the difference between density functionals defined by energy criterion and density functionals defined by density criterion: Exchange functionals

The difference between density functionals defined by energy criterion and density functionals defined by density criterion is studied for the exchange functional. It is shown that Slater potentials are exact exchange potentials in the sense that they yield the Hartree–Fock electron density if all operators are given by local expressions. © 2016 Wiley Periodicals, Inc.     

Self-interaction error in density functional theory: a mean-field correction for molecules and large systems

Corrections to the self-interaction error which is rooted in all standard exchange-correlation functionals in the density functional theory (DFT) have become the object of an increasing interest. After an introduction reminding the origin of the self-interaction error in the DFT formalism, and a brief review of the self-interaction free approximations, we present a simple, yet effective, self-consistent method to correct this error. The model is based on an average density self-interaction correction (ADSIC), where both exchange and Coulomb contributions are screened by a fraction of the electron density. The ansatz on which the method is built makes it particularly appealing, due to its simplicity and its favorable scaling with the size of the system. We have tested the ADSIC approach on one of the classical pathological problem for density functional theory: the direct estimation of the ionization potential from orbital eigenvalues. A large set of different chemical systems, ranging from simple atoms to large fullerenes, has been considered as test cases. Our results show that the ADSIC approach provides good numerical values for all the molecular systems, the agreement with the experimental values increasing, due to its average ansatz, with the size (conjugation) of the systems.     

Orbital functional theory of linear response and excitation

Orbital functional theory (OFT) is based on a rule that determines a single‐determinant reference state Φ for any exact N‐electron eigenstate Ψ. An OFT model postulates an explicit correlation energy functional E_c of occupied orbital functions {?_i} and occupation numbers {n_i}. The orbital Euler–Lagrange equations are analogous to Kohn–Sham equations, but do not in general contain local potential functions. Time‐dependent Hartree–Fock theory is generalized in OFT to a formally exact linear response theory that includes electronic correlation. In the exchange‐only limit, the theory reduces to the random‐phase approximation of many‐body theory. The formalism determines excitation energies. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Approximating the Pauli Potential in Bound Coulomb Systems

It is shown that the Pauli potential in bound Coulomb systems can in good approximation be composed from the corresponding atomic fragments. This provides a simple and fast procedure how to generate the Pauli potential in bound systems, which is needed to perform an orbital‐free density functional calculation. The method is applicable to molecules and solids. © 2016 Wiley Periodicals, Inc.     

One‐particle density matrix functional for correlation in molecular systems

Based on the analysis of the general properties for the one‐ and two‐particle reduced density matrices, a new natural orbital functional is obtained. It is shown that by partitioning the two‐particle reduced density matrix in an antisymmeterized product of one‐particle reduced density matrices and a correction Γ^c we can derive a corrected Hartree–Fock theory. The spin structure of the correction term from the improved Bardeen–Cooper–Schrieffer theory is considered to take into account the correlation between pairs of electrons with antiparallel spins. The analysis affords a nonidempotent condition for the one‐particle reduced density matrix. Test calculations of the correlation energy and the dipole moment of several molecules in the ground state demonstrate the reliability of the formalism. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 317–323, 2003     

PW86–PW91 density functional calculation of vertical ionization potentials: Some implications for present‐day functionals

A total of 181 vertical ionization potentials (VIPs) of 41 molecules were calculated by density functional theory (DFT) employing the Perdew–Wang 1986 (PW86) exchange and Perdew–Wang 1991 (PW91) correlation functionals and using the aug‐cc‐pV5Z basis and experimental ground‐state geometries. The overall average absolute deviation (AAD) from experiment was found to be 0.55 eV and only 0.31 eV for linear molecules but 0.86 eV for nonplanar molecules. A number of VIPs were in error by over 2 eV. In particular, DFT performed most poorly when ionization was from an orbital with highly varying density gradients (which arise from the orbital's shape or compactness or through its density being distributed over a number of atoms). Indications are that many or all present‐day functionals suffer from the same failings. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 34–52, 2001     

Grid‐based density functional calculations of many‐electron systems

Exploratory variational pseudopotential density functional calculations are performed for the electronic properties of many‐electron systems in the 3D cartesian coordinate grid (CCG). The atom‐centered localized gaussian basis set, electronic density, and the two‐body potentials are set up in the 3D cubic box. The classical Hartree potential is calculated accurately and efficiently through a Fourier convolution technique. As a first step, simple local density functionals of homogeneous electron gas are used for the exchange‐correlation potential, while Hay‐Wadt‐type effective core potentials are employed to eliminate the core electrons. No auxiliary basis set is invoked. Preliminary illustrative calculations on total energies, individual energy components, eigenvalues, potential energy curves, ionization energies, and atomization energies of a set of 12 molecules show excellent agreement with the corresponding reference values of atom‐centered grid as well as the grid‐free calculation. Results for three atoms are also given. Combination of CCG and the convolution procedure used for classical Coulomb potential can provide reasonably accurate and reliable results for many‐electron systems. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Conditions for accurate description of the electron density of atoms and molecules

Based on the Kato's cusp condition of the electron density and our recent relations for local strongly decaying properties in an electronic system, necessary conditions for trial electron densities of atomic and molecular systems are derived. These conditions take the form of integral‐differential equations, and their validity is verified numerically. The relevance of these conditions to the Thomas–Fermi problem in the orbital‐less density functional approach is discussed. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Determining the role of the underlying orbital‐dependence of PBE0‐DH and PBE‐QIDH double‐hybrid density functionals

We study the orbital‐dependence of three (parameter‐free) double‐hybrid density functionals, namely the PBE0‐DH, the PBE‐QIDH models, and the SOS1‐PBE‐QIDH spin‐opposite‐scaled variant of the latter. To do it, we feed all their energy terms with different sets of orbitals obtained previously from self‐consistent density functional theory calculations using several exchange‐correlation functionals (e.g., PBE, PBE0, PBEH&H), or directly with HF‐PBE orbitals, to see their effect on selected datasets for atomization and reaction energies, the latter proned to marked self‐interaction errors. We find that the PBE‐QIDH double‐hybrid model shows a great consistency, as the best results are always obtained for the set of orbitals corresponding to its hybrid scheme, which prompts us to recommend this model without any other fitting or reparameterization. © 2017 Wiley Periodicals, Inc.     

Exact electronic properties in the classically forbidden region of a metal surface

We derive exact properties of the inhomogeneous electron gas in the asymptotic classically forbidden region at a metal–vacuum interface within the framework of local effective potential energy theory. We derive a new expression for the asymptotic structure of the Kohn–Sham density functional theory (KS‐DFT) exchange‐correlation potential energy v_xc(r) in terms of the irreducible electron self‐energy. We also derive the exact asymptotic structure of the orbitals, density, the Dirac density matrix, the kinetic energy density, and KS exchange energy density. We further obtain the exact expression for the Fermi hole and demonstrate its structure in this asymptotic limit. The exchange‐correlation potential energy is derived to be v_xc(z → ∞) = ?α_KS,xc/z, and its exchange and correlation components to be v_x(z → ∞) = ?α_KS,x/z and v_c(z → ∞) = ?α_KS,c/z, respectively. The analytical expressions for the coefficients α_KS,xc and α_KS,x show them to be dependent on the bulk‐metal Wigner–Seitz radius and the barrier height at the surface. The coefficient α_KS,c = 1/4 is determined in the plasmon‐pole approximation and is independent of these metal parameters. Thus, the asymptotic structure of v_xc(z) in the vacuum region is image‐potential‐like but not the commonly accepted one of ?1/4z. Furthermore, this structure depends on the properties of the metal. Additionally, an analysis of these results via quantal density functional theory (Q‐DFT) shows that both the Pauli W_x(z → ∞) and lowest‐order correlation‐kinetic W(z → ∞) components of the exchange potential energy v_x(z → ∞), and the Coulomb W_c(z → ∞) and higher‐order correlation‐kinetic components of the correlation potential energy v_c(z → ∞), all contribute terms of O(1/z) to the structure. Hence correlations attributable to the Pauli exclusion principle, Coulomb repulsion, and correlation‐kinetic effects all contribute to the asymptotic structure of the effective potential energy at a metal surface. The relevance of the results derived to the theory of image states and to KS‐DFT is also discussed. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Examination of several density functionals in numerical Kohn–Sham calculations for atoms

We studied several exchange‐only and exchange–correlation energy density functionals in numerical, i.e., basis‐set‐free, nonrelativistic Kohn–Sham calculations for closed‐shell ¹S states of atoms and atomic ions with N electrons, where 2≤N≤120. Accurate total energies are presented to serve as reference data for algebraic approaches, as do the numerical Hartree–Fock results, which are also provided. Gradient‐corrected exchange‐only functionals considerably improve the total energies obtained from the usual local density approximation, when compared to the Hartree–Fock results. Such an improvement due to gradient corrections is not seen in general for highest orbital energies, neither for exchange‐only results (to be compared with Hartree–Fock results), nor for exchange–correlation results (to be compared with experimental ionization energies). © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 227–241, 2001     

Plane wave/pseudopotential implementation of excited state gradients in density functional linear response theory: a new route via implicit differentiation

This work presents the formalism and implementation of excited state nuclear forces within density functional linear response theory using a plane wave basis set. An implicit differentiation technique is developed for computing nonadiabatic coupling between Kohn-Sham molecular orbital wave functions as well as gradients of orbital energies which are then used to calculate excited state nuclear forces. The algorithm has been implemented in a plane wave/pseudopotential code taking into account only a reduced active subspace of molecular orbitals. It is demonstrated for the H(2) and N(2) molecules that the analytical gradients rapidly converge to the exact forces when the active subspace of molecular orbitals approaches completeness.     

Local potentials in independent‐electron models

Density functional theory, exact for ground states, is commonly assumed to imply an independent‐electron model in which only local potential functions appear. It has recently been shown that several paradoxes in different aspects of the theory can be resolved if this locality hypothesis is abandoned. However, the locality hypothesis itself appears to be implied by rigorous variational theory. This conflict is discussed and resolved here. The resolution involves embedding the density functional theory in an orbital functional theory whose functional derivatives are not confined to normalized ground states. The orbital Euler–Lagrange equations of this extended theory in general contain effective potentials that are linear operators acting on orbital wave functions. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 384–388, 2001