The one-electron self-interaction error in 74 density functional approximations: a case study on hydrogenic mono- and dinuclear systems

Do Optimally Tuned Range-Separated Hybrid Functionals Require a Reparametrization of the Dispersion Correction? It Depends

For ground- and excited-state studies of large molecules, it is the state of the art to combine (time-dependent) DFT with dispersion-corrected range-separated hybrid functionals (RSHs), which ensures an asymptotically correct description of exchange effects and London dispersion. Specifically for studying excited states, it is common practice to tune the range-separation parameter ω (optimal tuning), which can further improve the accuracy. However, since optimal tuning essentially changes the functional, it is unclear if and how much the parameters used for the dispersion correction depend on the chosen ω value. To answer this question, we explore this interdependency by refitting the DFT-D4 dispersion model for six established RSHs over a wide range of ω values (0.05-0.45 a0-1) using a set of noncovalently bound molecular complexes. The results reveal some surprising differences among the investigated functionals: While PBE-based RSHs and ωB97M-D4 generally exhibit a weak interdependency and robust performance over a wide range of ω values, B88-based RSHs, specifically LC-BLYP, are strongly affected. For these, even a minor reduction of ω from the default value manifests in strong systematic overbinding and poor performance in the typical range of optimally tuned ω values. Finally, we discuss strategies to mitigate these issues and reflect the results in the context of the employed D4 parameter optimization algorithm and fit set, outlining strategies for future improvements.

Spin state and magnetic coupling in polynuclear Ni(II) complexes from density functional theory: is there an optimal amount of Fock exchange?

Reliable prediction of the ground-state spin and magnetic coupling constants in transition-metal complexes is a well-known challenge for density functional theory (DFT). One popular strategy for addressing this long-standing issue involves the modification of the fraction of Fock exchange in a hybrid functional. Here we explore the viability of this approach using three polynuclear metal-organic complexes based on a Ni4O4 cubane motif, having different ground state spin values (S = 0, 2, 4) owing to the use of different ligands. We systematically search for an optimum fraction of Fock exchange, across various global, range-separated, and double hybrid functionals. We find that for all functionals tested, at best there only exists a very narrow range of Fock exchange fractions which results in a correct prediction of the ground-state spin for all three complexes. The useful range is functional dependent, but general trends can be identified. Typically, at least two similar systems must be used in order to determine both an upper and lower limit of the optimal range. This is likely owing to conflicting demands of minimizing delocalization errors, which typically requires a higher percentage of Fock exchange, and addressing static correlation, which typically requires a lower one. Furthermore, we find that within the optimal range of Fock exchange, the sign and relative magnitude of Ni-Ni magnetic coupling constants are reasonably well reproduced, but there is still room for quantitative improvement in the prediction. Thus, the prediction of spin state and magnetic coupling in polynuclear complexes remains an ongoing challenge for DFT.

Density-Corrected Density Functional Theory for Molecular Properties

Density-corrected (DC) density functional theory (DFT) has been proposed to overcome difficulties related to the self-interaction error. The procedure uses the Hartree-Fock electron density (matrix) non-self-consistently in conjunction with an approximate functional. DC-DFT has so far mainly been tested for total energy differences, whereas other types of molecular properties have not been evaluated systematically. This work focuses on the performance of DC-DFT for molecular properties, namely, dipole moments, static polarizabilities, and electric field gradients (EFGs) at atomic nuclei. Accurate reference data were generated from coupled-cluster theory to assess the performance of DC and self-consistent DFT calculations for twelve molecules, including diatomics with transition metals. DC-DFT does no harm in dipole moment calculations, but it negatively impacts the polarizability in at least one case. DC-DFT performs well for EFGs, even for the difficult case of CuCl.

One-electron self-interaction error and its relationship to geometry and higher orbital occupation

Density Functional Theory (DFT) sees prominent use in computational chemistry and physics; however, problems due to the self-interaction error (SIE) pose additional challenges to obtaining qualitatively correct results. As an unphysical energy an electron exerts on itself, the SIE impacts most practical DFT calculations. We conduct an in-depth analysis of the one-electron SIE in which we replicate delocalization effects for simple geometries. We present a simple visualization of such effects, which may help in future qualitative analysis of the one-electron SIE. By increasing the number of nuclei in a linear arrangement, the SIE increases dramatically. We also show how molecular shape impacts the SIE. Two- and three-dimensional shapes show an even greater SIE stemming mainly from the exchange functional with some error compensation from the one-electron error, which we previously defined [D. R. Lonsdale and L. Goerigk, Phys. Chem. Chem. Phys. 22, 15805 (2020)]. Most tested geometries are affected by the functional error, while some suffer from the density error. For the latter, we establish a potential connection with electrons being unequally delocalized by the DFT methods. We also show how the SIE increases if electrons occupy higher-lying atomic orbitals; seemingly one-electron SIE free methods in a ground are no longer SIE free in excited states, which is an important insight for some popular, non-empirical density functional approximations (DFAs). We conclude that the erratic behavior of the SIE in even the simplest geometries shows that robust DFAs are needed. Our test systems can be used as a future benchmark or contribute toward DFT development.

Poisoning density functional theory with benchmark sets of difficult systems

Large benchmark sets like GMTKN55 [Goerigk et al., Phys. Chem. Chem. Phys., 2017, 19, 32184] let us analyse the performance of density functional theory over a diverse range of systems and bonding types. However, assessing over a large and diverse set can miss cases where approaches fail badly, and can give a misleading sense of security. To this end we introduce a series of 'poison' benchmark sets, P30-5, P30-10 and P30-20, comprising systems with up to 5, 10 and 20 atoms, respectively. These sets represent the most difficult-to-model systems in GMTKN55. We expect them to be useful in developing new approximations, identifying weak points in existing ones, and to aid in selecting appropriate DFAs for computational studies involving difficult physics, e.g. catalysis.

Linear fractional charge behavior in density functional theory through dielectric tuning of conductor-like polarizable continuum model

A property of exact density functional theory is linear fractional charge behavior as electrons are added or removed from a molecule. Typical density functional approximations (DFAs) exhibit delocalization error, which overstabilizes this fractional charge. Conversely, solvent corrections have been shown to erroneously destabilize this fractional charge. This work will show that an implicit solvent correction with a tuned dielectric can be used as an ad hoc correction to offset the delocalizing character of DFAs and achieve linear fractional charge behavior. While desirable, in principle, we find that this linear charge behavior degrades the vertical ionization energies reported by DFAs. Our results reveal that the localizing character of the solvent correction and the Hartree-Fock (HF) exchange offset each other. This helps explain the decreased ratios of HF exchange to DFA exchange in long-range hybrid tuning studies that use a solvent correction.

The smallest 4f-metalla-aromatic molecule of cyclo-PrB2− with Pr–B multiple bonds

The concept of metalla-aromaticity proposed by Thorn–Hoffmann (Nouv. J. Chim. 1979, 3, 39) has been expanded to organometallic molecules of transition metals that have more than one independent electron-delocalized system. Lanthanides, with highly contracted 4f atomic orbitals, are rarely found in multiply aromatic systems. Here we report the discovery of a doubly aromatic triatomic lanthanide-boron molecule PrB₂⁻ based on a joint photoelectron spectroscopy and quantum chemical investigation. Global minimum structural searches reveal that PrB₂⁻ has a C_2v triangular structure with a paramagnetic triplet ³B₂ electronic ground state, which can be viewed as featuring a trivalent Pr(III,f²) and B₂⁴⁻. Chemical bonding analyses show that this cyclo-PrB₂⁻ species is the smallest 4f-metalla-aromatic system exhibiting σ and π double aromaticity and multiple Pr–B bonding characters. It also sheds light on the formation of the rare B₂⁴⁻ tetraanion by the high-lying 5d orbitals of the 4f-elements, completing the isoelectronic B₂⁴⁻, C₂²⁻, N₂, and O₂²⁺ series.

We report the smallest 4f-metalla-aromatic molecule of PrB₂⁻ exhibiting σ and π double aromaticity and multiple Pr–B bond characters.

A Convenient DFT-Based Strategy for Predicting Transition Temperatures of Valence Tautomeric Molecular Switches

The ability to identify promising candidate switchable molecules computationally, prior to synthesis, represents a considerable advance in the development of switchable molecular materials. Even more useful would be the possibility of predicting the switching temperature. Cobalt-dioxolene complexes can exhibit thermally induced valence tautomeric switching between low-spin Co^III-catecholate and high-spin Co^II-semiquinonate forms, where the half-temperature (T_1/2) is the temperature at which there are equal amounts of the two tautomers. We report the first simple computational strategy for accurately predicting T_1/2 values for valence tautomeric complexes. Dispersion-corrected density functional theory (DFT) methods have been applied to the [Co(dbdiox)(dbsq)(N₂L)] (dbdiox/dbsq^•- = 3,5-di-tert-butyldioxolene/semiquinonate; N₂L = diimine) family of valence tautomeric complexes, including the newly reported [Co(dbdiox)(dbsq)(MeO-bpy)] (1) (MeO-bpy = 4,4'-dimethoxy-2,2'-bipyridine). The DFT strategy has been thoroughly benchmarked to experimental data, affording highly accurate spin-distributions and an excellent energy match between experimental and calculated spin-states. Detailed orbital analysis of the [Co(dbdiox)(dbsq)(N₂L)] complexes has revealed that the diimine ligand tunes the T_1/2 value primarily through π-acceptance. We have established an excellent correlation between experimental T_1/2(toluene) values for [Co(dbdiox)(dbsq)(N₂L)] complexes and the calculated lowest unoccupied molecular orbital energy of the corresponding diimine ligand. The model affords accurate T_1/2(toluene) values for [Co(dbdiox)(dbsq)(N₂L)] complexes, with an average error of only 3.7%. This quantitative and simple DFT strategy allows experimentalists to not only rapidly identify proposed VT complexes but also predict the transition temperature. This study lays the groundwork for future in silico screening of candidate switchable molecules prior to experimental investigation, with associated time, cost, and environmental benefits.

Extrapolating DFT Toward the Complete Basis Set Limit: Lessons from the PBE Family of Functionals

Extrapolation of density functional theory results from 2- and 3-ζ calculations is a promising method for extracting higher accuracy data from calculations of systems at the affordability limit. In this work, the author presents formulas for the determination of extrapolation parameters, which account for the makeup of the density functional approximation. The formulas are fitted to reproduce the complete basis set limit energies of PBE and related density functional approximations, using a set of 30 singlet diatomics. Their performance is extensively evaluated using standard benchmark data sets. The current systematically derived expressions are shown to be transferrable outside the PBE family of functional approximations, with the resulting extrapolation parameters outperforming the previous, less-systematic values. A good performance of [2,3]-ζ extrapolations for interaction energies of systems with significant noncovalent character is confirmed and holds even in systems of ∼100 atoms in size.

Benchmarking London dispersion corrected density functional theory for noncovalent ion-π interactions

The strongly attractive noncovalent interactions of charged atoms or molecules with π-systems are important binding motifs in many chemical and biological systems. These so-called ion-π interactions play a major role in enzymes, molecular recognition, and for the structure of proteins. In this work, a molecular test set termed IONPI19 is compiled for inter- and intramolecular ion-π interactions, which is well balanced between anionic and cationic systems. The IONPI19 set includes interaction energies of significantly larger molecules (up to 133 atoms) than in other ion-π test sets and covers a broad range of binding motifs. Accurate (local) coupled cluster values are provided as reference. Overall, 19 density functional approximations, including seven (meta-)GGAs, eight hybrid functionals, and four double-hybrid functionals combined with three different London dispersion corrections, are benchmarked for interaction energies. DFT results are further compared to wave function based methods such as MP2 and dispersion corrected Hartree-Fock. Also, the performance of semiempirical QM methods such as the GFNn-xTB and PMx family of methods is tested. It is shown that dispersion-uncorrected DFT underestimates ion-π interactions significantly, even though electrostatic interactions dominate the overall binding. Accordingly, the new charge dependent D4 dispersion model is found to be consistently better than the standard D3 correction. Furthermore, the functional performance trend along Jacob's ladder is generally obeyed and the reduction of the self-interaction error leads to an improvement of (double) hybrid functionals over (meta-)GGAs, even though the effect of the SIE is smaller than expected. Overall, the double-hybrids PWPB95-D4/QZ and revDSD-PBEP86-D4/QZ turned out to be the most reliable among all assessed methods for the description of ion-π interactions, which opens up new perspectives for systems where coupled cluster calculations are no longer computationally feasible.

CHAL336 Benchmark Set: How Well Do Quantum-Chemical Methods Describe Chalcogen-Bonding Interactions?

We present the CHAL336 benchmark set-the most comprehensive database for the assessment of chalcogen-bonding (CB) interactions. After careful selection of suitable systems and identification of three high-level reference methods, the set comprises 336 dimers each consisting of up to 49 atoms and covers both σ- and π-hole interactions across four categories: chalcogen-chalcogen, chalcogen-π, chalcogen-halogen, and chalcogen-nitrogen interactions. In a subsequent study of DFT methods, we re-emphasize the need for using proper London dispersion corrections when treating noncovalent interactions. We also point out that the deterioration of results and systematic overestimation of interaction energies for some dispersion-corrected DFT methods does not hint at problems with the chosen dispersion correction but is a consequence of large density-driven errors. We conclude this work by performing the most detailed DFT benchmark study for CB interactions to date. We assess 109 variations of dispersion-corrected and dispersion-uncorrected DFT methods and carry out a detailed analysis of 80 of them. Double-hybrid functionals are the most reliable approaches for CB interactions, and they should be used whenever computationally feasible. The best three double hybrids are SOS0-PBE0-2-D3(BJ), revDSD-PBEP86-D3(BJ), and B2NCPLYP-D3(BJ). The best hybrids in this study are ωB97M-V, PW6B95-D3(0), and PW6B95-D3(BJ). We do not recommend using the popular B3LYP functional nor the MP2 approach, which have both been frequently used to describe CB interactions in the past. We hope to inspire a change in computational protocols surrounding CB interactions that leads away from the commonly used, popular methods to the more robust and accurate ones recommended herein. We would also like to encourage method developers to use our set for the investigation and reduction of density-driven errors in new density functional approximations.

Explaining and Fixing DFT Failures for Torsional Barriers

Most torsional barriers are predicted with high accuracies (about 1 kJ/mol) by standard semilocal functionals, but a small subset was found to have much larger errors. We created a database of almost 300 carbon-carbon torsional barriers, including 12 poorly behaved barriers, that stem from the Y═C-X group, where Y is O or S and X is a halide. Functionals with enhanced exchange mixing (about 50%) worked well for all barriers. We found that poor actors have delocalization errors caused by hyperconjugation. These problematic calculations are density-sensitive (i.e., DFT predictions change noticeably with the density), and using HF densities (HF-DFT) fixes these issues. For example, conventional B3LYP performs as accurately as exchange-enhanced functionals if the HF density is used. For long-chain conjugated molecules, HF-DFT can be much better than exchange-enhanced functionals. We suggest that HF-PBE0 has the best overall performance.

Assessing challenging intra- and inter-molecular charge-transfer excitations energies with double-hybrid density functionals

We investigate the performance of a set of recently introduced range-separated double-hybrid functionals, namely ωB2-PLYP, ωB2GP-PLYP, RSX-0DH, and RSX-QIDH models for hard-to-calculate excitation energies. We compare with the parent (B2-PLYP, B2GP-PLYP, PBE0-DH, and PBE-QIDH) and other (DSD-PBEP86) double-hybrid models as well as with some of the most widely employed hybrid functionals (B3LYP, PBE0, M06-2X, and ωB97X). For this purpose, we select a number of medium-sized intra- and inter-molecular charge-transfer excitations, which are known to be challenging to calculate using time-dependent density-functional theory (TD-DFT) and for which accurate reference values are available. We assess whether the high accuracy shown by the newest double-hybrid models is also confirmed for those cases too. We find that asymptotically corrected double-hybrid models yield a superior performance, especially for the inter-molecular charge-transfer excitation energies, as compared to standard double-hybrid models. Overall, the PBE-QIDH and its corresponding range-separated RSX-QIDH functional are recommended for general-purpose TD-DFT applications, depending on whether long-range effects are expected to play a significant role.

What Types of Chemical Problems Benefit from Density-Corrected DFT? A Probe Using an Extensive and Chemically Diverse Test Suite

For the large and chemically diverse GMTKN55 benchmark suite, we have studied the performance of density-corrected density functional theory (HF-DFT), compared to self-consistent DFT, for several pure and hybrid GGA and meta-GGA exchange–correlation (XC) functionals (PBE, BLYP, TPSS, and SCAN) as a function of the percentage of HF exchange in the hybrid. The D4 empirical dispersion correction has been added throughout. For subsets dominated by dynamical correlation, HF-DFT is highly beneficial, particularly at low HF exchange percentages. This is especially true for noncovalent interactions where the electrostatic component is dominant, such as hydrogen and halogen bonds: for π-stacking, HF-DFT is detrimental. For subsets with significant nondynamical correlation (i.e., where a Hartree–Fock determinant is not a good zero-order wavefunction), HF-DFT may do more harm than good. While the self-consistent series show optima at or near 37.5% (i.e., 3/8) for all four XC functionals—consistent with Grimme’s proposal of the PBE38 functional—HF-BnLYP-D4, HF-PBEn-D4, and HF-TPSSn-D4 all exhibit minima nearer 25% (i.e., 1/4) as the use of HF orbitals greatly mitigates the error at 25% for barrier heights. Intriguingly, for HF-SCANn-D4, the minimum is near 10%, but the weighted mean absolute error (WTMAD2) for GMTKN55 is only barely lower than that for HF-SCAN-D4 (i.e., where the post-HF step is a pure meta-GGA). The latter becomes an attractive option, only slightly more costly than pure Hartree–Fock, and devoid of adjustable parameters other than the three in the dispersion correction. Moreover, its WTMAD2 is only surpassed by the highly empirical M06-2X and by the combinatorially optimized empirical range-separated hybrids ωB97X-V and ωB97M-V.

Density Functional Theory for Molecule-Metal Surface Reactions: When Does the Generalized Gradient Approximation Get It Right, and What to Do If It Does Not

While density functional theory (DFT) is perhaps the most used electronic structure theory in chemistry, many of its practical aspects remain poorly understood. For instance, DFT at the generalized gradient approximation (GGA) tends to fail miserably at describing gas-phase reaction barriers, while it performs surprisingly well for many molecule-metal surface reactions. GGA-DFT also fails for many systems in the latter category, and up to now it has not been clear when one may expect it to work. We show that GGA-DFT tends to work if the difference between the work function of the metal and the molecule's electron affinity is greater than ∼7 eV and to fail if this difference is smaller, with sticking of O₂ on Al(111) being a spectacular example. Using dynamics calculations we show that, for this system, the DFT problem may be solved as done for gas-phase reactions, i.e., by resorting to hybrid functionals, but using screening at long-range to obtain a correct description of the metal. Our results suggest the GGA error in the O₂ + Al(111) barrier height to be functional driven. Our results also suggest the possibility to compute potential energy surfaces for the difficult-to-treat systems with computationally cheap nonself-consistent calculations in which a hybrid functional is applied to a GGA density.

Piecewise linearity, freedom from self-interaction, and a Coulomb asymptotic potential: three related yet inequivalent properties of the exact density functional

Three properties of the exact energy functional of DFT are important in general and for spectroscopy in particular, but are not necessarily obeyed by approximate functionals. We explain what they are, why they are important, and how they are related yet inequivalent.

Role of hemibonding in the structure and ultraviolet spectroscopy of the aqueous hydroxyl radical

The presence of a two-center, three-electron hemibond in the solvation structure of the aqueous hydroxl radical has long been debated, as its appearance can be sensitive to self-interaction error in density functional theory.