Ions in solution: Density corrected density functional theory (DC-DFT)

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.

Triplet Excitation-Energy Transfer Couplings from Subsystem Time-Dependent Density-Functional Theory

We present an implementation of triplet excitation-energy transfer (TEET) couplings based on subsystem-based time-dependent density-functional theory (sTDDFT). TEET couplings are systematically investigated by comparing "exact" and approximate variants of sTDDFT. We demonstrate that, while sTDDFT utilizing explicit approximate non-additive kinetic energy (NAKE) density functionals is well-suited for describing singlet EET processes, it is inadequate for characterizing TEET. However, we show that projection-based embedding (PbE)-based sTDDFT addresses the challenges faced by NAKE-sTDDFT and emerges as a promising method for accurately describing electronic couplings in TEET processes. We also introduce the mixed PbE-/NAKE-embedding procedure to investigate the TEET effects in solvated pairs of chromophores. This approach offers a good balance between accuracy and efficiency, enabling comprehensive studies of TEET processes in complex environments.

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.

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).

Radicals in aqueous solution: assessment of density-corrected SCAN functional

We study self-interaction effects in solvated and strongly-correlated cationic molecular clusters, with a focus on the solvated hydroxyl radical. To address the self-interaction issue, we apply the DC-r2SCAN method, with the auxiliary density matrix approach. Validating our method through simulations of bulk liquid water, we demonstrate that DC-r2SCAN maintains the structural accuracy of r2SCAN while effectively addressing spin density localization issues. Extending our analysis to solvated cationic molecular clusters, we find that the hemibonded motif in the [CH3S∴CH3SH]+ cluster is disrupted in the DC-r2SCAN simulation, in contrast to r2SCAN that preserves the (three-electron-two-center)-bonded motif. Similarly, for the [SH∴SH2]+ cluster, r2SCAN restores the hemibonded motif through spin leakage, while DC-r2SCAN predicts a weaker hemibond formation influenced by solvent-solute interactions. Our findings demonstrate the potential of DC-r2SCAN combined with the auxiliary density matrix method to improve electronic structure calculations, providing insights into the properties of solvated cationic molecular clusters. This work contributes to the advancement of self-interaction corrected electronic structure theory and offers a computational framework for modeling condensed phase systems with intricate correlation effects.

Assessing the Role of the Kohn-Sham Density in the Calculation of the Low-Lying Bethe-Salpeter Excitation Energies

We adopt the GW many-body perturbation theory in conjunction with the Bethe-Salpeter equation (BSE) to compute 57 excitation energies of a set of 37 molecules. By using the PBEh global hybrid functional and a self-consistent scheme on the eigenvalues in GW, we show a strong dependence of the BSE energy on the starting Kohn-Sham (KS) density functional. This arises from both the quasiparticle energies and the spatial localization of the frozen KS orbitals employed to compute the BSE. In order to address the arbitrariness in the mean field choice, we adopt an orbital-tuning scheme where the amount of Fock exchange, α, is tuned to impose that the KS HOMO matches the GW quasiparticle eigenvalue, thus fulfilling the ionization potential theorem in DFT. The performance of the proposed scheme yields excellent results and it is similar to M06-2X and PBEh with α = 75%, consistent with tuned values of α ranging between 60% and 80%.

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.

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.

Subsystem density-functional theory: A reliable tool for spin-density based properties

Subsystem density-functional theory compiles a set of features that allow for efficiently calculating properties of very large open-shell radical systems such as organic radical crystals, proteins, or deoxyribonucleic acid stacks. It is computationally less costly than correlated ab initio wave function approaches and can pragmatically avoid the overdelocalization problem of Kohn-Sham density-functional theory without employing hard constraints on the electron-density. Additionally, subsystem density-functional theory calculations commonly start from isolated fragment electron densities, pragmatically preserving a priori specified subsystem spin-patterns throughout the calculation. Methods based on subsystem density-functional theory have seen a rapid development over the past years and have become important tools for describing open-shell properties. In this Perspective, we address open questions and possible developments toward challenging future applications in connection with subsystem density-functional theory for spin-dependent properties.

Wavefunction embedding for molecular polaritons

Polaritonic chemistry relies on the strong light-matter interaction phenomena for altering the chemical reaction rates inside optical cavities. To explain and to understand these processes, the development of reliable theoretical models is essential. While computationally efficient quantum electrodynamics self-consistent field (QED-SCF) methods, such as quantum electrodynamics density functional theory (QEDFT) needs accurate functionals, quantum electrodynamics coupled cluster (QED-CC) methods provide a systematic increase in accuracy but at much greater cost. To overcome this computational bottleneck, herein we introduce and develop the QED-CC-in-QED-SCF projection-based embedding method that inherits all the favorable properties from the two worlds, computational efficiency and accuracy. The performance of the embedding method is assessed by studying some prototypical but relevant reactions, such as methyl transfer reaction, proton transfer reaction, as well as protonation reaction in a complex environment. The results obtained with the new embedding method are in excellent agreement with more expensive QED-CC results. The analysis performed on these reactions indicate that the electron-photon correlation effects are local in nature and that only a small region should be treated at the QED-CC level for capturing important effects due to cavity. This work sets the stage for future developments of polaritonic quantum chemistry methods and it will serve as a guideline for development of other polaritonic embedding models.

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.

Assessing the Interplay between Functional-Driven and Density-Driven Errors in DFT Models of Water

We investigate the interplay between functional-driven and density-driven errors in different density functional approximations within density functional theory (DFT) and the implications of these errors for simulations of water with DFT-based data-driven potentials. Specifically, we quantify density-driven errors in two widely used dispersion-corrected functionals derived within the generalized gradient approximation (GGA), namely BLYP-D3 and revPBE-D3, and two modern meta-GGA functionals, namely strongly constrained and appropriately normed (SCAN) and B97M-rV. The effects of functional-driven and density-driven errors on the interaction energies are first assessed for the water clusters of the BEGDB dataset. Further insights into the nature of functional-driven errors are gained from applying the absolutely localized molecular orbital energy decomposition analysis (ALMO-EDA) to the interaction energies, which demonstrates that functional-driven errors are strongly correlated with the nature of the interactions. We discuss cases where density-corrected DFT (DC-DFT) models display higher accuracy than the original DFT models and cases where reducing the density-driven errors leads to larger deviations from the reference energies due to the presence of large functional-driven errors. Finally, molecular dynamics simulations are performed with data-driven many-body potentials derived from DFT and DC-DFT data to determine the effect that minimizing density-driven errors has on the description of liquid water. Besides rationalizing the performance of widely used DFT models of water, we believe that our findings unveil fundamental relations between the shortcomings of some common DFT approximations and the requirements for accurate descriptions of molecular interactions, which will aid the development of a consistent, DFT-based framework for the development of data-driven and machine-learned potentials for simulations of condensed-phase systems.

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.

Quantification of Geometric Errors Made Simple: Application to Main-Group Molecular Structures

Nearly all electronic structure simulations begin with obtaining approximate geometries, making a systematic quantification of errors in approximate molecular structures of key importance. Recently, the geometric energy offset (GEO) framework based on a single and natural measure for quantifying and analyzing these errors has been proposed ( J. Phys. Chem. Lett. 2020, 11, 99579964). An accurate and far less costly approximation to GEO is utilized here to readily quantify errors in main-group structures and analyze them in a chemically intuitive way. The use of semiexperimental geometries as a reference further simplifies the analysis. The analysis reveals new insights into the geometric performance of methods, their rankings, as well as patterns across different classes of methods and basis sets that arise from the analysis.

Elevating density functional theory to chemical accuracy for water simulations through a density-corrected many-body formalism

Density functional theory (DFT) has been extensively used to model the properties of water. Albeit maintaining a good balance between accuracy and efficiency, no density functional has so far achieved the degree of accuracy necessary to correctly predict the properties of water across the entire phase diagram. Here, we present density-corrected SCAN (DC-SCAN) calculations for water which, minimizing density-driven errors, elevate the accuracy of the SCAN functional to that of "gold standard" coupled-cluster theory. Building upon the accuracy of DC-SCAN within a many-body formalism, we introduce a data-driven many-body potential energy function, MB-SCAN(DC), that quantitatively reproduces coupled cluster reference values for interaction, binding, and individual many-body energies of water clusters. Importantly, molecular dynamics simulations carried out with MB-SCAN(DC) also reproduce the properties of liquid water, which thus demonstrates that MB-SCAN(DC) is effectively the first DFT-based model that correctly describes water from the gas to the liquid phase.

Density Functional Calculations Based on the Exponential Ansatz

This work explores the application of the singles-based exponential ansatz to density functional calculations. In contrast to the standard approach where Kohn-Sham (KS) orbitals are determined prior to computing molecular quantities of interest, we consider the single-reference Hartree-Fock wave function as a starting point. Applying the exponential ansatz to this single reference gives an auxiliary wave function that is employed to calculate the electronic properties of the system. This wave function is determined self-consistently through the standard KS Hamiltonian but evaluated over the Hartree-Fock molecular orbital basis. By using spin-symmetry breaking, we recover size-consistent results free of unphysical fractional charges in the dissociation limit. Our method shows consistency with standard KS density functional calculations when the system geometry is similar to the equilibrium one or in repulsive configurations. For moderately long distances between atoms, not at dissociation, because of self-interaction the exponential ansatz may give instabilities in the form of large cluster amplitudes. To avoid these, this work introduces a relatively simple regularization method that preserves size-consistency and penalizes high amplitudes of the cluster operator, whereas the results remain physically meaningful. We also present the time-dependent extension of our theory and show that it can feature quantum states where multiple electrons are excited.

General Many-Body Framework for Data-Driven Potentials with Arbitrary Quantum Mechanical Accuracy: Water as a Case Study

We present a general framework for the development of data-driven many-body (MB) potential energy functions (MB-QM PEFs) that represent the interactions between small molecules at an arbitrary quantum-mechanical (QM) level of theory. As a demonstration, a family of MB-QM PEFs for water is rigorously derived from density functionals belonging to different rungs across Jacob's ladder of approximations within density functional theory (MB-DFT) and from Møller-Plesset perturbation theory (MB-MP2). Through a systematic analysis of individual MB contributions to the interaction energies of water clusters, we demonstrate that all MB-QM PEFs preserve the same accuracy as the corresponding ab initio calculations, with the exception of those derived from density functionals within the generalized gradient approximation (GGA). The differences between the DFT and MB-DFT results are traced back to density-driven errors that prevent GGA functionals from accurately representing the underlying molecular interactions for different cluster sizes and hydrogen-bonding arrangements. We show that this shortcoming may be overcome, within the MB formalism, by using density-corrected functionals (DC-DFT) that provide a more consistent representation of each individual MB contribution. This is demonstrated through the development of a MB-DFT PEF derived from DC-PBE-D3 data, which more accurately reproduce the corresponding ab initio results.

Hidden Hemibonding in the Aqueous Hydroxyl Radical

The existence of a two-center, three-electron hemibond in the first solvation shell of ^•OH(aq) has long been a matter of debate. The hemibond manifests in ab initio molecular dynamics simulations as a small-r feature in the oxygen radial distribution function (RDF) for H₂O···^•OH, but that feature disappears when semilocal density functionals are replaced with hybrids, suggesting a self-interaction artifact. Using periodic simulations at the PBE0+D3 level, we demonstrate that the hemibond is actually still present (as evidenced by delocalization of the spin density) but is obscured by the hydrogen-bonded feature in the RDF due to a slight elongation of the hemibond. Computed electronic spectra for ^•OH(aq) are in excellent agreement with experiment and confirm that hemibond-like configurations play an outsized role in the spectroscopy due to an intense charge-transfer transition that is strongly attenuated in hydrogen-bonded configurations. Apparently, 25% exact exchange (as in PBE0) is insufficient to eliminate delocalization of unpaired spins.

Selfconsistent random phase approximation methods

This Perspective reviews recent efforts toward selfconsistent calculations of ground-state energies within the random phase approximation (RPA) in the (generalized) Kohn-Sham (KS) density functional theory context. Since the RPA correlation energy explicitly depends on the non-interacting KS potential, an additional condition to determine the energy as a functional of the density is necessary. This observation leads to the concept of functional selfconsistency (FSC), which requires that the KS density equals the interacting density defined as the functional derivative of the ground-state energy with respect to the external potential. While all existing selfconsistent RPA schemes violate FSC, the recent generalized KS semicanonical projected RPA (GKS-spRPA) method takes a step toward satisfying it. This leads to systematic improvements in densities, binding energy curves, reference state stability, and molecular properties compared to non-selfconsistent RPA as well as optimized effective potential RPA. GKS-spRPA orbital energies accurately approximate valence and core ionization potentials, and even electron affinities of non-valence bound anions. The computational cost and performance of GKS-spRPA are compared to those of related selfconsistent schemes, including GW and orbital optimization methods, and limitations are discussed. Large differences between KS and interacting densities observed in the absence of FSC and the well-rounded performance of GKS-spRPA suggest that the KS potential as a density functional should be defined via the FSC condition for explicitly potential-dependent density functionals.

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.

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.

Learning to Approximate Density Functionals

Density functional theory (DFT) calculations are used in over 40,000 scientific papers each year, in chemistry, materials science, and far beyond. DFT is extremely useful because it is computationally much less expensive than ab initio electronic structure methods and allows systems of considerably larger size to be treated. However, the accuracy of any Kohn-Sham DFT calculation is limited by the approximation chosen for the exchange-correlation (XC) energy. For more than half a century, humans have developed the art of such approximations, using general principles, empirical data, or a combination of both, typically yielding useful results, but with errors well above the chemical accuracy limit (1 kcal/mol). Over the last 15 years, machine learning (ML) has made major breakthroughs in many applications and is now being applied to electronic structure calculations. This recent rise of ML begs the question: Can ML propose or improve density functional approximations? Success could greatly enhance the accuracy and usefulness of DFT calculations without increasing the cost.In this work, we detail efforts in this direction, beginning with an elementary proof of principle from 2012, namely, finding the kinetic energy of several Fermions in a box using kernel ridge regression. This is an example of orbital-free DFT, for which a successful general-purpose scheme could make even DFT calculations run much faster. We trace the development of that work to state-of-the-art molecular dynamics simulations of resorcinol with chemical accuracy. By training on ab initio examples, one bypasses the need to find the XC functional explicitly. We also discuss how the exchange-correlation energy itself can be modeled with such methods, especially for strongly correlated materials. Finally, we show how deep neural networks with differentiable programming can be used to construct accurate density functionals from very few data points by using the Kohn-Sham equations themselves as a regularizer. All these cases show that ML can create approximations of greater accuracy than humans, and is capable of finding approximations that can deal with difficult cases such as strong correlation. However, such ML-designed functionals have not been implemented in standard codes because of one last great challenge: generalization. We discuss how effortlessly human-designed functionals can be applied to a wide range of situations, and how difficult that is for ML.

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.

Subsystem density-functional theory for interacting open-shell systems: spin densities and magnetic exchange couplings

We investigate the possibility of describing interacting open-shell systems in high-spin and broken-symmetry (BS) states with subsystem density-functional theory (sDFT). This subsystem method typically starts from the electronic-structure results obtained for individual systems, for which the spin states can be individually defined. Through the confining effect of the embedding potential and/or the use of monomer basis sets, these individual spin states can be preserved in sDFT calculations. This offers the possibility of easy convergence to broken-symmetry states with arbitrary local spin patterns. We show that the resulting spin densities are in very good agreement with successfully converged broken-symmetry Kohn-Sham density-functional theory (KS-DFT) calculations. Yet sDFT can even cure those BS cases where KS-DFT suffers from convergence problems or convergence to undesired spin states. In contrast to KS-DFT, the sDFT-results only show a mild exchange-correlation functional dependence. We also show that magnetic coupling constants from sDFT are not satisfactory with standard approximations for the non-additive kinetic energy. When this component is evaluated "exactly", i.e. based on potential reconstruction, however, the magnetic coupling constants derived from spin-state energy differences are greatly improved. Hence, the interacting radicals studied here represent cases where even (semi-)local approximations for the non-additive kinetic-energy potential work well, while the parent energy functionals do not yield satisfactory results for spin-state energy differences.

Third-Order Møller-Plesset Theory Made More Useful? The Role of Density Functional Theory Orbitals

The practical utility of Møller-Plesset (MP) perturbation theory is severely constrained by the use of Hartree-Fock (HF) orbitals. It has recently been shown that the use of regularized orbital-optimized MP2 orbitals and scaling of MP3 energy could lead to a significant reduction in MP3 error [Bertels, L. W.; J. Phys. Chem. Lett. 2019, 10, 4170 4176]. In this work, we examine whether density functional theory (DFT)-optimized orbitals can be similarly employed to improve the performance of MP theory at both the MP2 and MP3 levels. We find that the use of DFT orbitals leads to significantly improved performance for prediction of thermochemistry, barrier heights, noncovalent interactions, and dipole moments relative to the standard HF-based MP theory. Indeed, MP3 (with or without scaling) with DFT orbitals is found to surpass the accuracy of coupled-cluster singles and doubles (CCSD) for several data sets. We also found that the results are not particularly functional sensitive in most cases (although range-separated hybrid functionals with low delocalization error perform the best). MP3 based on DFT orbitals thus appears to be an efficient, noniterative O(N⁶) scaling wave-function approach for single-reference electronic structure computations. Scaled MP2 with DFT orbitals is also found to be quite accurate in many cases, although modern double hybrid functionals are likely to be considerably more accurate.

Extending scaled-interaction adaptive-partitioning QM/MM to covalently bonded systems

Quantum mechanics/molecular mechanics (QM/MM) is the method of choice for atomistic simulations of large systems that can be partitioned into active and environmental regions. Adaptive-partitioning (AP) methods extend the applicability of QM/MM, allowing active regions to change during the simulation. AP methods achieve continuous potential energy surface (PES) by introducing buffer regions in which atoms have both QM and MM characters. Most of the existing AP-QM/MM methods require multiple QM calculations per time step, which can be expensive for systems with many atoms in buffer regions. Although one can lower the computational cost by grouping atoms into fragments, this may not be possible for all systems, especially for applications in covalent solids. The SISPA method [Field, J. Chem. Theory Comput., 2017, 13, 2342] differs from other AP-QM/MM methods by only requiring one QM calculation per time step, but it has the flaw that the QM charge density and wavefunction near the buffer/MM boundary tend to those of isolated atoms/fragments. Besides, regular QM/MM methods for treating covalent bonds cut by the QM/MM boundary are incompatible with SISPA. Due to these flaws, SISPA in its original form cannot treat covalently bonded systems properly. In this work, I show that a simple modification to the SISPA method improves the treatment of covalently bonded systems. I also study the effect of correcting the charge density in SISPA by developing a density-corrected pre-scaled algorithm. I demonstrate the methods with simple molecules and bulk solids.

Measuring Density-Driven Errors Using Kohn-Sham Inversion

Kohn-Sham (KS) inversion, that is, the finding of the exact KS potential for a given density, is difficult in localized basis sets. We study the precision and reliability of several inversion schemes, finding estimates of density-driven errors at a useful level of accuracy. In typical cases of substantial density-driven errors, Hartree-Fock density functional theory (HF-DFT) is almost as accurate as DFT evaluated on CCSD(T) densities. A simple approximation in practical HF-DFT also makes errors much smaller than the density-driven errors being calculated. Two paradigm examples, stretched NaCl and the HO·Cl^- radical, illustrate just how accurate HF-DFT is.

The Electron Affinity as the Highest Occupied Anion Orbital Energy with a Sufficiently Accurate Approximation of the Exact Kohn-Sham Potential

Negative ions are not accurately represented in density functional approximations (DFAs) such as (semi)local density functionals (LDA or GGA or meta-GGA). This is caused by the much too high orbital energies (not negative enough) with these DFAs compared to the exact Kohn–Sham values. Negative ions very often have positive DFA HOMO energies, hence they are unstable. These problems do not occur with the exact Kohn–Sham potential, the anion HOMO energy then being equal to minus the electron affinity. It is therefore desirable to develop sufficiently accurate approximations to the exact Kohn–Sham potential. There are further beneficial effects on the orbital shapes and the density of using a good approximation to the exact KS potential. Notably the unoccupied orbitals are not unduly diffuse, as they are in the Hartree–Fock model, with hybrid functionals, and even with (semi)local density functional approximations (LDFAs). We show that the recently developed B-GLLB-VWN approximation [Gritsenko et al. J. Chem. Phys.2016, 144, 204114] to the exact KS potential affords stable negative ions with HOMO orbital energy close to minus the electron affinity.

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

The self-interaction error (SIE), i.e. unphysical interactions of electrons with themselves, has plagued developers and users of Density Functional Approximations (DFAs) since the inception of Density Functional Theory (DFT). Formally, it can be separated into the one-electron and many-electron SIE; herein we present one of the most comprehensive studies of the first. While we focus mostly on the total SIE, we also make use of two different decompositions. The first is a separation into functional and density-driven errors as championed by Sim, Burke and co-workers [J. Phys. Chem. Lett., 2018, 9, 6385-6392]; the second separates the error into exchange, correlation, and one-electron components, with the latter being a density error that has not been discussed in this form before. After investigating the familiar hydrogen atom and dihydrogen cation, we establish a relationship between the SIE and the nuclear charge with the help of a series of heavier hydrogenic analogues. For the mononuclear systems and the diatomics at the dissociation limit, this relationship is linear in nature with prominent exceptions, mostly belonging to the Minnesota and range-separated (double-)hybrid DFAs. For the first time, we also show how the magnitude of the SIE depends on the underlying atomic-orbital basis set and how DFAs that rely on a popular van-der-Waals DFT type London-dispersion term exhibit "self-dispersion". We find that range separation is not a panacea for solving the one-electron SIE. DFAs that have been developed to be one-electron SIE free for one system, such as the hydrogen atom, show larger errors for heavier hydrogenic systems. Often, one-electron SIE-free DFAs rely on fortuitous error cancellation between their exchange and correlation components. An analysis of the most robust methods for general applications to date reveals that they suffer moderately from the one-electron SIE, while DFAs that are nearly SIE-free do not perform well in applications. Implicit in the continued existence of the one-electron SIE is that well-performing DFAs continue to suffer insufficiencies at their fundamental levels that are being compensated for by the SIE. Our analysis includes more than 250 000 datapoints, resulting in multiple insights that may drive future developments of new DFAs or SIE corrections.

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.

What is “many-body” dispersion and should I worry about it?

“Many-body” dispersion can refer to two distinct phenomena, here termed electronic and atomic many-body effects, both of which cause the dispersion energy to be non-additive.

Quantum Chemical Methods for Modeling Covalent Modification of Biological Thiols

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