Quantifying Density Errors in DFT

Exploring the role of mean-field potentials and short-range wave function behavior in the adiabatic connection

In this article, we explore the construction of Hamiltonians with long-range interactions and their corrections using the short-range behavior of the wave function. A key aspect of our investigation is the examination of the one-particle potential, kept constant in our previous work, and the effects of its optimization on the adiabatic connection. Our methodology involves the use of a parameter-dependent potential dependent on a single parameter to facilitate practical computations. We analyze the energy errors and densities in a two-electron system (harmonium) under various conditions, employing different confinement potentials and interaction parameters. The study reveals that while the mean-field potential improves the expectation value of the physical Hamiltonian, it does not necessarily improve the energy of the system within the bounds of chemical accuracy. We also delve into the impact of density variations in adiabatic connections, challenging the common assumption that a mean field improves results. Our findings indicate that as long as energy errors remain within chemical accuracy, the mean field does not significantly outperform a bare potential. This observation is attributed to the effectiveness of corrections based on the short-range behavior of the wave function, a universal characteristic that diminishes the distinction between using a mean field or not.

Dissociative chemisorption of O2 on Al(111): dynamics on a potential energy surface computed with a non-self-consistent screened hybrid density functional approach

Density functional theory (DFT) at the generalized gradient approximation (GGA) level is often considered the best compromise between feasibility and accuracy for reactions of molecules on metal surfaces. Recent work, however, strongly suggests that density functionals (DFs) based on GGA exchange are not able to describe molecule-metal surface reactions for which the work function of the metal surface minus the electron affinity of the molecule is less than 7 eV. Systems for which this is true exhibit an increased charge transfer from the metal to the molecule at the transition state, increasing the delocalisation of the electron density. This enlarged delocalisation can cause GGA-DFT to underestimate energy values relative to the gas-phase and thus underestimate the barrier height, similar to what has been observed for several gas-phase reactions. An example of such a molecule-metal surface system is O2 + Al(111). Following a similar strategy as for gas-phase reactions, previous work showed results of increased accuracy when using a screened hybrid DF for O2 + Al(111). However, even screened hybrid DFs are computationally expensive to use for metal surfaces. To resolve this, we test a non-self-consistent field (NSCF) screened hybrid DF approach. This approach computes screened hybrid DFT energies based on self-consistent-field (SCF) GGA electronic densities. Here, we explore the accuracy of the NSCF screened hybrid DF approach by implementing the NSCF HSE03-1/3x@RPBE DF for O2 + Al(111). We compute and analyse molecular beam sticking probabilities as well as a set of sticking probabilities for rotationally aligned O2. Our results show that the NSCF approach results in reaction probability curves that reproduce SCF results with near-chemical accuracy, suggesting that the NSCF approach can be used advantageously for exploratory purposes. An analysis of the potential energy surface and the barriers gives insight into the cause of the disagreement between the SCF and NSCF reaction probabilities and into the changes needed in theoretical modelling to further improve the description of the O2 + Al(111) system. Finally, the hole model yields fair agreement with dynamics results for the reaction probability curve, but results in an increased slope of the reaction probability curve compared to the molecular dynamics, with a shift to lower or higher energies depending on whether the vibrational energy of the molecule is included in the initial energy of the molecule or not.

Correcting Dispersion Corrections with Density-Corrected DFT

Almost all empirical parametrizations of dispersion corrections in DFT use only energy errors, thereby mixing functional and density-driven errors. We introduce density and dispersion-corrected DFT (D2C-DFT), a dual-calibration approach that accounts for density delocalization errors when parametrizing dispersion interactions. We simply exclude density-sensitive reactions from the training data. We find a significant reduction in both errors and variation among several semilocal functionals and their global hybrids when tailored dispersion corrections are employed with Hartree-Fock densities.

Targeting spectroscopic accuracy for dispersion bound systems from ab initio techniques: Translational eigenstates of Ne@C70

We investigate the endofullerene system Ne@C70 by constructing a three-dimensional Potential Energy Surface (PES) describing the translational motion of the Ne atom. This is constructed from electronic structure calculations from a plethora of methods, including MP2, SCS-MP2, SOS-MP2, RPA@PBE, and C(HF)-RPA, which were previously used for He@C60 in Panchagnula et al. [J. Chem. Phys. 160, 104303 (2024)], alongside B86bPBE-25X-XDM and B86bPBE-50X-XDM. The reduction in symmetry moving from C60 to C70 introduces a double well potential along the anisotropic direction, which forms a test of the sensitivity and effectiveness of the electronic structure methods. The nuclear Hamiltonian is diagonalized using a symmetrized double minimum basis set outlined in Panchagnula and Thom [J. Chem. Phys. 159, 164308 (2023)], with translational energies having error bars ±1 and ±2 cm-1. We find no consistency between electronic structure methods as they find a range of barrier heights and minima positions of the double well and different translational eigenspectra, which also differ from the Lennard-Jones (LJ) PES given in Mandziuk and Bačić [J. Chem. Phys. 101, 2126-2140 (1994)]. We find that generating effective LJ parameters for each electronic structure method cannot reproduce the full PES nor recreate the eigenstates, and this suggests that the LJ form of the PES, while simple, may not be best suited to describe these systems. Even though MP2 and RPA@PBE performed best for He@C60, due to the lack of concordance between all electronic structure methods, we require more experimental data in order to properly validate the choice.

How Does HF-DFT Achieve Chemical Accuracy for Water Clusters?

Bolstered by recent calculations of exact functional-driven errors (FEs) and density-driven errors (DEs) of semilocal density functionals in the water dimer binding energy [Kanungo, B. J. Phys. Chem. Lett. 2024, 15, 323-328], we investigate approximate FEs and DEs in neutral water clusters containing up to 20 monomers, charged water clusters, and alkali- and halide-water clusters. Our proxy for the exact density is r2SCAN 50, a 50% global hybrid of exact exchange with r2SCAN, which may be less correct than r2SCAN for the compact water monomer but importantly more correct for long-range electron transfers in the noncompact water clusters. We show that SCAN makes substantially larger FEs for neutral water clusters than r2SCAN, while both make essentially the same DEs. Unlike the case for barrier heights, these FEs are small in a relative sense and become large in an absolute sense only due to an increase in cluster size. SCAN@HF, short for SCAN evaluated on the Hartree-Fock (HF) density, produces a cancellation of errors that makes it chemically accurate for predicting the absolute binding energies of water clusters. Likewise, adding a long-range dispersion correction to r2SCAN@HF, as in the composite method HF-r2SCAN-DC4, makes its FE more negative than in r2SCAN@HF, permitting a near-perfect cancellation of FE and DE. r2SCAN by itself (and even more so, r2SCAN evaluated on the r2SCAN 50 density), is almost perfect for the energy differences between water hexamers, and thus probably also for liquid water away from the boiling point. Thus, the accuracy of composite methods like SCAN@HF and HF-r2SCAN-DC4 is not due to the HF density being closer to the exact density, but to a compensation of errors from its greater degree of localization. We also give an argument for the approximate reliability of this unconventional error cancellation for diverse molecular properties. Finally, we confirm this unconventional error cancellation for the SCAN description of the water trimer via Kohn-Sham inversion of the CCSD(T) density.

The rise and fall of stretched bond errors: Extending the analysis of Perdew-Zunger self-interaction corrections of reaction barrier heights beyond the LSDA

Incorporating self-interaction corrections (SIC) significantly improves chemical reaction barrier height predictions made using density functional theory methods. We present a detailed orbital-by-orbital analysis of these corrections for three semi-local density functional approximations (DFAs) situated on the three lowest rungs of Jacob's ladder of approximations. The analysis is based on Fermi-Löwdin Orbital Self-Interaction Correction (FLOSIC) calculations performed at several steps along the reaction pathway from the reactants (R) to the transition state (TS) to the products (P) for four representative reactions selected from the BH76 benchmark set. For all three functionals, the major contribution to self-interaction corrections of the barrier heights can be traced to stretched bond orbitals that develop near the TS configuration. The magnitude of the ratio of the self-exchange-correlation energy to the self-Hartree energy (XC/H) for a given orbital is introduced as an indicator of one-electron self-interaction error. XC/H = 1.0 implies that an orbital's self-exchange-correlation energy exactly cancels its self-Hartree energy and that the orbital, therefore, makes no contribution to the SIC in the FLOSIC scheme. For the practical DFAs studied here, XC/H spans a range of values. The largest values are obtained for stretched or strongly lobed orbitals. We show that significant differences in XC/H for corresponding orbitals in the R, TS, and P configurations can be used to identify the major contributors to the SIC of barrier heights and reaction energies. Based on such comparisons, we suggest that barrier height predictions made using the strongly constrained and appropriately normed meta-generalized gradient approximation may have attained the best accuracy possible for a semi-local functional using the Perdew-Zunger SIC approach.

Molecular driving forces for water adsorption in MOF-808: A comparative analysis with UiO-66

Metal-organic frameworks (MOFs), with their unique porous structures and versatile functionality, have emerged as promising materials for the adsorption, separation, and storage of diverse molecular species. In this study, we investigate water adsorption in MOF-808, a prototypical MOF that shares the same secondary building unit (SBU) as UiO-66, and elucidate how differences in topology and connectivity between the two MOFs influence the adsorption mechanism. To this end, molecular dynamics simulations were performed to calculate several thermodynamic and dynamical properties of water in MOF-808 as a function of relative humidity (RH), from the initial adsorption step to full pore filling. At low RH, the μ3-OH groups of the SBUs form hydrogen bonds with the initial water molecules entering the pores, which triggers the filling of these pores before the μ3-OH groups in other pores become engaged in hydrogen bonding with water molecules. Our analyses indicate that the pores of MOF-808 become filled by water sequentially as the RH increases. A similar mechanism has been reported for water adsorption in UiO-66. Despite this similarity, our study highlights distinct thermodynamic properties and framework characteristics that influence the adsorption process differently in MOF-808 and UiO-66.

Unconventional Error Cancellation Explains the Success of Hartree-Fock Density Functional Theory for Barrier Heights

Energy barriers, which control the rates of chemical reactions, are seriously underestimated by computationally efficient semilocal approximations for the exchange-correlation energy. The accuracy of a semilocal density functional approximation is strongly boosted for reaction barrier heights by evaluating that approximation non-self-consistently on Hartree-Fock electron densities, which has been known for ∼30 years. The conventional explanation is that the Hartree-Fock theory yields the more accurate density. This work presents a benchmark Kohn-Sham inversion of accurate coupled-cluster densities for the reaction H2 + F → HHF → H + HF and finds a strong, understandable cancellation between positive (excessively overcorrected) density-driven and large negative functional-driven errors (expected from stretched radical bonds in the transition state) within this Hartree-Fock density functional theory. This confirms earlier conclusions (Kaplan, A. D., et al. J. Chem. Theory Comput. 2023, 19, 532-543) based on 76 barrier heights and three less reliable, but less expensive, fully nonlocal density functional proxies for the exact density.

Total and orbital density-based analyses of molecules revealing long-range interaction regions

Total and orbital electron densities of molecules are explored for the effect of the long-range correction (LC) for density functional theory (DFT) exchange functionals by comparing to the effect of the ab initio coupled cluster singles and doubles (CCSD) method. Calculating the LC effect on the total electron densities shows that the LC stabilizes the electrons around the long-range interaction regions of kinetic energy density, which are assumed to be electrons other than free electrons and self-interacting electrons, while the CCSD method stabilizes the electrons in the long-range interaction regions in the vertical molecular planes. As a more precise test, the LC effect on orbital densities are compared to the CCSD effect on Dyson orbital densities. Surprisingly, these effects are similar for the unoccupied orbitals, indicating that the LC covers the effects required to reproduce the CCSD Dyson unoccupied orbitals. For exploring the discrepancies between these effects on the occupied orbitals, the photoionization cross sections are calculated as a direct test for the shapes of the HOMOs to investigate the differences between these effects on the occupied orbitals. Consequently, the LC clearly produces the canonical HOMOs close to the CCSD Dyson and experimental ones, except for the HOMO of benzene molecule that mixes with the HOMO - 1 for the CCSD Dyson orbitals. This indicates that the orbital analyses using the photoionization cross sections are available as a direct test for the quality of DFT functionals.

Consistent density functional theory-based description of ion hydration through density-corrected many-body representations

Delocalization error constrains the accuracy of density functional theory in describing molecular interactions in ion-water systems. Using Na+ and Cl- in water as model systems, we calculate the effects of delocalization error in the SCAN functional for describing ion-water and water-water interactions in hydrated ions, and demonstrate that density-corrected SCAN (DC-SCAN) predicts n-body and interaction energies with an accuracy approaching coupled cluster theory. The performance of DC-SCAN is size-consistent, maintaining an accurate description of molecular interactions well beyond the first solvation shell. Molecular dynamics simulations at ambient conditions with many-body MB-SCAN(DC) potentials, derived from the many-body expansion, predict the solvation structure of Na+ and Cl- in quantitative agreement with reference data, while simultaneously reproducing the structure of liquid water. Beyond rationalizing the accuracy of density-corrected models of ion hydration, our findings suggest that our unified density-corrected MB formalism holds great promise for efficient DFT-based simulations of condensed-phase systems with chemical accuracy.

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.

Density-Corrected Density Functional Theory for Open Shells: How to Deal with Spin Contamination

Density functional theory (DFT) is usually used self-consistently to predict chemical properties, but the use of the Hartree-Fock (HF) density improves energetics in certain, well-characterized cases. Density-corrected (DC) DFT provides the theory behind this, but unrestricted Hartree-Fock (UHF) densities yield poor energetics in cases of strong spin contamination. Here we compare with restricted open-shell HF (ROHF) across 13 different functionals and two DC-DFT methods. For significant spin contamination, ROHF densities outperform UHF densities by as much as a factor of 3, depending on the energy functional, and ROHF-DFT improves over self-consistent DFT for most of the tested functionals. We refine the DC(HF)-DFT algorithm to use ROHF densities in cases of severe spin contamination.

Simple and Efficient Route toward Improved Energetics within the Framework of Density-Corrected Density Functional Theory

The crucial step in density-corrected Hartree-Fock density functional theory (DC(HF)-DFT) is to decide whether the density produced by the density functional for a specific calculation is erroneous and, hence, should be replaced by, in this case, the HF density. We introduce an indicator, based on the difference in noninteracting kinetic energies between DFT and HF calculations, to determine when the HF density is the better option. Our kinetic energy indicator directly compares the self-consistent density of the analyzed functional with the HF density, is size-intensive, reliable, and most importantly highly efficient. Moreover, we present a procedure that makes best use of the computed quantities necessary for DC(HF)-DFT by additionally evaluating a related hybrid functional and, in that way, not only "corrects" the density but also the functional itself; we call that procedure corrected Hartree-Fock density functional theory (C(HF)-DFT).

The Good, the Bad, and the Ugly: Pseudopotential Inconsistency Errors in Molecular Applications of Density Functional Theory

The pseudopotential (PP) approximation is one of the most common techniques in computational chemistry. Despite its long history, the development of custom PPs has not tracked with the explosion of different density functional approximations (DFAs). As a result, the use of PPs with exchange/correlation models for which they were not developed is widespread, although this practice is known to be theoretically unsound. The extent of PP inconsistency errors (PPIEs) associated with this practice has not been systematically explored across the types of energy differences commonly evaluated in chemical applications. We evaluate PPIEs for a number of PPs and DFAs across 196 chemically relevant systems of both transition-metal and main-group elements, as represented by the W4-11, TMC34, and S22 data sets. Near the complete basis set limit, these PPs are found to cleanly approach all-electron (AE) results for noncovalent interactions but introduce root-mean-squared errors (RMSEs) upwards of 15 kcal mol^-1 into predictions of covalent bond energies for a number of popular DFAs. We achieve significant improvements through the use of empirical atom- and DFA-specific PP corrections, indicating considerable systematicity of the PPIEs. The results of this work have implications for chemical modeling in both molecular contexts and for DFA design, which we discuss.

Interaction energy of Cl2 and Br2 with H2 O: Exchange, dispersion and density the crucial ingredients

Modern Density Functional Theory models are now suitable for many molecular and condensed phase studies. The study of noncovalent interactions, a well-known drawback, is no longer an insurmountable obstacle through design and empirical corrections. However, using empirical corrections as in the DFT-D methods might not be an all-in-one solution. This work uses a simple system, X₂ -H₂ O with X = Cl or Br, with two different interactions, halogen-bonded (XB) and hydrogen-halogen (HX), to investigate the capability of current density functional approximations (DFA) in predicting interaction energies with eight different exchange-correlation functionals. SAPT(DFT) provides, for all the studied cases, better predictions than the widely used supermolecular approach. In addition, the components of the interaction energy suggest where some of the shortcomings originate in each DFA. The analysis of the functionals used confirms that PBE0 and ω-B97X-D have a physically correct behavior. Using SAPT(DFT) and PBE0, and ω-B97X-D, we obtained the interaction energy of Cl₂ and Br₂ inside different clathrate cages and satisfactorily compared with wavefunction results; hence, the lower and upper limits of this value are defined: Cl₂ @5¹² , -5.3 ± 0.3 kcal/mol; Cl₂ @5¹² 6² , -5.5 ± 0.1 kcal/mol; Br₂ @5¹² 6² , -7.6 ± 1.0 kcal/mol; Br₂ @5¹² 6³ , -10.6 ± 1.0 kcal/mol; Br₂ @5¹² 6⁴ , -10.9 ± 0.8 kcal/mol.

How Do Self-Interaction Errors Associated with Stretched Bonds Affect Barrier Height Predictions?

Density functional theory (DFT) suffers from self-interaction errors (SIEs) that generally result in the underestimation of chemical reaction barrier heights. This is commonly attributed to the tendency of density functional approximations to overstabilize delocalized densities that typically occur in the stretched bonds of transition state structures. The Perdew-Zunger self-interaction correction (PZSIC) and locally scaled self-interaction correction (LSIC) improve the prediction of barrier heights of chemical reactions, with LSIC giving better accuracy than PZSIC on average. These methods employ an orbital-by-orbital correction scheme to remove the one-electron SIE. In the context of barrier heights, this allows an analysis of how the self-interaction correction (SIC) for each orbital contributes to the calculated barriers using Fermi-Löwdin orbitals (FLOs). We hypothesize that the SIC contribution to the reaction barrier comes mainly from a limited number of orbitals that are directly involved in bond-breaking and bond-making in the reaction transition state. We call these participant orbitals (POs), in contrast to spectator orbitals (SOs) which are not directly involved in changes to the bonding. Our hypothesis is that ΔE_Total^SIC ≈ ΔE_PO^SIC, where ΔE_Total^SIC is the difference in the SIC corrections for the reactants or products and the transition state. We test this hypothesis for the reaction barriers of the BH76 benchmark set of reactions. We find that the stretched-bond orbitals indeed make the largest individual SIC contributions to the barriers. These contributions increase the barrier heights relative to LSDA, which underpredicts the barrier. However, the full stretched-bond hypothesis does not hold in all cases for either PZSIC or LSIC. There are many cases where the total SIC contribution from the SOs is significant and cannot be ignored. The size of the SIC contribution to the barrier height is a key indicator. A large SIC correction is correlated to a large LSDA error in the barrier, showing that PZSIC properly gives larger corrections when corrections are needed most. A comparison of the performance of PZSIC and LSIC shows that the two methods have similar accuracy for reactions with large LSDA errors, but LSIC is clearly better for reactions with small errors. We trace this to an improved description of reaction energies in LSIC.

Extending density functional theory with near chemical accuracy beyond pure water

Density functional simulations of condensed phase water are typically inaccurate, due to the inaccuracies of approximate functionals. A recent breakthrough showed that the SCAN approximation can yield chemical accuracy for pure water in all its phases, but only when its density is corrected. This is a crucial step toward first-principles biosimulations. However, weak dispersion forces are ubiquitous and play a key role in noncovalent interactions among biomolecules, but are not included in the new approach. Moreover, naïve inclusion of dispersion in HF-SCAN ruins its high accuracy for pure water. Here we show that systematic application of the principles of density-corrected DFT yields a functional (HF-r²SCAN-DC4) which recovers and not only improves over HF-SCAN for pure water, but also captures vital noncovalent interactions in biomolecules, making it suitable for simulations of solutions.

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.

Understanding Density-Driven Errors for Reaction Barrier Heights

Delocalization errors, such as charge-transfer and some self-interaction errors, plague computationally efficient and otherwise accurate density functional approximations (DFAs). Evaluating a semilocal DFA non-self-consistently on the Hartree-Fock (HF) density is often recommended as a computationally inexpensive remedy for delocalization errors. For sophisticated meta-GGAs like SCAN, this approach can achieve remarkable accuracy. This HF-DFT (also known as DFA@HF) is often presumed to work, when it significantly improves over the DFA, because the HF density is more accurate than the self-consistent DFA density in those cases. By applying the metrics of density-corrected density functional theory (DFT), we show that HF-DFT works for barrier heights by making a localizing charge-transfer error or density overcorrection, thereby producing a somewhat reliable cancellation of density- and functional-driven errors for the energy. A quantitative analysis of the charge-transfer errors in a few randomly selected transition states confirms this trend. We do not have the exact functional and electron densities that would be needed to evaluate the exact density- and functional-driven errors for the large BH76 database of barrier heights. Instead, we have identified and employed three fully nonlocal proxy functionals (SCAN 50% global hybrid, range-separated hybrid LC-ωPBE, and SCAN-FLOSIC) and their self-consistent proxy densities. These functionals are chosen because they yield reasonably accurate self-consistent barrier heights and because their self-consistent total energies are nearly piecewise linear in fractional electron number─two important points of similarity to the exact functional. We argue that density-driven errors of the energy in a self-consistent density functional calculation are second order in the density error and that large density-driven errors arise primarily from incorrect electron transfers over length scales larger than the diameter of an atom.

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.

Electronic-Structure Properties from Atom-Centered Predictions of the Electron Density

The electron density of a molecule or material has recently received major attention as a target quantity of machine-learning models. A natural choice to construct a model that yields transferable and linear-scaling predictions is to represent the scalar field using a multicentered atomic basis analogous to that routinely used in density fitting approximations. However, the nonorthogonality of the basis poses challenges for the learning exercise, as it requires accounting for all the atomic density components at once. We devise a gradient-based approach to directly minimize the loss function of the regression problem in an optimized and highly sparse feature space. In so doing, we overcome the limitations associated with adopting an atom-centered model to learn the electron density over arbitrarily complex data sets, obtaining very accurate predictions using a comparatively small training set. The enhanced framework is tested on 32-molecule periodic cells of liquid water, presenting enough complexity to require an optimal balance between accuracy and computational efficiency. We show that starting from the predicted density a single Kohn-Sham diagonalization step can be performed to access total energy components that carry an error of just 0.1 meV/atom with respect to the reference density functional calculations. Finally, we test our method on the highly heterogeneous QM9 benchmark data set, showing that a small fraction of the training data is enough to derive ground-state total energies within chemical accuracy.

Best-Practice DFT Protocols for Basic Molecular Computational Chemistry

Nowadays, many chemical investigations are supported by routine calculations of molecular structures, reaction energies, barrier heights, and spectroscopic properties. The lion's share of these quantum-chemical calculations applies density functional theory (DFT) evaluated in atomic-orbital basis sets. This work provides best-practice guidance on the numerous methodological and technical aspects of DFT calculations in three parts: Firstly, we set the stage and introduce a step-by-step decision tree to choose a computational protocol that models the experiment as closely as possible. Secondly, we present a recommendation matrix to guide the choice of functional and basis set depending on the task at hand. A particular focus is on achieving an optimal balance between accuracy, robustness, and efficiency through multi-level approaches. Finally, we discuss selected representative examples to illustrate the recommended protocols and the effect of methodological choices.

How Good Is the Density-Corrected SCAN Functional for Neutral and Ionic Aqueous Systems, and What Is So Right about the Hartree-Fock Density?

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 (O_A) and donor (O_D) 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.

Detection and Correction of Delocalization Errors for Electron and Hole Polarons Using Density-Corrected DFT

Modeling polaron defects is an important aspect of computational materials science, but the description of unpaired spins in density functional theory (DFT) often suffers from delocalization error. To diagnose and correct the overdelocalization of spin defects, we report an implementation of density-corrected (DC-)DFT and its analytic energy gradient. In DC-DFT, an exchange-correlation functional is evaluated using a Hartree-Fock density, thus incorporating electron correlation while avoiding self-interaction error. Results for an electron polaron in models of titania and a hole polaron in Al-doped silica demonstrate that geometry optimization with semilocal functionals drives significant structural distortion, including the elongation of several bonds, such that subsequent single-point calculations with hybrid functionals fail to afford a localized defect even in cases where geometry optimization with the hybrid functional does localize the polaron. This has significant implications for traditional workflows in computational materials science, where semilocal functionals are often used for structure relaxation. DC-DFT calculations provide a mechanism to detect situations where delocalization error is likely to affect the results.

Searching for a Reliable Density Functional for Molecule-Environment Interactions, Found B97M-V/def2-mTZVP

In the present study, we have examined density functional theory methods for the calculation of the interaction energy between a small molecule and its environment. For simple systems such as a neutral solute in a neutral solvent, good accuracy can be attained using low-cost "3c" methods, in particular r²SCAN-3c. When part(s) of the system is charged, the accurate computation of the interactions is more challenging. In these cases, we find the B97M-V/def2-mTZVP method to agree well with reference values; it also shows good accuracy for the more straightforward neutral systems. Thus, B97M-V/def2-mTZVP provides a means for accurate and low-cost computation of interaction energies, notably the binding between a substrate or a drug molecule and an enzyme, which may facilitate rational drug design.

Improving Results by Improving Densities: Density-Corrected Density Functional Theory

Density functional theory (DFT) calculations have become widespread in both chemistry and materials, because they usually provide useful accuracy at much lower computational cost than wavefunction-based methods. All practical DFT calculations require an approximation to the unknown exchange-correlation energy, which is then used self-consistently in the Kohn-Sham scheme to produce an approximate energy from an approximate density. Density-corrected DFT is simply the study of the relative contributions to the total energy error. In the vast majority of DFT calculations, the error due to the approximate density is negligible. But with certain classes of functionals applied to certain classes of problems, the density error is sufficiently large as to contribute to the energy noticeably, and its removal leads to much better results. These problems include reaction barriers, torsional barriers involving π-conjugation, halogen bonds, radicals and anions, most stretched bonds, etc. In all such cases, use of a more accurate density significantly improves performance, and often the simple expedient of using the Hartree-Fock density is enough. This Perspective explains what DC-DFT is, where it is likely to improve results, and how DC-DFT can produce more accurate functionals. We also outline challenges and prospects for the field.

Study of self-interaction-errors in barrier heights using locally scaled and Perdew-Zunger self-interaction methods

We study the effect of self-interaction errors on the barrier heights of chemical reactions. For this purpose, we use the well-known Perdew-Zunger self-interaction-correction (PZSIC) [J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981)] as well as two variations of the recently developed, locally scaled self-interaction correction (LSIC) [Zope et al., J. Chem. Phys. 151, 214108 (2019)] to study the barrier heights of the BH76 benchmark dataset. Our results show that both PZSIC and especially the LSIC methods improve the barrier heights relative to the local density approximation (LDA). The version of LSIC that uses the iso-orbital indicator z as a scaling factor gives a more consistent improvement than an alternative version that uses an orbital-dependent factor w based on the ratio of orbital densities to the total electron density. We show that LDA energies evaluated using the self-consistent and self-interaction-free PZSIC densities can be used to assess density-driven errors. The LDA reaction barrier errors for the BH76 set are found to contain significant density-driven errors for all types of reactions contained in the set, but the corrections due to adding SIC to the functional are much larger than those stemming from the density for the hydrogen transfer reactions and of roughly equal size for the non-hydrogen transfer reactions.

The potential energy profile of the decomposition of 1,1-diamino-2,2-dinitroethylene (FOX-7) in the gas phase

The decomposition products of 1,1-diamino-2,2-dinitroethylene (FOX-7) in the gas phase.

Electronic Energy and Local Property Errors at QTAIM Critical Points while Climbing Perdew's Ladder of Density-Functional Approximations

We investigate the relationships between electron-density and electronic-energy errors produced by modern exchange-correlation density-functional approximations belonging to all of the rungs of Perdew's ladder. To this aim, a panel of relevant (semi)local properties evaluated at critical points of the electron-density field (as defined within the framework of Bader's atoms-in-molecules theory) are computed on a large selection of molecular systems involved in thermodynamic, kinetic, and noncovalent interaction chemical databases using density functionals developed in a nonempirical and minimally and highly parametrized fashion. The comparison of their density- and energy-based performance, also discussed in terms of density-driven errors, casts light on the strengths and weaknesses of the most recent and efficient density-functional approximations.

Seeking the Optimal Descriptor for SN2 Reactions through Statistical Analysis of Density Functional Theory Results

Bimolecular nucleophilic substitution is one of the fundamental reactions in organic chemistry, yet there is still knowledge to be gained on the role of the nucleophile and the substrate. A statistical treatment of over 600 density functional theory (DFT)-computed barriers for bimolecular nucleophilic substitution at methyl derivatives (S_N2@C) leads to the identification of numerical descriptors that best represent the entering and leaving ability of 26 different nucleophiles. The treatment is based on singular value decomposition (SVD) of a matrix of computed energy barriers. The current work represents the extension to a problem of reactivity of the hidden descriptor methodology that we had previously developed for the thermodynamic problem of bond dissociation energies in transition-metal complexes. The analysis of the results shows that a single descriptor is sufficient. This hidden descriptor has different values for nucleophilic and leaving abilities and, contrary to expectation, does not correlate especially well with either frontier molecular orbital descriptors or solvation descriptors. In contrast, it correlates with other thermodynamic and geometric parameters. This statistical procedure can be in principle extended to additional chemical fragments and other reactions.

The effect of noise on the predictive limit of QSAR models

A key challenge in the field of Quantitative Structure Activity Relationships (QSAR) is how to effectively treat experimental error in the training and evaluation of computational models. It is often assumed in the field of QSAR that models cannot produce predictions which are more accurate than their training data. Additionally, it is implicitly assumed, by necessity, that data points in test sets or validation sets do not contain error, and that each data point is a population mean. This work proposes the hypothesis that QSAR models can make predictions which are more accurate than their training data and that the error-free test set assumption leads to a significant misevaluation of model performance. This work used 8 datasets with six different common QSAR endpoints, because different endpoints should have different amounts of experimental error associated with varying complexity of the measurements. Up to 15 levels of simulated Gaussian distributed random error was added to the datasets, and models were built on the error laden datasets using five different algorithms. The models were trained on the error laden data, evaluated on error-laden test sets, and evaluated on error-free test sets. The results show that for each level of added error, the RMSE for evaluation on the error free test sets was always better. The results support the hypothesis that, at least under the conditions of Gaussian distributed random error, QSAR models can make predictions which are more accurate than their training data, and that the evaluation of models on error laden test and validation sets may give a flawed measure of model performance. These results have implications for how QSAR models are evaluated, especially for disciplines where experimental error is very large, such as in computational toxicology.

Combining Machine Learning and Computational Chemistry for Predictive Insights Into Chemical Systems

Machine learning models are poised to make a transformative impact on chemical sciences by dramatically accelerating computational algorithms and amplifying insights available from computational chemistry methods. However, achieving this requires a confluence and coaction of expertise in computer science and physical sciences. This Review is written for new and experienced researchers working at the intersection of both fields. We first provide concise tutorials of computational chemistry and machine learning methods, showing how insights involving both can be achieved. We follow with a critical review of noteworthy applications that demonstrate how computational chemistry and machine learning can be used together to provide insightful (and useful) predictions in molecular and materials modeling, retrosyntheses, catalysis, and drug design.

Assessment of the performance of DFT functionals in the fulfillment of off-diagonal hypervirial relationships

Off-diagonal hypervirial relationships, combined with quantum mechanical sum rules of charge-current conservation, offer a way to test electronic excited-state transition energies and moments, which does not need any external reference. A number of fundamental relationships were recast into absolute deviations from zero, which have been used to assess the performance of some popular DFT functionals. Extended TD-DFT calculations have been carried out for a pool of molecules chosen for this purpose, adopting a large basis set to ensure high quality results. A partial agreement with previous benchmarks is observed.

Requirements for an accurate dispersion-corrected density functional

Post-self-consistent dispersion corrections are now the norm when applying density-functional theory to systems where non-covalent interactions play an important role. However, there is a wide range of base functionals and dispersion corrections available from which to choose. In this work, we opine on the most desirable requirements to ensure that both the base functional and dispersion correction, individually, are as accurate as possible for non-bonded repulsion and dispersion attraction. The base functional should be dispersionless, numerically stable, and involve minimal delocalization error. Simultaneously, the dispersion correction should include finite damping, higher-order pairwise dispersion terms, and electronic many-body effects. These criteria are essential for avoiding reliance on error cancellation and obtaining correct results from correct physics.

On the Use of Normalized Metrics for Density Sensitivity Analysis in DFT

In the past years, there has been a discussion about how the errors in density functional theory might be related to errors in the self-consistent densities obtained from different density functional approximations. This, in turn, brings up the discussion about the different ways in which we can measure such errors and develop metrics that assess the sensitivity of calculated energies to changes in the density. It is important to realize that there cannot be a unique metric in order to look at this density sensitivity, simultaneously needing size-extensive and size-intensive metrics. In this study, we report two metrics that are widely applicable to any density functional approximation. We also show how they can be used to classify different chemical systems of interest with respect to their sensitivity to small variations in the density.

Improved Spin-State Energy Differences of Fe(II) Molecular and Crystalline Complexes via the Hubbard U-Corrected Density

We recently showed that the DFT+U approach with a linear-response U yields adiabatic energy differences biased toward high spin [Mariano et al. J. Chem. Theory Comput. 2020, 16, 6755-6762]. Such bias is removed here by employing a density-corrected DFT approach where the PBE functional is evaluated on the Hubbard U-corrected density. The adiabatic energy differences of six Fe(II) molecular complexes computed using this approach, named PBE[U] here, are in excellent agreement with coupled cluster-corrected CASPT2 values for both weak- and strong-field ligands resulting in a mean absolute error (MAE) of 0.44 eV, smaller than that of the recently proposed Hartree-Fock density-corrected DFT (1.22 eV) and any other tested functional, including the best performer TPSSh (0.49 eV). We take advantage of the computational efficiency of this approach and compute the adiabatic energy differences of five molecular crystals using PBE[U] with periodic boundary conditions. The results show, again, an excellent agreement (MAE = 0.07 eV) with experimentally extracted values and a superior performance compared with the best performers M06-L (MAE = 0.08 eV) and TPSSh (MAE = 0.31 eV) computed on molecular fragments.

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.

Density Sensitivity of Empirical Functionals

Empirical fitting of parameters in approximate density functionals is common. Such fits conflate errors in the self-consistent density with errors in the energy functional, but density-corrected DFT (DC-DFT) separates these two. We illustrate with catastrophic failures of a toy functional applied to H₂⁺ at varying bond lengths, where the standard fitting procedure misses the exact functional; Grimme's D3 fit to noncovalent interactions, which can be contaminated by large density errors such as in the WATER27 and B30 data sets; and double-hybrids trained on self-consistent densities, which can perform poorly on systems with density-driven errors. In these cases, more accurate results are found at no additional cost by using Hartree-Fock (HF) densities instead of self-consistent densities. For binding energies of small water clusters, errors are greatly reduced. Range-separated hybrids with 100% HF at large distances suffer much less from this effect.

The Trip to the Density Functional Theory Zoo Continues: Making a Case for Time-Dependent Double Hybrids for Excited-State Problems

This account is written for general users of time-dependent density functional theory (TD-DFT) methods as well as chemists who are unfamiliar with the field. It includes a brief overview of conventional TD-DFT approaches and recommendations for applications to organic molecules based on our own experience. The main emphasis of this work, however, lies in providing the first in-depth review of time-dependent double-hybrid density functionals. They were first established in 2007 with very promising follow-up studies in the subsequent four years before developments or applications became scarce. The topic has regained more interest since 2017, and this account reviews those latest developments led by our group. These developments have shown unprecedented robustness for a variety of different types of electronic excitations when compared to more conventional TD-DFT methods. In particular, time-dependent double hybrids do not suffer from artificial ghost states and are able to reproduce exciton-coupled absorption spectra. Our latest methods include range separation and belong to the currently best TD-DFT methods for singlet-singlet excitations in organic molecules. While there is still room for improvement and further development in this space, we hope that this account encourages users to adjust their computational protocols to such new methods to provide more real-life testing and scenarios.

Representing Exact Electron Densities by a Single Slater Determinant in Finite Basis Sets

The premise for Kohn-Sham density functional theory is that the exact electron density can be generated by a set of orbitals in a single Slater determinant. While this is ensured in a complete basis set, it has been shown that it cannot hold in small basis sets. The present work probes how accurately a reference electron density of the full-CI type can be reproduced by a set of orbitals in a single Slater determinant, as a function of the basis set used for the fitting electron density. The key finding is that the fitting error may be significant for basis sets of double- or triple-ζ quality. It is also shown that it is important that the fitting basis set includes the same basis functions as used for generating the reference electron density. The main limitation in a given basis set is the lack of higher order polarization functions. The error for practical purposes becomes insignificant for basis sets of quadruple-ζ or better quality, and this should be the choice when assessing the accuracy of exchange-correlation functionals by comparing electron densities to accurate reference results generated by wave function methods. The methodology in the present work can be used to transform an electron density from a multideterminant wave function into a set of orbitals in a single Slater determinant, and this may be useful for developing and testing new exchange-correlation functionals.

Mean-field density matrix decompositions

We introduce new and robust decompositions of mean-field Hartree-Fock and Kohn-Sham density functional theory relying on the use of localized molecular orbitals and physically sound charge population protocols. The new lossless property decompositions, which allow for partitioning one-electron reduced density matrices into either bond-wise or atomic contributions, are compared to alternatives from the literature with regard to both molecular energies and dipole moments. Besides commenting on possible applications as an interpretative tool in the rationalization of certain electronic phenomena, we demonstrate how decomposed mean-field theory makes it possible to expose and amplify compositional features in the context of machine-learned quantum chemistry. This is made possible by improving upon the granularity of the underlying data. On the basis of our preliminary proof-of-concept results, we conjecture that many of the structure-property inferences in existence today may be further refined by efficiently leveraging an increase in dataset complexity and richness.

Improving the exchange and correlation potential in density-functional approximations through constraints

We review and expand on our work to impose constraints on the effective Kohn-Sham (KS) potential of local and semi-local density-functional approximations. Constraining the minimisation of the approximate total energy density-functional invariably leads to an optimised effective potential (OEP) equation, the solution of which yields the KS potential. We review briefly our previous work on this and demonstrate with numerous examples that despite the well-known mathematical issues of the OEP with finite basis sets, our OEP equations are numerically robust. We demonstrate that appropriately constraining the 'screening charge' which corresponds to the Hartree, exchange and correlation potential not only corrects its asymptotic behaviour but also allows the exchange and correlation potential to exhibit a non-zero derivative discontinuity, a feature of the exact KS potential that is necessary for the accurate prediction of band-gaps in solids but very hard to capture with semi-local approximations.

Quantum chemical accuracy from density functional approximations via machine learning

Kohn-Sham density functional theory (DFT) is a standard tool in most branches of chemistry, but accuracies for many molecules are limited to 2-3 kcal ⋅ mol-1 with presently-available functionals. Ab initio methods, such as coupled-cluster, routinely produce much higher accuracy, but computational costs limit their application to small molecules. In this paper, we leverage machine learning to calculate coupled-cluster energies from DFT densities, reaching quantum chemical accuracy (errors below 1 kcal ⋅ mol-1) on test data. Moreover, density-based Δ-learning (learning only the correction to a standard DFT calculation, termed Δ-DFT ) significantly reduces the amount of training data required, particularly when molecular symmetries are included. The robustness of Δ-DFT  is highlighted by correcting "on the fly" DFT-based molecular dynamics (MD) simulations of resorcinol (C6H4(OH)2) to obtain MD trajectories with coupled-cluster accuracy. We conclude, therefore, that Δ-DFT  facilitates running gas-phase MD simulations with quantum chemical accuracy, even for strained geometries and conformer changes where standard DFT fails.