Approximate Density Functionals: Which Should I Choose?

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.

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

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.

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.

Delocalization error is one of the most fundamental and dominant errors that plagues presently used density functional approximations. It is responsible for a large class of problems in the density functional theory calculations. For an effective and universal alleviation of the delocalization error, we develop a local scaling correction scheme by imposing the Perdew-Parr-Levy- Balduz linearity condition to local regions of a system. Our novel scheme is applicable to various mainstream density functional approximations. It substantially reduces the delocalization error, as exemplified by the significantly improved description of dissociating molecules, transition-state species, and charge-transfer systems. The usefulness of our novel scheme affirms that the explicit treatment of fractional electron distributions is essentially important for reducing the intrinsic delocalization error associated with approximate density functionals.

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 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.

The calculation of accurate reaction energies and barrier heights is essential in computational studies of reaction mechanisms and thermochemistry. To assess methods regarding their ability to predict these two properties, high-quality benchmark sets are required that comprise a reasonably large and diverse set of organic reactions. Due to the time-consuming nature of both locating transition states and computing accurate reference energies for reactions involving large molecules, previous benchmark sets have been limited in scope, the number of reactions considered, and the size of the reactant and product molecules. Recent advances in coupled-cluster theory, in particular local correlation methods like DLPNO-CCSD(T), now allow the calculation of reaction energies and barrier heights for relatively large systems. In this work, we present a comprehensive and diverse benchmark set of barrier heights and reaction energies based on DLPNO-CCSD(T)/CBS called BH9. BH9 comprises 449 chemical reactions belonging to nine types common in organic chemistry and biochemistry. We examine the accuracy of DLPNO-CCSD(T) vis-a-vis canonical CCSD(T) for a subset of BH9 and conclude that, although there is a penalty in using the DLPNO approximation, the reference data are accurate enough to serve as a benchmark for density functional theory (DFT) methods. We then present two applications of the BH9 set. First, we examine the performance of several density functional approximations commonly used in thermochemical and mechanistic studies. Second, we assess our basis set incompleteness potentials regarding their ability to mitigate basis set incompleteness errors. The number of data points, the diversity of the reactions considered, and the relatively large size of the reactant molecules make BH9 the most comprehensive thermochemical benchmark set to date and a useful tool for the development and assessment of computational methods.

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.

ContextCopper is an essential trace element that plays a central role in redox chemistry and electron transfer processes in biological systems. To gain a deeper understanding of the electronic behavior of copper species, we carried out a comparative evaluation of the Cu2 and CuO molecules, focusing on key properties such as ionization energy, electron affinity, vibrational frequencies, bond lengths, and dissociation energies. Cu2 serves as a model for dinuclear copper sites present in metalloproteins like tyrosinase and hemocyanin, while CuO captures the essential features of copper-oxygen bonding relevant to copper-dependent oxidases and oxygen-activating enzymes. By systematically benchmarking density functional approximations (DFAs) against high-level CCSD(T) reference calculations or experimental data, we identify the methodologies that best reproduce the electronic and structural properties of these prototypical copper systems. The functional PBE, in particular, demonstrates the most consistent performance across both species. Insights obtained from Cu2 and CuO serve as a foundation for understanding more complex copper coordination environments. In this context, we extend our analysis to the Cu(II)/indomethacin complex, illustrating how the lessons learned from the fundamental systems can be applied to biologically relevant copper-ligand interactions. Overall, this study provides a systematic assessment of the accuracy of different DFAs for describing copper-containing species, establishing a solid framework for future investigations of bioinorganic copper chemistry and copper-based drug candidates.MethodsThis study employed density functional theory (DFT) alongside thermodynamic cycles to assess the stability of Cu(Indo)₂ and Cu₂(Indo)₄ complexes in ethanol solution. To select suitable computational methods, a benchmark was conducted using Cu(II)/acetate complexes as reference systems. A total of fifteen DFT functionals—BPW91, PBE, B97D, revTPSS, M06-L, M11-L, B3LYP, BHandHLYP, PBE0, ωB97XD, APDF, M06, M06-2X, M06-HF, and TPSSh—were tested in combination with four basis sets: Def2-SVP, Def2-TZVP, 6–31 + G(d,p), and 6–311 + G(d,p). The most reliable functional-basis set combinations were then applied to the copper-indomethacin complexes. In addition, electronic and structural properties of Cu₂ and CuO—such as ionization potentials, electron affinities, vibrational frequencies, equilibrium bond lengths, and spin or magnetic coupling constants—were calculated. Computational results were validated through comparison with available experimental data.Supplementary InformationThe online version contains supplementary material available at 10.1007/s00894-025-06538-x.

An old and yet unsolved problem in density-functional theory is the strong dependence of degenerate open-shell atomic energies on the occupancy of the atomic orbitals. This arises from the fact that degenerate atomic orbitals of different ml do not have equivalent densities. Approximate density functionals therefore give energies depending strongly on which orbitals are occupied. This problem is solved in the present work by incorporating current density into the calculations using a current-density dependent functional previously published by the author.

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.

Density functional theory (DFT) is increasingly used for computational screening procedures with the aim of finding new catalysts. To achieve this, it is critical that relative differences between materials are predicted with high accuracy. How DFT at the generalized gradient approximation (GGA) level performs in this respect is investigated here for catalytic reactions employing acidic zeotypes using highly accurate DLPNO-CCSD(T) calculations as the reference. This is studied for 65 reaction energies and 130 reaction barriers related to zeolite catalysis. Our results obtained for the PBE-D3 and BEEF-vdW functionals show that while these functionals are prone to large errors, they predict trends occurring from one catalyst to another with an accuracy of about 5 kJ/mol, strongly supporting the widespread use of DFT calculations for the computational screening and design of new catalytic materials.

Tungsten diselenide (WSe2) is a material that has captured the attention of researchers due to its unique properties, the proposed study focuses on the optical and electronic characteristics of WSe2, by making use of the density functional theory (DFT) and employing Perdew-Burke-Ernzerhof (PBE) Generalized Gradient Approximation (GGA) density function and PseudoDojo pseudopotential. The study investigates the properties of both bulk and a monolayer of WSe2 to determine how the number of layers affects the properties of the material. To further examine the peculiarities of WSe2, four different systems based on Vanadium doped-WSe2 have been investigated from the optical and electronic points of view. The systems contain different percentages of V: (i) WSe2:V 1.4%, (ii) WSe2:V 2.8%, (iii) WSe2:V 5.6%, and (iv) WSe2:V 11.2%. The introduction of Vanadium results in a reduction in the bandgap and a global shift of the projected density of the states. The Valence Band Maximum (VBM) also crosses the Fermi level, which is consistent with the p-type nature of vanadium doping. Furthermore, the absorption spectra change in terms of the position and the intensity of the optical transition as a result of the Vanadium introduction.