Fractional Occupations Research Articles

Universal Generalization of Density Functional Theory for Static Correlation.

A major challenge for density functional theory (DFT) is its failure to treat static correlation, yielding errors in predicted charges, band gaps, van der Waals forces, and reaction barriers. Here we combine one- and two-electron reduced density matrix (1- and 2-RDM) theories with DFT to obtain a universal O(N^{3}) generalization of DFT for static correlation. Using the lowest unitary invariant of the cumulant 2-RDM, we generate a 1-RDM functional theory that corrects the convexity of any DFT functional to capture static correlation in its fractional orbital occupations. Importantly, the unitary invariant yields a predictive theory by revealing the dependence of the correction's strength upon the trace of the two-electron repulsion matrix. We apply the theory to the barrier to rotation in ethylene, the relative energies of the benzynes, as well as an 11-molecule, dissociation benchmark. By inheriting the computational efficiency of DFT without sacrificing the treatment of static correlation, the theory opens new possibilities for the prediction and interpretation of significant quantum molecular effects and phenomena.

What can lattice DFT teach us about real-space DFT?

In this paper we establish a connection between density functional theory (DFT) for lattice models and common real-space DFT. We consider the lattice DFT description of a two-level model subject to generic interactions in Mermin's DFT formulation in the grand canonical ensemble at finite temperature. The case of only density-density and Hund's rule interaction studied in earlier work is shown to be equivalent to an exact-exchange description of DFT in the real-space picture. In addition, we also include the so-called pair-hopping interaction which can be treated analytically and, crucially, leads to non-integer occupations of the Kohn-Sham (KS) levels even in the limit of zero temperature. Treating the hydrogen molecule in a minimal basis is shown to be equivalent to our two-level lattice DFT model. By means of the fractional occupations of the KS orbitals (which, in this case, are identical to the many-body ones) we reproduce the results of full configuration interaction, even in the dissociation limit and without breaking the spin symmetry. Beyond the minimal basis, we embed our HOMO-LUMO model into a standard DFT calculation and, again, obtain results in overall good agreement with exact ones without the need of breaking the spin symmetry.

Meta-GGA Density Functional Calculations on Atoms with Spherically Symmetric Densities in the Finite Element Formalism.

Density functional calculations on atoms are often used for determining accurate initial guesses as well as generating various types of pseudopotential approximations and efficient atomic-orbital basis sets for polyatomic calculations. To reach the best accuracy for these purposes, the atomic calculations should employ the same density functional as the polyatomic calculation. Atomic density functional calculations are typically carried out employing spherically symmetric densities, corresponding to the use of fractional orbital occupations. We have described their implementation for density functional approximations (DFAs) belonging to the local density approximation (LDA) and generalized gradient approximation (GGA) levels of theory as well as Hartree-Fock (HF) and range-separated exact exchange [Lehtola, S. Phys. Rev. A 2020, 101, 012516]. In this work, we describe the extension to meta-GGA functionals using the generalized Kohn-Sham scheme, in which the energy is minimized with respect to the orbitals, which in turn are expanded in the finite element formalism with high-order numerical basis functions. Furnished with the new implementation, we continue our recent work on the numerical well-behavedness of recent meta-GGA functionals [Lehtola, S.; Marques, M. A. L. J. Chem. Phys. 2022, 157, 174114]. We pursue complete basis set (CBS) limit energies for recent density functionals and find many to be ill-behaved for the Li and Na atoms. We report basis set truncation errors (BSTEs) of some commonly used Gaussian basis sets for these density functionals and find the BSTEs to be strongly functional dependent. We also discuss the importance of density thresholding in DFAs and find that all of the functionals studied in this work yield total energies converged to 0.1 μEh when densities smaller than 10-11a0-3 are screened out.

Particle-Breaking Hartree-Fock Theory for Open Molecular Systems.

In this work we present the particle-breaking Hartree-Fock (PBHF) model which is a mean-field approach to open molecular systems. The interaction of a system with the environment is parametrized through a particle-breaking term in the molecular Hamiltonian. The PBHF wave function is constructed through an exponential unitary transformation of a Slater determinant with a given number of electrons. We consider only the closed-shell formalism. The parametrization results in a linear combination of Slater determinants with different numbers of electrons, i.e., the PBHF wave function is not an eigenfunction of the number operator. As a result, the density matrix may have fractional occupations which are, because of the unitary parametrization, always between 0.0 and 2.0. The occupations are optimized simultaneously with the orbitals, using the trust-region optimization procedure. In the limit of a particle-conserving Hamiltonian, the PBHF optimization will converge to a standard Hartree-Fock wave function. We show that the average number of electrons may be decreased or increased depending on whether the particle-breaking term affects occupied or virtual orbitals.

Localized orbital scaling correction for periodic systems.

Density functional theory offers accurate structure prediction at acceptable computational cost, but commonly used approximations suffer from delocalization error; this results in inaccurate predictions of quantities such as energy band gaps of finite and bulk systems, energy level alignments, and electron distributions at interfaces. The localized orbital scaling correction (LOSC) was developed to correct delocalization error by using orbitals localized in space and energy. These localized orbitals span both the occupied and unoccupied spaces and can have fractional occupations in order to correct both the total energy and the one-electron energy eigenvalues. We extend the LOSC method to periodic systems, in which the localized orbitals employed are dually localized Wannier functions. In light of the effect of the bulk environment on the electrostatic interaction between localized orbitals, we modify the LOSC energy correction to include a screened Coulomb kernel. For a test set of semiconductors and large-gap insulators, we show that the screened LOSC method consistently improves the band gap compared to the parent density functional approximation.

Conservative Potentials for a Lattice-Mapped, Coarse-Grain Scheme with Fuzzy Switching Functions.

We extend our previous work (Luo, S.; Thachuk, M. J. Phys. Chem. A 2021, 125, 64866497) on determining conservative potentials for lattice-like, coarse-grain (CG) mapping schemes to the case where the boundaries between different spatial regions are not sharply defined but are fuzzy. In other words, the system is divided into interpenetrating "subcells" such that atomistic particles continuously change their memberships as they move through space. This is done by using fuzzy switching functions to define overlapping regions between subcells with fractional particle occupations. In this case, a full mass matrix is required to describe the system, and its off-diagonal elements are nonzero and contribute to the CG potential. As the overlapping region increases in size, we observe the mass distribution transitions from a discrete spectrum, through an intermediate state, and finally to a continuous Gaussian-like function. We interpret this as a quantitative measure for signaling when a continuum-theory description of the system is appropriate. Nonzero correlations among all CG variables are calculated and are found to depend strongly on the degree of overlap. In particular, those for the diagonal mass elements decrease in magnitude, and there exists a specific value of the overlap for which the correlations are zero. Other correlations are strong only when the overlap is quite large, so there is a trade-off between the complexity of the interactions in the system and the degree of fuzziness between the subcells. However, if the number of particles in a subcell is large enough and the overlap is moderate, then the CG potential is found to be well-approximated by a generalized quadratic function. These results demonstrate the transition between atomistic and continuum resolutions in a system and have implications for designing CG schemes with mixed atomistic and continuum character.

Electronic energies from coupled fermionic "Zombie" states' imaginary time evolution.

Zombie states are a recently introduced formalism to describe coupled coherent fermionic states that address the fermionic signproblem in a computationally tractable manner. Previously, it has been shown that Zombie states with fractional occupations of spin orbitals obeyed the correct fermionic creation and annihilation algebra and presented results for real-time evolution [D. V. Shalashilin, J. Chem. Phys. 148, 194109 (2018)]. In this work, we extend and build on this formalism by developing efficient algorithms for evaluating the Hamiltonian and other operators between Zombie states and address their normalization. We also show how imaginary time propagation can be used to find the ground state of a system. We also present a biasing method, for setting up a basis set of random Zombie states, that allows much smaller basis sizes to be used while still accurately describing the electronic structure Hamiltonian and its ground state and describe a technique of wave function "cleaning" that removes the contributions of configurations with the wrong number of electrons, improving the accuracy further. We also show how low-lying excited states can be calculated efficiently using a Gram-Schmidt orthogonalization procedure. The proposed algorithm of imaginary time propagation on biased random grids of Zombie states may present an alternative to the existing quantum Monte Carlo methods.

Approximate functionals in hypercomplex Kohn–Sham theory

The recently developed hypercomplex Kohn–Sham (HCKS) theory shows great potential to overcome the static/strong correlation issue in density functional theory (DFT), which highlights the necessity of further exploration of the HCKS theory toward better handling many-electron problem. This work mainly focuses on approximate functionals in HCKS, seeking to gain more insights into functional development from the comparison between Kohn–Sham (KS) DFT and HCKS. Unlike KS-DFT, HCKS can handle different correlation effects by resorting to a set of auxiliary orbitals with dynamically varying fractional occupations. These orbitals of hierarchical correlation (HCOs) thus contain distinct electronic information for better considering the exchange–correlation effect in HCKS. The test on the triplet–singlet gaps shows that HCKS has much better performance as compared to KS-DFT in use of the same functionals, and the systematic errors of semi-local functionals can be effectively reduced by including appropriate amount of the HCO-dependent Hartree–Fock exchange. In contrast, KS-DFT shows large systematic errors, which are hardly reduced by the functionals tested in this work. Therefore, HCKS creates new channels to address to the strong correlation issue, and further development of functionals that depend on HCOs and their occupations is necessary for the treatment of strongly correlated systems.

Unity of Kohn-Sham density-functional theory and reduced-density-matrix-functional theory

This work presents a rigorous theory to unify the two independent theoretical frameworks of Kohn-Sham (KS) density-functional theory (DFT) and reduced-density-matrix-functional theory (RDMFT). The generalization of the KS orbitals to hypercomplex number systems leads to the hypercomplex KS (HCKS) theory, which extends the search space for the electron density in KS-DFT and reformulates the kinetic energy. The HCKS theory provides a general framework, and different dimensions of the HCKS orbitals lead to different HCKS methods, with KS-DFT and RDMFT being two cases corresponding to the smallest and largest dimensions. Furthermore, a series of tests show that HCKS can capture the multireference nature of strong correlation by dynamically varying fractional occupations, while maintaining the same computational scaling as the KS method. With great potential to overcome the fundamental limitations of the KS method, HCKS creates new possibilities for the development and application of DFT.

Excitation energies from thermally assisted-occupation density functional theory: Theory and computational implementation.

The time-dependent density functional theory (TDDFT) has been broadly used to investigate the excited-state properties of various molecular systems. However, the current TDDFT heavily relies on outcomes from the corresponding ground-state DFT calculations, which may be prone to errors due to the lack of proper treatment in the non-dynamical correlation effects. Recently, thermally assisted-occupation DFT (TAO-DFT) [J.-D. Chai, J. Chem. Phys. 136, 154104 (2012)], a DFT with fractional orbital occupations, was proposed, explicitly incorporating the non-dynamical correlation effects in the ground-state calculations with low computational complexity. In this work, we develop TDTAO-DFT, which is a TD, linear-response theory for excited states within the framework of TAO-DFT. With tests on the excited states of H2, the first triplet excited state (13Σu +) was described well, with non-imaginary excitation energies. TDTAO-DFT also yields zero singlet-triplet gap in the dissociation limit for the ground singlet (11Σg +) and the first triplet state (13Σu +). In addition, as compared to traditional TDDFT, the overall excited-state potential energy surfaces obtained from TDTAO-DFT are generally improved and better agree with results from the equation-of-motion coupled-cluster singles and doubles.

COMPORTAMIENTO FERROMAGNÉTICO CON SATURACIÓN INCOMPLETA DEL COMPUESTO INTERMETÁLICO Ƞ-Al5Fe2 SINTETIZADO MEDIANTE LA TÉCNICA DE FUSIÓN POR DESCARGA DE ARCO

El presente trabajo presenta un estudio del proceso de síntesis y de caracterización estructural y magnética del intermetálico ƞ-Al5Fe2 preparado mediante la técnica de fusión por descarga eléctrica de arco. Con la finalidad de obtener una fase homogénea, todas las muestras obtenidas por fusión fueron sometidas a un tratamiento de recocido térmico durante 2 días a 847 °C y, finalmente, enfriadas rápidamente en agua.La caracterización estructural y micro-estructural de las muestras se realizó mediante un análisis combinado de difracción de rayos X y espectroscopia Mössbauer de transmisión del Fe57; en tanto que, el estudio morfológico de los granos se realizó usando microscopía electrónica de barrido. La caracterización magnética de las muestras se llevó a cabo empleando la técnica de magnetometría de muestra vibrante. Medidas de fluorescencia de rayos X fueron realizadas para el análisis de composición de elementos.Los resultados indican que el método de síntesis empleado permite obtener directamente la fase intermetálica ƞ-Al5Fe2 con alta calidad estructural: estructura cristalina ortorrómbica y grupo espacial Cmcm También se ha encontrado que la aleación ƞ-Al5Fe2 muestra un comportamiento ferromagnético con saturación incompleta. Por otro lado, estudios Mössbauer de la estructura local alrededor de los átomos de hierro muestran desorden topológico debido a las ocupancias fraccionales inherentes en la estructura del sistema intermetálico bajo estudio. Finalmente, el análisis composicional indica que el material se encuentra dentro del rango de homogeneidad permitido.

Active-Space Pair Two-Electron Reduced Density Matrix Theory for Strong Correlation

An active-space variational calculation of the two-electron reduced density matrix (2-RDM) is derived and implemented where the active orbitals are correlated within the pair approximation. The pair approximation considers only doubly occupied configurations of the wave function, which enables the calculation of the 2-RDM at a computational cost of [Formula: see text]. Calculations were performed both with the pair active-space configuration interaction (PASCI) method and the pair active-space self-consistent field (PASSCF) method. The latter includes a mixing of the active and inactive orbitals through unitary transformations. The active-space pair 2-RDM method is applied to the nitrogen molecule, the p-benzyne diradical, a newly synthesized biscobalt complex, and the nitrogenase cofactor FeMoco. The FeMoco molecule is treated in a [120,120] active space. Fractional occupations are recovered in each of these systems, indicating the presence of strong electron correlation.

Fully numerical calculations on atoms with fractional occupations and range-separated exchange functionals

A recently developed finite element approach for fully numerical atomic structure calculations [S. Lehtola, Int. J. Quantum Chem. 119, e25945 (2019)] is extended to the description of atoms with spherically symmetric densities via fractionally occupied orbitals. Specialized versions of Hartree-Fock as well as local density and generalized gradient approximation density functionals are developed, allowing extremely rapid calculations at the basis set limit on the ground and low-lying excited states even for heavy atoms. The implementation of range-separation based on the Yukawa or complementary error function (erfc) kernels is also described, allowing complete basis set benchmarks of modern range-separated hybrid functionals with either integer or fractional occupation numbers. Finally, computation of atomic effective potentials at the local density or generalized gradient approximation levels for the superposition of atomic potentials (SAP) approach [S. Lehtola, J. Chem. Theory Comput. 15, 1593 (2019)] that has been shown to be a simple and efficient way to initialize electronic structure calculations is described. The present numerical approach is shown to afford beyond microhartree accuracy with a small number of numerical basis functions, and to reproduce literature results for the ground states of atoms and their cations for $1 \leq Z \leq 86 $. Our results indicate that the literature values deviate by up to 10 {\mu}Eh from the complete basis set limit. The numerical scheme for the erfc kernel is shown to work by comparison to results from large Gaussian basis set calculations from the literature. Spin-restricted ground states are reported for Hartree-Fock and Hartree-Fock-Slater calculations with fractional occupations for $1 \leq Z \leq 118$.

Strong Correlation and Charge Localization in Kohn-Sham Theories with Fractional Orbital Occupations.

We study static correlation and delocalization errors and show that even methods with good energies can yield significant delocalization errors that affect the density, leading to large errors in predicting, e.g., dipole moments. We illustrate this point by comparing existing state-of-the-art approaches with an accurate exchange-correlation functional based on a generalized valence-bond ansatz in which orbitals and fractional occupations are treated as variational parameters via an optimized effective potential (OEP). We show that the OEP exhibits step and peak features that, similar to the exact Kohn-Sham potential of density functional theory (DFT), are crucial to prevent charge delocalization. We further show that the step is missing in common approximations within reduced density matrix functional theory, resulting in delocalization errors comparable to those found in DFT approximations. Finally, we explain the delocalization error as coming from artificial mixing of the ground state with a charge-transfer excited state that is avoided if the occupation numbers exhibit discontinuities.

Density embedding with constrained chemical potential

We formulate a chemical-potential constrained density embedding method for systems where different fragments can have either continuum or discrete electronic states. We illustrate the method with the simplest model system designed to mimic an atom near a metal surface. It is trivial to separate the full system into two fragments (metal and atom) only when the distance between them is infinite. In this case, a range of metallic chemical potentials, μ, will lead to an identical number of electrons, N, on the atom. Our density embedding method can be used to define fragments even at finite separations. We show that using these definitions for fragments, the typical staircase function is partially smoothed out due to the finite-distance interactions, resembling finite-temperature effects. Fractional occupations on the atom occur only for sharply-defined μ's. Because calculating fractional charges is important in various fields, from electrolysis to catalysis, solar cells and organic electronics, we anticipate several potential uses of the proposed approach.

On extending Kohn-Sham density functionals to systems with fractional number of electrons.

We analyze four ways of formulating the Kohn-Sham (KS) density functionals with a fractional number of electrons, through extending the constrained search space from the Kohn-Sham and the generalized Kohn-Sham (GKS) non-interacting v-representable density domain for integer systems to four different sets of densities for fractional systems. In particular, these density sets are (I) ensemble interacting N-representable densities, (II) ensemble non-interacting N-representable densities, (III) non-interacting densities by the Janak construction, and (IV) non-interacting densities whose composing orbitals satisfy the Aufbau occupation principle. By proving the equivalence of the underlying first order reduced density matrices associated with these densities, we show that sets (I), (II), and (III) are equivalent, and all reduce to the Janak construction. Moreover, for functionals with the ensemble v-representable assumption at the minimizer, (III) reduces to (IV) and thus justifies the previous use of the Aufbau protocol within the (G)KS framework in the study of the ground state of fractional electron systems, as defined in the grand canonical ensemble at zero temperature. By further analyzing the Aufbau solution for different density functional approximations (DFAs) in the (G)KS scheme, we rigorously prove that there can be one and only one fractional occupation for the Hartree Fock functional, while there can be multiple fractional occupations for general DFAs in the presence of degeneracy. This has been confirmed by numerical calculations using the local density approximation as a representative of general DFAs. This work thus clarifies important issues on density functional theory calculations for fractional electron systems.

The quest for best suited references for configuration interaction singles calculations of core excited states.

Near edge X-ray absorption fine structure (NEXAFS) simulations based on the conventional configuration interaction singles (CIS) lead to excitation energies, which are systematically blue shifted. Using a (restricted) open shell core hole reference instead of the Hartree Fock (HF) ground state orbitals improves (Decleva et al., Chem. Phys., 1992, 168, 51) excitation energies and the shape of the spectra significantly. In this work, we systematically vary the underlying SCF approaches, that is, based on HF or density functional theory, to identify best suited reference orbitals using a series of small test molecules. We compare the energies of the K edges and NEXAFS spectra to experimental data. The main improvement compared to conventional CIS, that is, using HF ground state orbitals, is due to the electrostatic influence of the core hole. Different SCF approaches, density functionals, or the use of fractional occupations lead only to comparably small changes. Furthermore, to account for bigger systems, we adapt the core-valence separation for our approach. We demonstrate that the good quality of the spectrum is not influenced by this approximation when used together with the non-separated ground state wave function. Simultaneously, the computational demands are reduced remarkably. © 2016 Wiley Periodicals, Inc.

Orbitals from local RDMFT: Are they Kohn-Sham or natural orbitals?

Recently, an approximate theoretical framework was introduced, called local reduced density matrix functional theory (local-RDMFT), where functionals of the one-body reduced density matrix (1-RDM) are minimized under the additional condition that the optimal orbitals satisfy a single electron Schrödinger equation with a local potential. In the present work, we focus on the character of these optimal orbitals. In particular, we compare orbitals obtained by local-RDMFT with those obtained with the full minimization (without the extra condition) by contrasting them against the exact NOs and orbitals from a density functional calculation using the local density approximation (LDA). We find that the orbitals from local-RMDFT are very close to LDA orbitals, contrary to those of the full minimization that resemble the exact NOs. Since local RDMFT preserves the good quality of the description of strong static correlation, this finding opens the way to a mixed density/density matrix scheme, where Kohn-Sham orbitals obtain fractional occupations from a minimization of the occupation numbers using 1-RDM functionals. This will allow for a description of strong correlation at a cost only minimally higher than a density functional calculation.

Synthesis, Structural, Spectroscopic, and Magnetic Characterization of Two-Coordinate Cobalt(II) Aryloxides with Bent or Linear Coordination

Treatment of the cobalt(II) amide, [Co{N(SiMe3)2}2]2, with four equivalents of the sterically crowded terphenyl phenols, HOAr(Me6) (Ar(Me6) = C6H3-2,6(C6H2-2,4,6-Me3)2) or HOAr(iPr4) (Ar(iPr4) = C6H3-2,6(C6H3-2,6-Pr(i)2)2), produced the first well-characterized, monomeric two-coordinate cobalt(II) bisaryloxides, Co(OAr(Me6))2 (1) and Co(OAr(iPr4))2 (2a and 2b), as red solids in good yields with elimination of HN(SiMe3)2. The compounds were characterized by electronic spectroscopy, X-ray crystallography, and direct current magnetization measurements. The O-Co-O interligand angles in 2a and 2b are 180°, whereas the O-Co-O angle in 1 is bent at 130.12(8)° and its cobalt(II) ion has a highly distorted pseudotetrahedral geometry with close interactions to the ipso-carbons of the two flanking aryl rings. The Co-O distances in 1, 2a, and 2b are 1.858(2), 1.841(1), and 1.836(2) Å respectively. Structural refinement revealed that 1, 2a, and 2b have different fractional occupations of the cobalt site in their crystal structures: 1, 95.0%, 2a, 93.5%, and 2b, 84.6%. Correction of the magnetic data for the different cobalt(II) occupancies showed that the magnetization of 2a and 2b was virtually identical. The effective magnetic moments for 1, 2a, and 2b, 5.646(5), 5.754(5), and 5.636(3) μB respectively, were indicative of significant spin-orbit coupling. The differences in magnetic properties between 1 and 2a/2b are attributed to their different cobalt coordination geometries.

Extensions of the Time-Dependent Density Functional Based Tight-Binding Approach.

The time-dependent density functional based tight-binding (TD-DFTB) approach is generalized to account for fractional occupations. In addition, an on-site correction leads to marked qualitative and quantitative improvements over the original method. Especially, the known failure of TD-DFTB for the description of σ → π* and n → π* excitations is overcome. Benchmark calculations on a large set of organic molecules also indicate a better description of triplet states. The accuracy of the revised TD-DFTB method is found to be similar to first principles TD-DFT calculations at a highly reduced computational cost. As a side issue, we also discuss the generalization of the TD-DFTB method to spin-polarized systems. In contrast to an earlier study, we obtain a formalism that is fully consistent with the use of local exchange-correlation functionals in the ground state DFTB method.