Improved self-consistent and resolution-of-identity approximated Becke'05 density functional model of nondynamic electron correlation

Multiconfigurational Correlation at DFT + U Cost: On-Site Electron-Electron Interactions Yield a Block-Localized Configuration Interaction Hamiltonian

This work presents a first-principles wavefunction-in-DFT approach based on the Hubbard density functional theory (DFT) + U method. This approach begins with the standard DFT reference system of noninteracting electrons and introduces an electron-electron interaction projected onto DFT+U-type atomic states. The reference system's configuration interaction Hamiltonian is block-localized to these states and can be expressed in terms of state occupation numbers, state self-energies (which correspond to unscreened Hubbard U values), and the promotion energies of doubly excited Slater determinants. Simple approximations for the promotion energies provide multiconfigurational correlation energies without requiring explicit orbital localization/transform. Numerical results for fractionally occupied chromium atom, bonded chromium dimer, dissociating covalent bonds, and large active spaces show that the approach provides beyond-zero-sum accuracy at computational cost comparable to standard DFT+U.

Density-functional theory vs density-functional fits: The best of both

In a recent paper [A. D. Becke, J. Chem. Phys. 156, 214101 (2022)], we compared two Kohn-Sham density functionals based on physical modeling and theory with the best density-functional power series fits in the literature. With only a handful of physically motivated pre-factors, our functionals matched, and even slightly exceeded, the performance of the best power-series functionals on the general main group thermochemistry, kinetics, and noncovalent interactions (GMTKN55) chemical database of Goerigk et al. [Phys. Chem. Chem. Phys. 19, 32184 (2017)]. This begs the question: how much can their performance be improved by adding power-series terms of our own? We address this question in the present work. First, we describe a series expansion variable that we believe contains more local physics than any other variable considered to date. Then we undertake modest, one-dimensional fits to the GMTKN55 data with our theory-based functional corrected by power-series exchange and dynamical correlation terms. We settle on 12 power-series terms (plus six parent terms) and achieve the lowest GMTKN55 "WTMAD2" error yet reported, by a substantial margin, for a hybrid Kohn-Sham density functional. The new functional is called "B22plus."

Local Hybrid Functional Applicable to Weakly and Strongly Correlated Systems

The recent idea (Wodyński, A.; Arbuznikov, A. V.; Kaupp M. J. Chem. Phys. 2021, 155, 144101) to augment local hybrid functionals by a strong-correlation (sc) factor obtained from the adiabatic connection in the spirit of the KP16 model has been extended and applied to generate the accurate sc-corrected local hybrid functional scLH22t. By damping small values of the ratio between nondynamical and dynamical correlation entering the correction factor, it has become possible to avoid double counting of nondynamical correlation for weakly correlated situations and thereby preserve the excellent accuracy of the underlying LH20t local hybrid for such cases almost perfectly. On the other hand, scLH22t improves substantially over LH20t in reducing fractional-spin errors (FSEs), in providing improved spin-restricted bond dissociation curves, and in treating some typical systems with multireference character. The obtained FSEs are similar to those of the KP16/B13 model and slightly larger than for B13, but performance for weakly correlated systems is better than for these two related methods, which are also difficult to use self-consistently. The recent DM21 functional based on the training of a deep neural network still performs somewhat better than scLH22t but allows no physical insights into the origins of reduced FSEs. Examination of local mixing functions (LMFs) for the corrected scLH22t and uncorrected LH20t functionals provides further insights: in weakly correlated situations, the LMF remains essentially unchanged. Strong-correlation effects manifest in a reduction of the LMF values in certain regions of space, even to the extent of producing negative LMF values. It is suggested that this is the mechanism by which also DM21, which may be viewed as a range-separated local hybrid, is able to reduce FSEs.

Recent Advances in Cartesian-Grid DFT in Atoms and Molecules

In the past several decades, density functional theory (DFT) has evolved as a leading player across a dazzling variety of fields, from organic chemistry to condensed matter physics. The simple conceptual framework and computational elegance are the underlying driver for this. This article reviews some of the recent developments that have taken place in our laboratory in the past 5 years. Efforts are made to validate a viable alternative for DFT calculations for small to medium systems through a Cartesian coordinate grid- (CCG-) based pseudopotential Kohn–Sham (KS) DFT framework using LCAO-MO ansatz. In order to legitimize its suitability and efficacy, at first, electric response properties, such as dipole moment (μ), static dipole polarizability (α), and first hyperpolarizability (β), are calculated. Next, we present a purely numerical approach in CCG for proficient computation of exact exchange density contribution in certain types of orbital-dependent density functionals. A Fourier convolution theorem combined with a range-separated Coulomb interaction kernel is invoked. This takes motivation from a semi-numerical algorithm, where the rate-deciding factor is the evaluation of electrostatic potential. Its success further leads to a systematic self-consistent approach from first principles, which is desirable in the development of optimally tuned range-separated hybrid and hyper functionals. Next, we discuss a simple, alternative time-independent DFT procedure, for computation of single-particle excitation energies, by means of “adiabatic connection theorem” and virial theorem. Optical gaps in organic chromophores, dyes, linear/non-linear PAHs, and charge transfer complexes are faithfully reproduced. In short, CCG-DFT is shown to be a successful route for various practical applications in electronic systems.

Density-functional theory vs density-functional fits

Kohn–Sham density-functional theory (DFT), the predominant framework for electronic structure computations in chemistry today, has undergone considerable evolution in the past few decades. The earliest DFT approximations were based on uniform electron gas models completely free of empirical parameters. Tremendous improvements were made by incorporating density gradients and a small number of parameters, typically one or two, obtained from fits to atomic data. Incorporation of exact exchange and fitting to molecular data, such as experimental heats of formation, allowed even further improvements. This, however, opened a Pandora’s Box of fitting possibilities, given the limitless choices of chemical reactions that can be fit. The result is a recent explosion of DFT approximations empirically fit to hundreds, or thousands, of chemical reference data. These fitted density functionals may contain several dozen empirical parameters. What has been lost in this fitting trend is physical modeling based on theory. In this work, we present a density functional comprising our best efforts to model exchange–correlation in DFT using good theory. We compare its performance to that of heavily fit density functionals using the GMTKN55 chemical reference data of Goerigk and co-workers [Phys. Chem. Chem. Phys. 19, 32184 (2017)]. Our density-functional theory, using only a handful of physically motivated pre-factors, competes with the best heavily fit Kohn–Sham functionals in the literature.

Reformulation of Thermally-Assisted-Occupation Density Functional Theory in the Kohn-Sham Framework

We reformulate the thermally-assisted-occupation density functional theory (TAO-DFT) into the Kohn-Sham single-determinant framework and construct two new post-self-consistent field (post-SCF) static correlation correction schemes, named rTAO and rTAO-1. In contrast to the original TAO-DFT with the density in an ensemble form, in which each orbital density is weighted with a fractional occupation number, the ground-state density is given by a single-determinant wavefunction, a regular Kohn-Sham density, and total ground state energy is expressed in the normal Kohn-Sham form with a static correlation energy formulated in terms of the Kohn-Sham orbitals. In post-SCF calculations with rTAO functionals, an efficient energy scanning to quantitatively determine $\theta$ is also proposed. The rTAOs provide a promising method to simulate systems with strong static correlation as original TAO, but simpler and more efficient.We show that both rTAO and rTAO-1 is capable of reproducing most results from TAO-DFT without the additional functional Eθ used in TAO-DFT. Furthermore, our numerical results support that, without the functional Eθ, both rTAO and rTAO-1 can capture correct static correlation profiles in various systems.

Local hybrid functionals augmented by a strong-correlation model

The strong-correlation factor of the recent KP16/B13 exchange-correlation functional has been adapted and applied to the framework of local hybrid (LH) functionals. The expression identifiable as nondynamical (NDC) and dynamical (DC) correlations in LHs is modified by inserting a position-dependent KP16/B13-style strong-correlation factor q_AC(r) based on a local version of the adiabatic connection. Different ways of deriving this factor are evaluated for a simple one-parameter LH based on the local density approximation. While the direct derivation from the LH NDC term fails due to known deficiencies, hybrid approaches, where the factor is determined from the B13 NDC term as in KP16/B13 itself, provide remarkable improvements. In particular, a modified B13 NDC expression using Patra's exchange-hole curvature showed promising results. When applied to the simple LH as a first attempt, it reduces atomic fractional-spin errors and deficiencies of spin-restricted bond dissociation curves to a similar extent as the KP16/B13 functional itself while maintaining the good accuracy of the underlying LH for atomization energies and reaction barriers in weakly correlated situations. The performance of different NDC expressions in deriving strong-correlation corrections is analyzed, and areas for further improvements of strong-correlation corrected LHs and related approaches are identified. All the approaches evaluated in this work have been implemented self-consistently into a developers' version of the Turbomole program.

Correcting the Charge Delocalization Error of Density Functional Theory

The charge delocalization error, besides nondynamic correlation, has been a major challenge to density functional theory. Contemporary functionals undershoot the dissociation of symmetric charged dimers A₂⁺, a simple but stringent test, predict a spurious barrier, and improperly delocalize charges for charged molecular clusters. We extend a functional designed for nondynamic correlation to treat the charge delocalization error by modifying the nondynamic correlation for parallel spins. The modified functional eliminates those problems and reduces the multielectron self-interaction error. Furthermore, its results are the closest to those of CCSD(T) in the whole range of the dissociation compared with contemporary functionals. It correctly localizes the net positive charge in (CH₄)_n⁺ clusters and predicts a nearly constant ionization potential as a result. Testing of the SIE4x4 set shows that the new functional outperforms a wide variety of functionals assessed for this set in the literature. Overall, we show the feasibility of treating charge delocalization together with nondynamic correlation.

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.

Analysis and visualization of energy densities. I. Insights from real-time time-dependent density functional theory simulations

In this article, we report a scheme to analyze and visualize the energy density fluctuations during the real-time time-dependent density functional theory (RT-TDDFT) simulations. Using Ag4-N2 complexes as examples, it is shown that the grid-based Kohn-Sham energy density can be computed at each time step using a procedure from Nakai and coworkers. Then the instantaneous energy of each molecular fragment (such as Ag4 and N2) can be obtained by partitioning the Kohn-Sham energy densities using Becke or fragment-based Hirshfeld (FBH) scheme. A strong orientation-dependence is observed for the energy flow between the Ag4 cluster and a nearby N2 molecule in the RT-TDDFT simulations. Future applications of such an energy density analysis in electron dynamics simulations are discussed.

Analysis and visualization of energy densities. II. Insights from linear-response time-dependent density functional theory calculations

Inspired by the analysis of Kohn-Sham energy densities by Nakai and coworkers, we extended the energy density analysis to linear-response time-dependent density functional theory (LR-TDDFT) calculations. Using ethylene-tetrafluoroethylene and oxyluciferin-water complexes as examples, distinctive distribution patterns were demonstrated for the excitation energy densities of local excitations (within a molecular fragment) and charge-transfer excitations (between molecular fragments). It also provided a simple way to compute the effective energy of both hot carriers (particle and hole) from charge-transfer excitations via an integration of the excitation energy density over the donor and acceptor grid points.

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 (1³Σ_u ⁺) was described well, with non-imaginary excitation energies. TDTAO-DFT also yields zero singlet-triplet gap in the dissociation limit for the ground singlet (1¹Σ_g ⁺) and the first triplet state (1³Σ_u ⁺). 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.

Quantum Chemical Methods for Modeling Covalent Modification of Biological Thiols

Targeted covalent inhibitor drugs require computational methods that go beyond simple molecular-mechanical force fields in order to model the chemical reactions that occur when they bind to their targets. Here, several semiempirical and density-functional theory (DFT) methods are assessed for their ability to describe the potential energy surface and reaction energies of the covalent modification of a thiol by an electrophile. Functionals such as PBE and B3LYP fail to predict a stable enolate intermediate. This is largely due to delocalization error, which spuriously stabilizes the prereaction complex, in which excess electron density is transferred from the thiolate to the electrophile. Functionals with a high-exact exchange component, range-separated DFT functionals, and variationally optimized exact exchange (i.e., the LC-B05minV functional) correct this issue to various degrees. The large gradient behavior of the exchange enhancement factor is also found to significantly affect the results, leading to the improved performance of PBE0. While ωB97X-D and M06-2X were reasonably accurate, no method provided quantitative accuracy for all three electrophiles, making this a very strenuous test of functional performance. Additionally, one drawback of M06-2X was that molecular dynamics (MD) simulations using this functional were only stable if a fine integration grid was used. The low-cost semiempirical methods, PM3, AM1, and PM7, provide a qualitatively correct description of the reaction mechanism, although the energetics is not quantitatively reliable. As a proof of concept, the potential of mean force for the addition of methylthiolate to methylvinyl ketone was calculated using quantum mechanical/molecular mechanical MD in an explicit polarizable aqueous solvent. © 2019 Wiley Periodicals, Inc.

Practical Density Functionals beyond the Overdelocalization-Underbinding Zero-Sum Game

Density functional theory (DFT) uses a density functional approximation (DFA) to add electron correlation to mean-field electronic structure calculations. Standard strategies (generalized gradient approximations GGAs, meta-GGAs, hybrids, etc.) for building DFAs, no matter whether based on exact constraints or empirical parametrization, all face a zero-sum game between overdelocalization (fractional charge error, FC) and underestimation of covalent bonding (fractional spin error, FS). This work presents an alternative strategy. Practical "Rung 3.5" ingredients are used to implement insights from hyper-GGA DFAs that reduce both FS and FC errors. Prototypes of this strategy qualitatively improve FS and FC error over 40 years of standard DFAs while maintaining low cost and practical evaluation of properties. Numerical results ranging from transition metal thermochemistry to absorbance peaks and excited-state geometry optimizations highlight this strategy's promise and indicate areas requiring further development.

Perspective: Fifty years of density-functional theory in chemical physics

Since its formal inception in 1964-1965, Kohn-Sham density-functional theory (KS-DFT) has become the most popular electronic structure method in computational physics and chemistry. Its popularity stems from its beautifully simple conceptual framework and computational elegance. The rise of KS-DFT in chemical physics began in earnest in the mid 1980s, when crucial developments in its exchange-correlation term gave the theory predictive power competitive with well-developed wave-function methods. Today KS-DFT finds itself under increasing pressure to deliver higher and higher accuracy and to adapt to ever more challenging problems. If we are not mindful, however, these pressures may submerge the theory in the wave-function sea. KS-DFT might be lost. I am hopeful the Kohn-Sham philosophical, theoretical, and computational framework can be preserved. This Perspective outlines the history, basic concepts, and present status of KS-DFT in chemical physics, and offers suggestions for its future development.

Comparison of the performance of exact-exchange-based density functional methods

How to describe nondynamic electron correlation is still a major challenge to density functional theory (DFT). Recent models designed particularly for this problem, such as Becke'05 (B05) and Perdew-Staroverov-Tao-Scuseria (PSTS) functionals employ the exact-exchange density, the efficient calculation of which is technically quite challenging. We have recently implemented self-consistently the B05 functional based on an efficient resolution-identity (RI) technique. In this study, we report a self-consistent RI implementation of the PSTS functional. In contrast to its original implementation, our version brings no limitation on the choice of the basis set. We have also implemented the Mori-Sanchez-Cohen-Yang-2 (MCY2) functional, another recent DFT method that includes full exact exchange. The performance of PSTS, B05, and MCY2 is validated on thermochemistry, reaction barriers, and dissociation energy curves, with an emphasis on nondynamic correlation effects in the discussion. All three methods perform rather well in general, B05 and MCY2 being on average somewhat better than PSTS. We include also results with other functionals that represent various aspects of the development in this field in recent years, including B3LYP, M06-HF, M06-2X, ωB97X, and TPSSh. The performance of the heavy-parameterized functionals M06-2X and ωB97X is on average better than that of B05, MCY2, and PSTS for standard thermodynamic properties and reactions, while the latter functionals do better in hydrogen abstraction reactions and dissociation processes. In particular, B05 is found to be the only functional that yields qualitatively correct dissociation curves for two-center symmetric radicals like He(2)(+). Finally, we compare the performance of all these functionals on a strongly correlated exemplary case system, the NO dimer. Only PSTS, B05, and MCY2 describe the system qualitatively correctly. Overall, this new type of functionals show good promise of overcoming some of the difficulties DFT encounters for systems with strong nondynamic correlation.

Modified Becke'05 method of nondynamic correlation in density functional theory with self-consistent implementation

Becke's B05 method for nondynamic correlation is simplified for self-consistent implementation. An alternative form is proposed for the nondynamic correlation factors that do not require solving a complicated nonlinear algebraic equation. The four linear parameters of B05 are re-optimized together with one extra parameter entering a modified expression for the second-order same-spin energy contribution. The latter is co-linear with the exact-exchange energy density and does not require higher moments of the relaxed exchange hole. Preliminary tests of this method show that it leads to a slight improvement over the resolution-of-identity B05 results reported previously for atomization energies, and to a definite improvement for reaction barriers of Hydrogen abstraction.