Preserving Symmetry and Degeneracy in the Localized Orbital Scaling Correction Approach

Nonempirical Prediction of the Length-Dependent Ionization Potential in Molecular Chains

The ionization potential of molecular chains is well-known to be a tunable nanoscale property that exhibits clear quantum confinement effects. State-of-the-art methods can accurately predict the ionization potential in the small molecule limit and in the solid-state limit, but for intermediate, nanosized systems prediction of the evolution of the electronic structure between the two limits is more difficult. Recently, optimal tuning of range-separated hybrid functionals has emerged as a highly accurate method for predicting ionization potentials. This was first achieved for molecules using the ionization potential theorem (IPT) and more recently extended to solid-state systems, based on an ansatz that generalizes the IPT to the removal of charge from a localized Wannier function. Here, we study one-dimensional molecular chains of increasing size, from the monomer limit to the infinite polymer limit using this approach. By comparing our results with other localization-based methods and where available with experiment, we demonstrate that Wannier-localization-based optimal tuning is highly accurate in predicting ionization potentials for any chain length, including the nanoscale regime.

Combining Local Range Separation and Local Hybrids: A Step in the Quest for Obtaining Good Energies and Eigenvalues from One Functional

Some of the most successful exchange-correlation approximations in density functional theory are "hybrids", i.e., they rely on combining semilocal density functionals with exact nonlocal Fock exchange. In recent years, two classes of hybrid functionals have emerged as particularly promising: range-separated hybrids on the one hand, and local hybrids on the other hand. These functionals offer the hope to overcome a long-standing "observable dilemma", i.e., the fact that density functionals typically yield either a good description of binding energies, as obtained, e.g., in global and local hybrids, or physically interpretable eigenvalues, as obtained, e.g., in optimally tuned range-separated hybrids. Obtaining both of these characteristics from one and the same functional with the same set of parameters has been a long-standing challenge. We here discuss combining the concepts of local range separation and local hybrids as part of a constraint-guided quest for functionals that overcome the observable dilemma.

Excited-State Charge Transfer Coupling from Quasiparticle Energy Density Functional Theory

The recently developed Quasiparticle Energy (QE) scheme, based on a DFT calculation with one more (or less) electron, offers a good description of excitation energies, even with charge transfer characters. In this work, QE is further extended to calculate electron transfer (ET) couplings involving two excited states. We tested it with a donor-acceptor complex, consisting of a furan and a 1,1-dicyanoethylene (DCNE), in which two low lying charge transfer and local excitation states are involved. With generalized Mülliken-Hush and fragment charge-difference schemes, couplings from the QE approach generally agree well with those obtained from TDDFT, except that QE couplings exhibit better exponential distance dependence. Couplings from half-energy gaps with an external field are also calculated and reported. Our results show that the QE scheme is robust in calculating ET couplings with greatly reduced computational time.

How well do one-electron self-interaction-correction methods perform for systems with fractional electrons?

Recently developed locally scaled self-interaction correction (LSIC) is a one-electron SIC method that, when used with a ratio of kinetic energy densities (zσ) as iso-orbital indicator, performs remarkably well for both thermochemical properties as well as for barrier heights overcoming the paradoxical behavior of the well-known Perdew-Zunger self-interaction correction (PZSIC) method. In this work, we examine how well the LSIC method performs for the delocalization error. Our results show that both LSIC and PZSIC methods correctly describe the dissociation of H2+ and He2+ but LSIC is overall more accurate than the PZSIC method. Likewise, in the case of the vertical ionization energy of an ensemble of isolated He atoms, the LSIC and PZSIC methods do not exhibit delocalization errors. For the fractional charges, both LSIC and PZSIC significantly reduce the deviation from linearity in the energy vs number of electrons curve, with PZSIC performing superior for C, Ne, and Ar atoms while for Kr they perform similarly. The LSIC performs well at the endpoints (integer occupations) while substantially reducing the deviation. The dissociation of LiF shows both LSIC and PZSIC dissociate into neutral Li and F but only LSIC exhibits charge transfer from Li+ to F- at the expected distance from the experimental data and accurate ab initio data. Overall, both the PZSIC and LSIC methods reduce the delocalization errors substantially.

Stretching Bonds without Breaking Symmetries in Density Functional Theory

Kohn-Sham density functional theory (KS-DFT) stands out among electronic structure methods due to its balance of accuracy and computational efficiency. However, to achieve chemically accurate energies, standard density functional approximations in KS-DFT often need to break underlying symmetries, a long-standing "symmetry dilemma". By employing fragment spin densities as the main variables in calculations (rather than total molecular densities, as in KS-DFT), we present an embedding framework in which this symmetry dilemma is understood and partially resolved. The spatial overlap between fragment densities is used as the main ingredient to construct a simple, physically motivated approximation to a universal functional of the fragment densities. This "overlap approximation" is shown to significantly improve semilocal KS-DFT binding energies of molecules without artificially breaking either charge or spin symmetries. The approach is shown to be applicable to covalently bonded molecules and to systems of the "strongly correlated" type.

Corrected density functional theory and the random phase approximation: Improved accuracy at little extra cost

We recently introduced an efficient methodology to perform density-corrected Hartree-Fock density functional theory [DC(HF)-DFT] calculations and an extension to it we called "corrected" HF DFT [C(HF)-DFT] [Graf and Thom, J. Chem. Theory Comput. 19 5427-5438 (2023)]. In this work, we take a further step and combine C(HF)-DFT, augmented with a straightforward orbital energy correction, with the random phase approximation (RPA). We refer to the resulting methodology as corrected HF RPA [C(HF)-RPA]. We evaluate the proposed methodology across various RPA methods: direct RPA (dRPA), RPA with an approximate exchange kernel, and RPA with second-order screened exchange. C(HF)-dRPA demonstrates very promising performance; for RPA with exchange methods, on the other hand, we often find over-corrections.

koopmans: An Open-Source Package for Accurately and Efficiently Predicting Spectral Properties with Koopmans Functionals

Over the past decade we have developed Koopmans functionals, a computationally efficient approach for predicting spectral properties with an orbital-density-dependent functional framework. These functionals impose a generalized piecewise linearity condition to the entire electronic manifold, ensuring that orbital energies match the corresponding electron removal/addition energy differences (in contrast to semilocal DFT, where a mismatch between the two lies at the heart of the band gap problem and, more generally, the unreliability of Kohn-Sham orbital energies). This strategy has proven to be very powerful, yielding molecular orbital energies and solid-state band structures with comparable accuracy to many-body perturbation theory but at greatly reduced computational cost while preserving a functional formulation. This paper reviews the theory of Koopmans functionals, discusses the algorithms necessary for their implementation, and introduces koopmans, an open-source package that contains all of the code and workflows needed to perform Koopmans functional calculations and obtain reliable spectral properties of molecules and materials.

Fractional Charge Density Functional Theory and Its Application to the Electro-inductive Effect

We employed the chemical potential equalization principle to demonstrate that fractional electrons are involved in the electro-inductive effect as well as the vibrational Stark effect. By the chemical potential model, we were able to deduce that the frontier molecular orbitals of immobilized molecules can provide valuable insight into these effects. To further understand and quantify these findings, we introduced fractional charge density functional theory (FC-DFT), a canonical ensemble approach for open systems. This method allows for the calculation of electronic energies, nuclear gradients, and the Hessian matrix of fractional electronic systems. To correct the spurious delocalization error commonly found in approximate density functionals for small systems, we imposed the Perdew-Parr-Levy-Balduz (PPLB) condition through linear interpolation of two adjacent integer points (LI-FC-DFT). Although this approach is relatively simple in terms of molecular modeling, the results obtained through LI-FC-DFT calculations predict the same trend seen in experimental reactivity and the frequency change of immobilized molecules.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

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.

Exact Analytical Form of Diatomic Molecular Orbitals

We provide the exact analytical form of diatomic molecular orbitals, as given by the solutions of a single-electron diatomic molecule with arbitrary nuclear charges, using our recently developed method for solving Schrödinger equations. We claim that the best representation of the wave function is a factorized form including a power prefactor, an exponentially decaying term, a modulator function on the exponential, and additional factors accounting for nodal surfaces and the magnetic quantum number. Applying our method, we have identified unexpected extreme points along the potential energy curves, hence revealing the limitations of the well-known concepts of bonding and antibonding.

Describing Chemical Reactivity with Frontier Molecular Orbitalets

Locality in physical space is critical in understanding chemical reactivity in the analysis of various phenomena and processes in chemistry, biology, and materials science, as exemplified in the concepts of reactive functional groups and active sites. Frontier molecular orbitals (FMOs) pinpoint the locality of chemical bonds that are chemically reactive because of the associated orbital energies and thus have achieved great success in describing chemical reactivity, mainly for small systems. For large systems, however, the delocalization nature of canonical molecular orbitals makes it difficult for FMOs to highlight the locality of the chemical reactivity. To obtain localized molecular orbitals that also reflect the frontier nature of the chemical processes, we develop the concept of frontier molecular orbitalets (FMOLs) for describing the reactivity of large systems. The concept of orbitalets was developed recently in the localized orbital scaling correction method, which aims for eliminating the delocalization error in common density functional approximations. Orbitalets are localized in both physical and energy spaces and thus contain both orbital locality and energy information. The FMOLs are thus the orbitalets with energies highest among occupied orbitalets and lowest among unoccupied ones. The applications of FMOLs to hexadeca-1,3,5,7,9,11,13,15-octaene in its equilibrium geometry, inter- and intra-molecular charge-transfer systems, and two transition states of a bifurcating reaction demonstrate that FMOLs can connect quantum mechanical treatments of chemical systems and chemical reactivities by locating the reactive region of large chemical systems. Therefore, FMOLs extend the role of FMOs for small systems and describe the chemical reactivity of large systems with energy and locality insight, with potentially broad applications.

DFT+U within the framework of linear combination of numerical atomic orbitals

We present a formulation and implementation of the density functional theory (DFT)+U method within the framework of linear combination of numerical atomic orbitals (NAO). Our implementation not only enables single-point total energy and electronic-structure calculations but also provides access to atomic forces and cell stresses, hence allowing for full structure relaxations of periodic systems. Furthermore, our implementation allows one to deal with non-collinear spin texture, with the spin-orbit coupling (SOC) effect treated self-consistently. The key aspect behind our implementation is a suitable definition of the correlated subspace when multiple atomic orbitals with the same angular momentum are used, and this is addressed via the "Mulliken charge projector" constructed in terms of the first (most localized) atomic orbital within the d/f angular momentum channel. The important Hubbard U and Hund J parameters can be estimated from a screened Coulomb potential of the Yukawa type, with the screening parameter either chosen semi-empirically or determined from the Thomas-Fermi screening model. Benchmark calculations are performed for four late transition metal monoxide bulk systems, i.e., MnO, FeO, CoO, and NiO, and for the 5d-electron compounds IrO₂. For the former type of systems, we check the performance of our DFT+U implementation for calculating bandgaps, magnetic moments, electronic band structures, as well as forces and stresses; for the latter, the efficacy of our DFT+U+SOC implementation is assessed. Systematic comparisons with available experimental results, especially with the results from other implementation schemes, are carried out, which demonstrate the validity of our NAO-based DFT+U formalism and implementation.

Implementation and Validation of Constrained Density Functional Theory Forces in the CP2K Package

Constrained density functional theory (CDFT) is a powerful tool for the prediction of electron transfer parameters in condensed phase simulations at a reasonable computational cost. In this work we present an extension to CDFT in the popular mixed Gaussian/plane wave electronic structure package CP2K, implementing the additional force terms arising from a constraint based on Hirshfeld charge partitioning. This improves upon the existing Becke partitioning scheme, which is prone to give unphysical atomic charges. We verify this implementation for a variety of systems: electron transfer in (H₂O)₂⁺ in a vacuum, electron tunnelling between oxygen vacancy centers in solid MgO, and electron self-exchange in aqueous Ru²⁺–Ru³⁺. We find good agreement with previous plane-wave CDFT results for the same systems, but at a significantly lower computational cost, and we discuss the general reliability of condensed phase CDFT calculations.

Understanding the chemical bonding of ground and excited states of HfO and HfB with correlated wavefunction theory and density functional approximations

Knowledge of the chemical bonding of HfO and HfB ground and low-lying electronic states provides essential insights into a range of catalysts and materials that contain Hf-O or Hf-B moieties. Here, we carry out high-level multi-reference configuration interaction theory and coupled cluster quantum chemical calculations on these systems. We compute full potential energy curves, excitation energies, ionization energies, electronic configurations, and spectroscopic parameters with large quadruple-ζ and quintuple-ζ quality correlation consistent basis sets. We also investigate equilibrium chemical bonding patterns and effects of correlating core electrons on property predictions. Differences in the ground state electron configuration of HfB(X⁴Σ^-) and HfO(X¹Σ⁺) lead to a significantly stronger bond in HfO than HfB, as judged by both dissociation energies and equilibrium bond distances. We extend our analysis to the chemical bonding patterns of the isovalent HfX (X = O, S, Se, Te, and Po) series and observe similar trends. We also note a linear trend between the decreasing value of the dissociation energy (D_e) from HfO to HfPo and the singlet-triplet energy gap (ΔE_S-T) of the molecule. Finally, we compare these benchmark results to those obtained using density functional theory (DFT) with 23 exchange-correlation functionals spanning multiple rungs of "Jacob's ladder." When comparing DFT errors to coupled cluster reference values on dissociation energies, excitation energies, and ionization energies of HfB and HfO, we observe semi-local generalized gradient approximations to significantly outperform more complex and high-cost functionals.

Molecular orbital projectors in non-empirical jmDFT recover exact conditions in transition-metal chemistry

Low-cost, non-empirical corrections to semi-local density functional theory are essential for accurately modeling transition-metal chemistry. Here, we demonstrate the judiciously modified density functional theory (jmDFT) approach with non-empirical U and J parameters obtained directly from frontier orbital energetics on a series of transition-metal complexes. We curate a set of nine representative Ti(III) and V(IV) d1 transition-metal complexes and evaluate their flat-plane errors along the fractional spin and charge lines. We demonstrate that while jmDFT improves upon both DFT+U and semi-local DFT with the standard atomic orbital projectors (AOPs), it does so inefficiently. We rationalize these inefficiencies by quantifying hybridization in the relevant frontier orbitals. To overcome these limitations, we introduce a procedure for computing a molecular orbital projector (MOP) basis for use with jmDFT. We demonstrate this single set of d1 MOPs to be suitable for nearly eliminating all energetic delocalization error and static correlation error. In all cases, MOP jmDFT outperforms AOP jmDFT, and it eliminates most flat-plane errors at non-empirical values. Unlike DFT+U or hybrid functionals, jmDFT nearly eliminates energetic delocalization error and static correlation error within a non-empirical framework.

Combining localized orbital scaling correction and Bethe-Salpeter equation for accurate excitation energies

We applied localized orbital scaling correction (LOSC) in Bethe-Salpeter equation (BSE) to predict accurate excitation energies for molecules. LOSC systematically eliminates the delocalization error in the density functional approximation and is capable of approximating quasiparticle (QP) energies with accuracy similar to or better than GW Green's function approach and with much less computational cost. The QP energies from LOSC, instead of commonly used G0W0 and evGW, are directly used in BSE. We show that the BSE/LOSC approach greatly outperforms the commonly used BSE/G0W0 approach for predicting excitations with different characters. For the calculations of Truhlar-Gagliardi test set containing valence, charge transfer, and Rydberg excitations, BSE/LOSC with the Tamm-Dancoff approximation provides a comparable accuracy to time-dependent density functional theory (TDDFT) and BSE/evGW. For the calculations of Stein CT test set and Rydberg excitations of atoms, BSE/LOSC considerably outperforms both BSE/G0W0 and TDDFT approaches with a reduced starting point dependence. BSE/LOSC is, thus, a promising and efficient approach to calculate excitation energies for molecular systems.

Exact Second-Order Corrections and Accurate Quasiparticle Energy Calculations in Density Functional Theory

We develop a second-order correction to commonly used density functional approximations (DFAs) to eliminate the systematic delocalization error. The method, based on the previously developed global scaling correction (GSC), is an exact quadratic correction to the DFA for the fractional charge behavior and uses the analytical second derivatives of the total energy with respect to fractional occupation numbers of the canonical molecular orbitals. For small and medium-size molecules, this correction leads to ground-state orbital energies that are a highly accurate approximation to the corresponding quasiparticle energies. It provides excellent predictions of ionization potentials, electron affinities, photoemission spectrum, and photoexcitation energies beyond previous approximate second-order approaches, thus showing potential for broad applications in computational spectroscopy.

Replacing hybrid density functional theory: motivation and recent advances

Density functional theory (DFT) is the most widely-used electronic structure approximation across chemistry, physics, and materials science. Every year, thousands of papers report hybrid DFT simulations of chemical structures, mechanisms, and spectra. Unfortunately, hybrid DFT's accuracy is ultimately limited by tradeoffs between over-delocalization and under-binding. This review summarizes these tradeoffs, and introduces six modern attempts to go beyond them while maintaining hybrid DFT's relatively low computational cost: DFT+U, self-interaction corrections, localized orbital scaling corrections, local hybrid functionals, real-space nondynamical correlation, and our rung-3.5 approach. The review concludes with practical suggestions for DFT users to identify and mitigate these tradeoffs' impact on their simulations.

Charge-Transfer Steps in Density Functional Theory from the Perspective of the Exact Electron Factorization

When a molecule dissociates, the exact Kohn-Sham (KS) and Pauli potentials may form step structures. Reproducing these steps correctly is central for the description of dissociation and charge-transfer processes in density functional theory (DFT): The steps align the KS eigenvalues of the dissociating subsystems relative to each other and determine where electrons localize. While the step height can be calculated from the asymptotic behavior of the KS orbitals, this provides limited insight into what causes the steps. We give an explanation of the steps with an exact mapping of the many-electron problem to a one-electron problem, the exact electron factorization (EEF). The potentials appearing in the EEF have a clear physical meaning that translates to the DFT potentials by replacing the interacting many-electron system with the KS system. With a simple model of a diatomic, we illustrate that the steps are a consequence of spatial electron entanglement and are the result of a charge transfer. From this mechanism, the step height can immediately be deduced. Moreover, two methods to approximately reproduce the potentials during dissociation are proposed. One is based on the states of the dissociated system, while the other one is based on an analogy to the Born-Oppenheimer treatment of a molecule. The latter method also shows that the steps connect adiabatic potential energy surfaces. The view of DFT from the EEF thus provides a better understanding of how many-electron effects are encoded in a one-electron theory and how they can be modeled.

Describing polymer polarizability with localized orbital scaling correction in density functional theory

Polarizability reflects the response of the molecular charge distribution to an applied external electric field and thus closely relates to the molecular electron density. For the calculation of polarizability within density functional theory (DFT), it is well known that conventional density functional approximations (DFAs) greatly overestimate the results for polymers with long chains and the π-conjugated system. This is a manifestation of the delocalization error of the commonly used DFAs-they normally produce too delocalized electron density and underestimate the total energy for systems with fractional charge character, which occurs for long molecules in a longitudinal electric field. Thus, to achieve an accurate description of polarizabilities for polymeric molecular systems from DFT, applying DFAs with minimal delocalization error is very important. In this work, we use the recently developed localized orbital scaling correction (LOSC) to the conventional DFAs, which has been shown to largely eliminate the delocalization error, to calculate and study the polarizabilities of three classic polymers, polyyne, polyacetylene, and hydrogen chain. The results from this work demonstrate that applying LOSC to conventional DFAs with self-consistent field calculations can largely improve the description of polarizability from DFT calculations and the improved quality of electron density in LOSC leads to the improved results of polarizability of the polymers. However, the improvement is not complete and adjustment of the parameters in the LOSC method can further improve the accuracy to reach the level similar to the MP2 method. This work also points to the direction for the further development of LOSC in self-consistent calculations.

Density Functional Prediction of Quasiparticle, Excitation, and Resonance Energies of Molecules With a Global Scaling Correction Approach

Molecular quasiparticle and excitation energies determine essentially the spectral characteristics measured in various spectroscopic experiments. Accurate prediction of these energies has been rather challenging for ground-state density functional methods, because the commonly adopted density function approximations suffer from delocalization error. In this work, by presuming a quantitative correspondence between the quasiparticle energies and the generalized Kohn–Sham orbital energies, and employing a previously developed global scaling correction approach, we achieve substantially improved prediction of molecular quasiparticle and excitation energies. In addition, we also extend our previous study on temporary anions in resonant states, which are associated with negative molecular electron affinities. The proposed approach does not require any explicit self-consistent field calculation on the excited-state species, and is thus highly efficient and convenient for practical purposes.

Introductory lecture: when the density of the noninteracting reference system is not the density of the physical system in density functional theory

A major challenge in density functional theory (DFT) is the development of density functional approximations (DFAs) to overcome errors in existing DFAs, leading to more complex functionals. For such functionals, we consider roles of the noninteracting reference systems. The electron density of the Kohn-Sham (KS) reference with a local potential has been traditionally defined as being equal to the electron density of the physical system. This key idea has been applied in two ways: the inverse calculation of such a local KS potential for the reference from a given density and the direct calculation of density and energy based on given DFAs. By construction, the inverse calculation can yield a KS reference with the density equal to the input density of the physical system. In application of DFT, however, it is the direct calculation of density and energy from a DFA that plays a central role. For direct calculations, we find that the self-consistent density of the KS reference defined by the optimized effective potential (OEP), is not the density of the physical system, when the DFA is dependent on the external potential. This inequality holds also for the density of generalized KS (GKS) or generalized OEP reference, which allows a nonlocal potential, when the DFA is dependent on the external potential. Instead, the density of the physical system, consistent with a given DFA, is given by the linear response of the total energy with respect to the variation of the external potential. This is a paradigm shift in DFT on the use of noninteracting references: the noninteracting KS or GKS references represent the explicit computational variables for energy minimization, but not the density of the physical system for external potential-dependent DFAs. We develop the expressions for the electron density so defined through the linear response for general DFAs, demonstrate the results for orbital functionals and for many-body perturbation theory within the second-order and the random-phase approximation, and explore the connections to developments in DFT.

Self-Consistent Calculation of the Localized Orbital Scaling Correction for Correct Electron Densities and Energy-Level Alignments in Density Functional Theory

The recently developed localized orbital scaling correction (LOSC) method shows the ability to systematically and size-consistently reduce the delocalization error existing in conventional density functional approximations (DFAs). However, the application of LOSC to DFAs was mainly through a post self-consistent field (SCF) manner, and few results from applying LOSC to DFAs in an SCF manner have been reported. The reason is that the originally proposed SCF approach to SCF-LOSC calculation uses an approximate Hamiltonian and encounters convergence problems easily in practice. In this work, we develop a new SCF approach with a correct Hamiltonian and achieve reliable SCF-LOSC calculations. We demonstrate the capability of the new SCF approach for SCF-LOSC to correctly describe the electron densities, total energies, and energy-level alignment for the molecular dissociation process, while conventional DFAs or LOSC-DFAs with post-SCF calculations show large errors. This work demonstrates that the new SCF approach for SCF-LOSC would be a promising method for studying problems for correct electron densities and energy-level alignments in large systems.