Local scaling correction for reducing delocalization error in density functional approximations.

The delocalization error of popular density functional approximations (DFAs) leads to diversified problems in present-day density functional theory calculations. For achieving a universal elimination of delocalization error, we develop a localized orbital scaling correction (LOSC) framework, which unifies our previously proposed global and local scaling approaches. The LOSC framework accurately characterizes the distributions of global and local fractional electrons, and is thus capable of correcting system energy, energy derivative and electron density in a self-consistent and size-consistent manner. The LOSC–DFAs lead to systematically improved results, including the dissociation of cationic species, the band gaps of molecules and polymer chains, the energy and density changes upon electron addition and removal, and photoemission spectra.

Density functional theory (DFT) is the most widely used electronic structure method, due to its simplicity and cost effectiveness. The accuracy of a DFT calculation depends not only on the choice of the density functional approximation (DFA) adopted but also on the electron density produced by the DFA. SCAN is a modern functional that satisfies all known constraints for meta-GGA functionals. The density-driven errors, defined as energy errors arising from errors of the self-consistent DFA electron density, can hinder SCAN from achieving chemical accuracy in some systems, including water. Density-corrected DFT (DC-DFT) can alleviate this shortcoming by adopting a more accurate electron density which, in most applications, is the electron density obtained at the Hartree-Fock level of theory due to its relatively low computational cost. In this work, we present extensive calculations aimed at determining the accuracy of the DC-SCAN functional for various aqueous systems. DC-SCAN (SCAN@HF) shows remarkable consistency in reproducing reference data obtained at the coupled cluster level of theory, with minimal loss of accuracy. Density-driven errors in the description of ionic aqueous clusters are thoroughly investigated. By comparison with the orbital-optimized CCD density in the water dimer, we find that the self-consistent SCAN density transfers a spurious fraction of an electron across the hydrogen bond to the hydrogen atom (H*, covalently bound to the donor oxygen atom) from the acceptor (OA) and donor (OD) oxygen atoms, while HF makes a much smaller spurious transfer in the opposite direction, consistent with DC-SCAN (SCAN@HF) reduction of SCAN overbinding due to delocalization error. While LDA seems to be the conventional extreme of density delocalization error, and HF the conventional extreme of (usually much smaller) density localization error, these two densities do not quite yield the conventional range of density-driven error in energy differences. Finally, comparisons of the DC-SCAN results with those obtained with the Fermi-Löwdin orbital self-interaction correction (FLOSIC) method show that DC-SCAN represents a more accurate approach to reducing density-driven errors in SCAN calculations of ionic aqueous clusters. While the HF density is superior to that of SCAN for noncompact water clusters, the opposite is true for the compact water molecule with exactly 10 electrons.

Accurately describing intermolecular interactions within the framework of Kohn-Sham density functional theory (KS-DFT) has resulted in numerous benchmark databases over the past two decades. By far, the largest efforts have been spent on closed-shell, neutral dimers for which today, the interaction energies and geometries can be accurately reproduced by various combinations of dispersion-corrected density functional approximations (DFAs). In sharp contrast, charged, open-shell dimers remain a challenge as illustrated by the analysis of the OREL26rad benchmark set consisting of pi-dimer radical cations. Aside from the methodological aspect, achieving a proper description of radical cationic complexes is appealing due to their role as models for charge carriers in organic semiconductors. In the interest of providing an assessment of more realistic dimer systems, we construct a dataset of large radical cationic dimers (CryOrel) and jointly train the 19 parameters of a dispersion corrected, range-separated hybrid density functional (wB97X-dDsC), with the objective of providing the maximum balance between the treatment of long-range London dispersion and reduction of the delocalization error. These conditions are essential to obtain accurate energy profiles and binding energies of charged, open-shell dimers. Comparisons with the performance of the parent wB97X functional series and state-of-the-art wavefunction based methods are provided.

We present a benchmark study of density functional approximation (DFA) performances in predicting the two-photon-absorption strengths in π-conjugated molecules containing electron-donating/-accepting moieties. A set of 48 organic molecules is chosen for this purpose, for which the two-photon-absorption (2PA) parameters are evaluated using different DFAs, including BLYP, PBE, B3LYP, PBE0, CAM-B3LYP, LC-BLYP, and optimally tuned LC-BLYP. Minnesota functionals and ωB97X-D are also used, applying the two-state approximation, for a subset of molecules. The efficient resolution-of-identity implementation of the coupled-cluster CC2 model (RI-CC2) is used as a reference for the assessment of the DFAs. Two-state models within the framework of both DFAs and RI-CC2 are used to gain a deeper insight into the performance of different DFAs. Our results give a clear picture of the performance of the density functionals in describing the two-photon activity in dipolar π-conjugated systems. The results show that global hybrids are best suited to reproduce the absolute values of 2PA strengths of donor–acceptor molecules. The range-separated functionals CAM-B3LYP and optimally tuned LC-BLYP, however, show the highest linear correlations with the reference RI-CC2 results. Hence, we recommend the latter DFAs for structure–property studies across large series of dipolar compounds.

Noncovalent intermolecular interactions are very important in many research areas. Therefore, it is vital to understand the extent to which approximate density functionals give a proper description of noncovalent interactions. Previous research has demonstrated that some approximate density functionals can predict usefully accurate interaction energies for many noncovalent systems; however, most of that work is limited to small and moderate-sized molecules. Very recently though, accurate benchmarks have become available for some very large molecules. The present work applies 21 approximate density functionals to compute the binding energies of seven large molecular systems that have a number of atoms ranging from 200 to 910. The results are judged by comparison to the recently published CIM-DLPNO-CCSD(T) results, which are assumed to provide a reliable benchmark. The five most accurate methods among those tested are found to be PW6B95-D4, PW6B95-D3(BJ), revM11, M06-L, and MN15.

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.

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.

Starting from exceedingly accurate many-body wavefunctions, we have constructed essentially exact densities, exchange-correlation potentials and components of the total energy for helium and twoelectron ions. These density functional results are compared to the corresponding quantities obtained from a variety of commonly used approximate density functionals, namely the local density approximation and various generalized gradient approximations in order to test the accuracy of the approximate functionals. Although the generalized gradient approximations yield improved energies compared to the local density approximation, the exchange and correlation potentials (especially the latter) obtained from the generalized gradient approximations are in poor agreement with the corresponding exact potentials. The large-distance asymptotic behavior of the exact exchange-correlation potential to O(1/r4) is found to agree with theoretical predictions. The short range behaviour of the exchangecorrelation potential is very close to quadratic. The prospects for the improved generalized gradient approximations are discussed. Keywords: density functional theory, exchange-correlation potential, generalized gradient approximation

Density functional methods have a long tradition in inorganic and bioinorganic chemistry. We introduce the density functional machinery and give an overview of most popular approximate exchange‐correlation (XC) functionals. We present comparisons of density functionals for energies, structures, and reaction barriers of inorganic and bioinorganic systems, giving guidance on the title question. New development directions and current trends in density functional theory (DFT) are reviewed.

We have performed density functional calculations using a range of local and semi-local as well as hybrid density functional approximations of the structure and elastic constants of 18 semiconductors and insulators. We find that most of the approximations have a very small error in the lattice constants, of the order of 1%, while the errors in the elastic constants and bulk modulus are much larger, at about 10% or better. When comparing experimental and theoretical lattice constants and bulk modulus we have included zero-point phonon effects. These effects make the experimental reference lattice constants 0.019 Å smaller on average while making the bulk modulus 4.3 GPa stiffer on average. According to our study, the overall best performing density functional approximations for determining the structure and elastic properties are the PBEsol functional, the two hybrid density functionals PBE0 and HSE (Heyd, Scuseria, and Ernzerhof), as well as the AM05 functional.

One of the most important challenges in density functional theory (DFT) is the proper description of fractional charge systems relating to the self-interaction error (SIE). Traditionally, the SIE has been formulated as a one-electron problem, which has been addressed in several recent functionals. However, these recent one-electron SIE-free functionals, while greatly improving the description of thermochemistry and reaction barriers in general, still exhibit many of the difficulties associated with SIE. Thus we emphasize the need to surpass this limit and shed light on the many-electron SIE. After identifying the sufficient condition for functionals to be free from SIE, we focus on the symptoms and investigate the performance of most popular functionals. We show that these functionals suffer from many-electron SIE. Finally, we give a SIE classification of density functionals.

ConspectusThe quantum chemical modeling of organic crystals and other molecular condensed-phase problems requires computationally affordable electronic structure methods which can simultaneously describe intramolecular conformational energies and intermolecular interactions accurately. To achieve this, we have developed a spin-component-scaled, dispersion-corrected second-order Møller-Plesset perturbation theory (SCS-MP2D) model. SCS-MP2D augments canonical MP2 with a dispersion correction which removes the uncoupled Hartree-Fock dispersion energy present in canonical MP2 and replaces it with a more reliable coupled Kohn-Sham treatment, all evaluated within the framework of Grimme's D3 dispersion model. The spin-component scaling is then used to improve the description of the residual (nondispersion) portion of the correlation energy.The SCS-MP2D model improves upon earlier corrected MP2 models in a few ways. Compared to the highly successful dispersion-corrected MP2C model, which is based solely on intermolecular perturbation theory, the SCS-MP2D dispersion correction improves the description of both inter- and intramolecular interactions. The dispersion correction can also be evaluated with trivial computational cost, and nuclear analytic gradients are computed readily to enable geometry optimizations. In contrast to earlier spin-component scaling MP2 models, the optimal spin-component scaling coefficients are only mildly sensitive to the choice of training data, and a single global parametrization of the model can describe both thermochemistry and noncovalent interactions.The resulting dispersion-corrected, spin-component-scaled MP2 (SCS-MP2D) model predicts conformational energies and intermolecular interactions with accuracy comparable to or better than those of many range-separated and double-hybrid density functionals, as is demonstrated on a variety of benchmark tests. Among the functionals considered here, only the revDSD-PBEP86-D3(BJ) functional gives consistently smaller errors in benchmark tests. The results presented also hint that further improvements of SCS-MP2D may be possible through a more robust fitting procedure for the seven empirical parameters.To demonstrate the performance of SCS-MP2D further, several applications to molecular crystal problems are presented. The three chosen examples all represent cases where density-driven delocalization error causes GGA or hybrid density functionals to artificially stabilize crystals exhibiting more extended π-conjugation. Our pragmatic strategy addresses the delocalization error by combining a periodic density functional theory (DFT) treatment of the infinite lattice with intramolecular/conformational energy corrections computed with SCS-MP2D. For the anticancer drug axitinib, applying the SCS-MP2D conformational energy correction produces crystal polymorph stabilities that are consistent with experiment, in contrast to earlier studies. For the crystal structure prediction of the ROY molecule, so named for its colorful red, orange, and yellow crystals, this approach leads to the first plausible crystal energy landscape, and it reveals that the lowest-energy polymorphs have already been found experimentally. Finally, in the context of photomechanical crystals, which transform light into mechanical work, these techniques are used to predict the structural transformations and extract design principles for maximizing the work performed.

HF-DFT, the practice of evaluating approximate density functionals on Hartree-Fock densities, has long been used in testing density functional approximations. Density-corrected DFT (DC-DFT) is a general theoretical framework for identifying failures of density functional approximations by separating errors in a functional from errors in its self-consistent (SC) density. Most modern DFT calculations yield highly accurate densities, but important characteristic classes of calculation have large density-driven errors, including reaction barrier heights, electron affinities, radicals and anions in solution, dissociation of heterodimers, and even some torsional barriers. Here, the HF density (if not spin-contaminated) usually yields more accurate and consistent energies than those of the SC density. We use the term DC(HF)-DFT to indicate DC-DFT using HF densities only in such cases. A recent comprehensive study (J. Chem. Theory Comput. 2021, 17, 1368-1379) of HF-DFT led to many unfavorable conclusions. A reanalysis using DC-DFT shows that DC(HF)-DFT substantially improves DFT results precisely when SC densities are flawed.

Kohn-Sham density functional theory has become a standard method for modeling energetic, spectroscopic, and chemical reactivity properties of large molecules and solids. Density functional theory provides a rigorous theoretical framework for modeling the many-body exchange-correlation effects that dominate the computational cost of traditional wave function approaches. The advent of hybrid exchange-correlation functionals which incorporate a fraction of nonlocal exact exchange has solidified the prominence of density functional theory within computational chemistry. Hybrids provide accurate treatments of properties such as thermochemistry and molecular geometry. But they also exhibit some rather spectacular failures, and often contain multiple empirical parameters. This article reviews our work on developing novel exchange-correlation functionals that build upon the successes of global hybrids. We focus on more flexible functional forms, including local and range-separated hybrid functionals, constructed to obey known exact constraints and (ideally) to incorporate a minimum of empirical parametrization. The article places our work within the context of some other new approximate density functionals and discusses prospects for future work.

Broderick and Herbert's article (D. R. Broderick and J. M. Herbert, Chem. Sci., 2024, 15, 19893–19906, https://doi.org/10.1039/D4SC05955G) explores an open concern about using energies from density-functional approximations when developing force fields and machine learning potentials for large-scale simulations. The authors explicitly decomposed self-interaction errors (SIEs) from density-functional approximations (DFAs) and found how they behave in many-body expansions (MBEs) that are leveraged in large-scale simulations. For DFAs to be deemed reliable for developing many-body potentials, they would ideally provide stable energetics within the MBE terms that are most often used by force fields and machine learning potentials (MLPs), i.e., within their three- and four-body terms. It was instead found that many widely used DFAs produce wild oscillations in these MBE terms, whereby three-body terms can become problematically enormous. This raises concerns that any force field and/or MLP that appears well-fitted to DFA data on small systems might be poorly conditioned for large-scale simulations due to intrinsic SIEs. This commentary provides more context of Broderick and Herbert's work and its consequences for members of the multiscale modeling community.