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

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.

Revisiting Artifacts of Kohn-Sham Density Functionals for Biosimulation

We revisit the problem of unphysical charge density delocalization/fractionalization induced by the self-interaction error of common approximate Kohn-Sham (KS) density functional theory functionals on simulation of small to medium-sized proteins in a vacuum. Aside from producing unphysical electron densities and total energies, the vanishing of the HOMO-LUMO gap associated with the unphysical charge delocalization leads to an unphysical low-energy spectrum and catastrophic failure of most popular solvers for the KS self-consistent field (SCF) problem. We apply a robust quasi-Newton SCF solver [ Phys. Chem. Chem. Phys. 2024, 26, 6557] to obtain solutions for some of these difficult cases. The anatomy of the charge delocalization is revealed by the natural deformation orbitals obtained from the density matrix difference between the Hartree-Fock and KS solutions; the charge delocalization not only can occur between charged fragments (such as in zwitterionic polypeptides) but also involves neutral fragments. The vanishing-gap phenomenon and troublesome SCF convergence are both attributed to the unphysical KS Fock operator eigenspectra of molecular fragments (e.g., amino acids or their side chains). Analysis of amino acid pairs suggests that the unphysical charge delocalization can be partially ameliorated by the use of some range-separated hybrid functionals but not by semilocal or standard hybrid functionals. Last, we demonstrate that solutions without the unphysical charge delocalization can be located even for semilocal KS functionals highly prone to such defects, but such solutions have non-Aufbau character and are unstable with respect to mixing of the non-overlapping "frontier" orbitals. Caution should be exercised when unexpectedly small (or vanishing) HOMO-LUMO gaps and atypical SCF convergence patterns (e.g., oscillatory) are observed in KS DFT simulations in any context (bio or otherwise).

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.

Evaluating Noncovalent Interactions in Halogenated Molecules with Double-Hybrid Functionals and a Dedicated Small Basis Set

We present here an extension of our recently developed PBE-QIDH/DH-SVPD basis set to halogen atoms, with the aim of obtaining, for weakly interacting halogenated molecules, interaction energies close to those provided by a large basis set (def2-TZVPP) coupled to empirical dispersion potential. The core of our approach is the split-valence basis set, DH-SVPD, that has been developed for F, Cl, Br, and I atoms using a self-consistent formula, containing only energy terms computed for dimers and the corresponding monomers at the same level of theory. The basis set developed considering four systems, one for each halogen atoms, has been then tested on the X40, X4 × 10 benchmarks as well as on other two, less standard, data sets. Finally, a large system (380 atoms) has been also considered as a "crash" test. Our results show that the simple and nonempirical PBE-QIDH/DH-SVPD approach is able to provide accurate results for interaction energies of all the considered systems and can thus be considered as a cheaper alternative to DH functionals paired with empirical dispersion corrections and a large basis set of triple-ζ quality.

Identifying and embedding transferability in data-driven representations of chemical space

Transferability, especially in the context of model generalization, is a paradigm of all scientific disciplines. However, the rapid advancement of machine learned model development threatens this paradigm, as it can be difficult to understand how transferability is embedded (or missed) in complex models developed using large training data sets. Two related open problems are how to identify, without relying on human intuition, what makes training data transferable; and how to embed transferability into training data. To solve both problems for ab initio chemical modelling, an indispensable tool in everyday chemistry research, we introduce a transferability assessment tool (TAT) and demonstrate it on a controllable data-driven model for developing density functional approximations (DFAs). We reveal that human intuition in the curation of training data introduces chemical biases that can hamper the transferability of data-driven DFAs. We use our TAT to motivate three transferability principles; one of which introduces the key concept of transferable diversity. Finally, we propose data curation strategies for general-purpose machine learning models in chemistry that identify and embed the transferability principles.

Herbal Products-Powered Thermosensitive Hydrogel with Phototherapy and Microenvironment Reconstruction for Accelerating Multidrug-Resistant Bacteria-Infected Wound Healing

Wound healing and infection remain significant challenges due to the ineffectiveness against multidrug-resistant (MDR) bacteria and the complex oxidative wound microenvironments. To address these issues, thymoquinone-reinforced injectable and thermosensitive TQ@PEG-PAF-Cur hydrogels with dual functions of microenvironment reshaping and photodynamic therapy are developed. The hydrogel comprises natural compound thymoquinone (TQ) and poly (ethylene glycol)-block-poly (alanine-co-phenyl alanine) copolymers (PEG-PAF) conjugated with natural photosensitizer curcumin (Cur). The incorporation of TQ and Cur reduces the sol-to-gel transition temperature of TQ@PEG-PAF-Cur to 30°C, compared to PEG-PAF hydrogel (37°C), due to the formation of strong hydrogen bonding, matching the wound microenvironment temperature. Under blue light excitation, TQ@PEG-PAF-Cur generates significant amounts of reactive oxygen species such as H2O2, 1O2, and ·OH, exhibiting rapid and efficient bactericidal capacities against methicillin-resistant Staphylococcus aureus and broad spectrum β-lactamases Escherichia coli via photodynamic therapy (PDT). Additionally, Cur effectively inhibits the expressions of proinflammatory cytokines in skin tissue-forming cells. As a result, the composite hydrogel can rapidly transform into a gel to cover the wound, reshape the wound microenvironment, and accelerate wound healing in vivo. This collaborative antibacterial strategy provides valuable insights to guide the development of multifunctional materials for efficient wound healing.

The Potential of Neural Network Potentials

In the next half-century, physical chemistry will likely undergo a profound transformation, driven predominantly by the combination of recent advances in quantum chemistry and machine learning (ML). Specifically, equivariant neural network potentials (NNPs) are a breakthrough new tool that are already enabling us to simulate systems at the molecular scale with unprecedented accuracy and speed, relying on nothing but fundamental physical laws. The continued development of this approach will realize Paul Dirac's 80-year-old vision of using quantum mechanics to unify physics with chemistry and providing invaluable tools for understanding materials science, biology, earth sciences, and beyond. The era of highly accurate and efficient first-principles molecular simulations will provide a wealth of training data that can be used to build automated computational methodologies, using tools such as diffusion models, for the design and optimization of systems at the molecular scale. Large language models (LLMs) will also evolve into increasingly indispensable tools for literature review, coding, idea generation, and scientific writing.

Deep Mind 21 functional does not extrapolate to transition metal chemistry

The development of density functional approximations stands at a crossroads: while machine-learned functionals show potential to surpass their human-designed counterparts, their extrapolation to unseen chemistry lags behind. Here we assess how well the recent Deep Mind 21 (DM21) machine-learned functional [Science, 2021, 374, 1385-1389], trained on main-group chemistry, extrapolates to transition metal chemistry (TMC). We show that DM21 demonstrates comparable or occasionally superior accuracy to B3LYP for TMC, but consistently struggles with achieving self-consistent field convergence for TMC molecules. We also compare main-group and TMC machine-learning DM21 features to shed light on DM21's challenges in TMC. We finally propose strategies to overcome limitations in the extrapolative capabilities of machine-learned functionals in TMC.

Many-body interactions and deep neural network potentials for water

We present a detailed assessment of deep neural network potentials developed within the Deep Potential Molecular Dynamics (DeePMD) framework and trained on the MB-pol data-driven many-body potential energy function. Specific focus is directed at the ability of DeePMD-based potentials to correctly reproduce the accuracy of MB-pol across various water systems. Analyses of bulk and interfacial properties as well as many-body interactions characteristic of water elucidate inherent limitations in the transferability and predictive accuracy of DeePMD-based potentials. These limitations can be traced back to an incomplete implementation of the "nearsightedness of electronic matter" principle, which may be common throughout machine learning potentials that do not include a proper representation of self-consistently determined long-range electric fields. These findings provide further support for the "short-blanket dilemma" faced by DeePMD-based potentials, highlighting the challenges in achieving a balance between computational efficiency and a rigorous, physics-based representation of the properties of water. Finally, we believe that our study contributes to the ongoing discourse on the development and application of machine learning models in simulating water systems, offering insights that could guide future improvements in the field.

Translational eigenstates of He@C60 from four-dimensional ab initio potential energy surfaces interpolated using Gaussian process regression

We investigate the endofullerene system 3He@C60 with a four-dimensional potential energy surface (PES) to include the three He translational degrees of freedom and C60 cage radius. We compare second order Møller-Plesset perturbation theory (MP2), spin component scaled-MP2, scaled opposite spin-MP2, random phase approximation (RPA)@Perdew, Burke, and Ernzerhof (PBE), and corrected Hartree-Fock-RPA to calibrate and gain confidence in the choice of electronic structure method. Due to the high cost of these calculations, the PES is interpolated using Gaussian Process Regression (GPR), owing to its effectiveness with sparse training data. The PES is split into a two-dimensional radial surface, to which corrections are applied to achieve an overall four-dimensional surface. The nuclear Hamiltonian is diagonalized to generate the in-cage translational/vibrational eigenstates. The degeneracy of the three-dimensional harmonic oscillator energies with principal quantum number n is lifted due to the anharmonicity in the radial potential. The (2l + 1)-fold degeneracy of the angular momentum states is also weakly lifted, due to the angular dependence in the potential. We calculate the fundamental frequency to range between 96 and 110 cm-1 depending on the electronic structure method used. Error bars of the eigenstate energies were calculated from the GPR and are on the order of ∼±1.5 cm-1. Wavefunctions are also compared by considering their overlap and Hellinger distance to the one-dimensional empirical potential. As with the energies, the two ab initio methods MP2 and RPA@PBE show the best agreement. While MP2 has better agreement than RPA@PBE, due to its higher computational efficiency and comparable performance, we recommend RPA as an alternative electronic structure method of choice to MP2 for these systems.

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.

Effect of ion pair strategy on transdermal delivery of guanfacine: Which factor dominates drug permeation?

Ion pair is an effective chemical approach to promoting drug transdermal permeation, and the traditional interpretation for its enhanced permeation effect is mainly attributed to counterions altering the physicochemical properties of the drug (lipophilicity, melting point, etc.). In this work, guanfacine (GFC), a non-stimulant for anti-attention deficit and hyperactivity disorder (ADHD), was used as a model drug, and several organic or inorganic acids were designed thereby successfully constructing ion pairs. The transdermal permeation ability of ion pairs through isolated porcine skin was observed and ranked as follows: guanfacine caprylate (GFC-CA) > GFC > guanfacine laurate (GFC-LA) > guanfacine fumarate (GFC-FA) > guanfacine hydrochloride (GFC-HA) > guanfacine palmitate (GFC-PA). The effect of key physicochemical properties (octanol-water partition coefficient, molecular volume, melting point) on the transdermal permeation rate of the model drug was analyzed in detail. In addition, GFC-CA was observed to alter the lipid structure of the skin, suggesting the traditional explanation of the action of ion pair may be inadequate and underrated, and ion pair may also enhance permeation by disrupting skin structure. The intriguing phenomenon is expected to provide a novel approach to achieving precise transdermal drug delivery.

Near-infrared light-activatable, analgesic nanocomposite delivery system for comprehensive therapy of diabetic wounds in rats

Impaired angiogenesis, bacterial infection, persistent severe pain, exacerbated inflammation, and oxidative stress injury are intractable problems in the treatment of chronic diabetic ulcer wounds. A strategy that effectively targets all these issues has proven challenging. Herein, an in-situ sprayable nanoparticle-gel composite comprising platinum clusters (Pt) loaded-mesoporous polydopamine (MPDA) nanoparticle and QX-314-loaded fibrin gel (Pt@MPDA/QX314@Fibrin) was developed for diabetic wound analgesia and therapy. The composite shows good local analgesic effect of QX-314 mediated by near-infrared light (NIR) activation of transient receptor potential vanilloid 1 (TRPV1) channel, as well as multifunctional therapeutic effects of rapid hemostasis, anti-inflammation, antioxidation, and antibacterial properties that benefit the fast-healing of diabetic wounds. Furthermore, it demonstrates that the composite, with good biodegradability and biosafety, significantly relieved wound pain by inhibiting the expression of c-Fos in the dorsal root ganglion and the activation of glial cells in the spinal cord dorsal horn. Consequently, our designed sprayable Pt@MPDA/QX314@Fibrin composite with good biocompatibility, NIR activation of TRPV1 channel-mediated QX-314 local wound analgesia and comprehensive treatments, is promising for chronic diabetic wound therapy.

Molecular Insights into the Influence of Ions on the Water Structure. I. Alkali Metal Ions in Solution

In this study, we explore the impact of alkali metal ions (Li+, Na+, K+, Rb+, and Cs+) on the hydration structure of water using molecular dynamics simulations carried out with MB-nrg potential energy functions (PEFs). Our analyses include radial distribution functions, coordination numbers, dipole moments, and infrared spectra of water molecules, calculated as a function of solvation shells. The results collectively indicate a highly local influence of all of the alkali metal ions on the hydrogen-bond network established by the surrounding water molecules, with the smallest and most densely charged Li+ ion exerting the most pronounced effect. Remarkably, the MB-nrg PEFs demonstrate excellent agreement with available experimental data for the position and size of the first solvation shells, underscoring their potential as predictive models for realistic simulations of ionic aqueous solutions across various thermodynamic conditions and environments.

Inferring energy-composition relationships with Bayesian optimization enhances exploration of inorganic materials

Computational exploration of the compositional spaces of materials can provide guidance for synthetic research and thus accelerate the discovery of novel materials. Most approaches employ high-throughput sampling and focus on reducing the time for energy evaluation for individual compositions, often at the cost of accuracy. Here, we present an alternative approach focusing on effective sampling of the compositional space. The learning algorithm PhaseBO optimizes the stoichiometry of the potential target material while improving the probability of and accelerating its discovery without compromising the accuracy of energy evaluation.

A step toward density benchmarking-The energy-relevant "mean field error"

Since the development of generalized gradient approximations in the 1990s, approximations based on density functional theory have dominated electronic structure theory calculations. Modern approximations can yield energy differences that are precise enough to be predictive in many instances, as validated by large- and small-scale benchmarking efforts. However, assessing the quality of densities has been the subject of far less attention, in part because reliable error measures are difficult to define. To this end, this work introduces the mean-field error, which directly assesses the quality of densities from approximations. The mean-field error is contextualized within existing frameworks of density functional error analysis and understanding and shown to be part of the density-driven error. It is demonstrated in several illustrative examples. Its potential use in future benchmarking protocols is discussed, and some conclusions are drawn.

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.

Regularized and Opposite Spin-Scaled Functionals from Møller-Plesset Adiabatic Connection─Higher Accuracy at Lower Cost

Noncovalent interactions (NCIs) play a crucial role in biology, chemistry, material science, and everything in between. To improve pure quantum-chemical simulations of NCIs, we propose a methodology for constructing approximate correlation energies by combining an interpolation along the Møller-Plesset adiabatic connection (MP AC) with a regularization and spin-scaling strategy applied to MP2 correlation energies. This combination yields cosκos-SPL2, which exhibits superior accuracy for NCIs compared to any of the individual strategies. With the N4 formal scaling, cosκos-SPL2 is competitive or often outperforms more expensive dispersion-corrected double hybrids for NCIs. The accuracy of cosκos-SPL2 particularly shines for anionic halogen bonded complexes, where it surpasses standard dispersion-corrected DFT by a factor of 3 to 5.

Nonlocal Functionals Inspired by the Strongly Interacting Limit of DFT: Exact Constraints and Implementation

Capturing strong correlation effects remains a key challenge for the development of improved exchange-correlation (XC) functionals in density functional theory. The recently proposed multiple radii functional (MRF) [J. Phys. Chem. Lett. 2017, 8, 2799; J. Chem. Theory Comput. 2019, 15, 3580] was designed to capture strong correlation effects seamlessly, as its mathematical structure draws from that of the exact XC functional in the limit of infinite correlations. The MRF functional provides a framework for building approximations along the density-fixed adiabatic connection, delivers accurate XC energy densities in the standard DFT gauge (same as that of the exact exchange energy density), and is free of one-electron self-interaction errors. To facilitate the development of XC functionals based on the MRF, we examine the behavior of the MRF functional when applied to uniform and scaled densities and consider how it can be made exact for the uniform electron gas. These theoretical insights are then used to build improved forms for the fluctuation function, an object that defines XC energy densities within the MRF framework. We also show how the MRF fluctuation function for physical correlation can be easily readjusted to accurately capture the XC functional in the limit of infinite correlations, demonstrating the versatility of MRF for building approximations for different correlation regimes. We describe the implementation of MRF using densities expanded on Gaussian basis sets, which improves the efficiency of previous grid-based MRF implementations.

Better performance of Hartree-Fock over DFT: a quantum mechanical investigation on pyridinium benzimidazolate types of zwitterions in the light of localization/delocalization issues

CONTEXT

With the advent of fast computing facilities, combined with rapid emerges of many new and intricate quantum mechanical functionals, computations with pure Hartree-Fock (HF) theory are now-a-days regarded as trivial or obsolete, or even considered as not reliable by many researchers. Consequently, current trends in computational chemistry show extensive use of post-HF theories for smaller molecular systems and various DFT methods for organic and inorganic chemistry related problems (larger molecules/systems). In this contribution, I have tried to show that sometimes, HF might be more suitable over DFT methodologies in addressing structure-property correlations. Molecules studied here were previously synthesized by Boyd in 1966 and important experimental data were produced by Alcalde and co-workers in 1987. Comparison of computed and experimental results clearly shows that HF method was more effective in reproducing the experimental data compared to especially the DFT methodologies. Reliability of HF method was further assured from the very similar results shown by the CCSD, CASSCF, CISD and QCISD methods. Current study also indicates that the localization issue associated with HF proved to be advantageous over delocalization issue of DFT based methodologies, in correctly describing the structure-property correlation for zwitterion systems.

METHODS

All computations were performed with Gaussian 09. A wide-range of quantum mechanical methodologies, HF, B3LYP, CAM-B3LYP, BMK, B3PW91, TPSSh, LC-ωPBE, M06-2X, M06-HF, ωB97xD, MP2, CASSCF, CCSD, QCISD, CISD and semi-empirical methods like, Huckel, CNDO, AM1, PM3MM and PM6, were used for investigations.

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.

Realistic phase diagram of water from "first principles" data-driven quantum simulations

Since the experimental characterization of the low-pressure region of water's phase diagram in the early 1900s, scientists have been on a quest to understand the thermodynamic stability of ice polymorphs on the molecular level. In this study, we demonstrate that combining the MB-pol data-driven many-body potential for water, which was rigorously derived from "first principles" and exhibits chemical accuracy, with advanced enhanced-sampling algorithms, which correctly describe the quantum nature of molecular motion and thermodynamic equilibria, enables computer simulations of water's phase diagram with an unprecedented level of realism. Besides providing fundamental insights into how enthalpic, entropic, and nuclear quantum effects shape the free-energy landscape of water, we demonstrate that recent progress in "first principles" data-driven simulations, which rigorously encode many-body molecular interactions, has opened the door to realistic computational studies of complex molecular systems, bridging the gap between experiments and simulations.

A "short blanket" dilemma for a state-of-the-art neural network potential for water: Reproducing experimental properties or the physics of the underlying many-body interactions?

Deep neural network (DNN) potentials have recently gained popularity in computer simulations of a wide range of molecular systems, from liquids to materials. In this study, we explore the possibility of combining the computational efficiency of the DeePMD framework and the demonstrated accuracy of the MB-pol data-driven, many-body potential to train a DNN potential for large-scale simulations of water across its phase diagram. We find that the DNN potential is able to reliably reproduce the MB-pol results for liquid water, but provides a less accurate description of the vapor-liquid equilibrium properties. This shortcoming is traced back to the inability of the DNN potential to correctly represent many-body interactions. An attempt to explicitly include information about many-body effects results in a new DNN potential that exhibits the opposite performance, being able to correctly reproduce the MB-pol vapor-liquid equilibrium properties, but losing accuracy in the description of the liquid properties. These results suggest that DeePMD-based DNN potentials are not able to correctly "learn" and, consequently, represent many-body interactions, which implies that DNN potentials may have limited ability to predict the properties for state points that are not explicitly included in the training process. The computational efficiency of the DeePMD framework can still be exploited to train DNN potentials on data-driven many-body potentials, which can thus enable large-scale, "chemically accurate" simulations of various molecular systems, with the caveat that the target state points must have been adequately sampled by the reference data-driven many-body potential in order to guarantee a faithful representation of the associated properties.

Toward routine Kohn-Sham inversion using the "Lieb-response" approach

Kohn-Sham (KS) inversion, in which the effective KS mean-field potential is found for a given density, provides insights into the nature of exact density functional theory (DFT) that can be exploited for the development of density functional approximations. Unfortunately, despite significant and sustained progress in both theory and software libraries, KS inversion remains rather difficult in practice, especially in finite basis sets. The present work presents a KS inversion method, dubbed the "Lieb-response" approach, that naturally works with existing Fock-matrix DFT infrastructure in finite basis sets, is numerically efficient, and directly provides meaningful matrix and energy quantities for pure-state and ensemble systems. Some additional work yields potential. It thus enables the routine inversion of even difficult KS systems, as illustrated in a variety of problems within this work, and provides outputs that can be used for embedding schemes or machine learning of density functional approximations. The effect of finite basis sets on KS inversion is also analyzed and investigated.

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.

Spin-State Splittings in 3d Transition-Metal Complexes Revisited: Benchmarking Approximate Methods for Adiabatic Spin-State Energy Differences in Fe(II) Complexes

The CASPT2+δMRCI composite approach reported in a companion paper has been extended and used to provide high-quality reference data for a series of adiabatic spin gaps (defined as ΔE = E_quintet - E_singlet) of [Fe^IIL₆]²⁺ complexes (L = CNH, CO, NCH, NH₃, H₂O), either at nonrelativistic level or including scalar relativistic effects. These highly accurate data have been used to evaluate the performance of various more approximate methods. Coupled-cluster theory with singles, doubles, and perturbative triples, CCSD(T), is found to agree well with the new reference data for Werner-type complexes but exhibits larger underestimates by up to 70 kJ/mol for the π-acceptor ligands, due to appreciable static correlation in the low-spin states of these systems. Widely used domain-based local CCSD(T) calculations, DLPNO-CCSD(T), are shown to depend very sensitively on the cutoff values used to construct the localized domains, and standard values are not sufficient. A large number of density functional approximations have been evaluated against the new reference data. The B2PLYP double hybrid gives the smallest deviations, but several functionals from different rungs of the usual ladder hierarchy give mean absolute deviations below 20 kJ/mol. This includes the B97-D semilocal functional, the PBE0* global hybrid with 15% exact-exchange admixture, as well as the local hybrids LH07s-SVWN and LH07t-SVWN. Several further functionals achieve mean absolute errors below 30 kJ/mol (M06L-D4, SSB-D, B97-1-D4, LC-ωPBE-D4, LH12ct-SsirPW92-D4, LH12ct-SsifPW92-D4, LH14t-calPBE-D4, LHJ-HFcal-D4, and several further double hybrids) and thereby also still overall outperform CCSD(T) or uncorrected CASPT2. While exact-exchange admixture is a crucial factor in favoring high-spin states, the present evaluations confirm that other aspects can be important as well. A number of the better-performing functionals underestimate the spin gaps for the π-acceptor ligands but overestimate them for L = NH₃, H₂O. In contrast to a previous suggestion, non-self-consistent density functional theory (DFT) computations on top of Hartree-Fock orbitals are not a promising path to produce accurate spin gaps in such complexes.

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.

Hydration Structure of Na+ and K+ Ions in Solution Predicted by Data-Driven Many-Body Potentials

The hydration structure of Na⁺ and K⁺ ions in solution is systematically investigated using a hierarchy of molecular models that progressively include more accurate representations of many-body interactions. We found that a conventional empirical pairwise additive force field that is commonly used in biomolecular simulations is unable to reproduce the extended X-ray absorption fine structure (EXAFS) spectra for both ions. In contrast, progressive inclusion of many-body effects rigorously derived from the many-body expansion of the energy allows the MB-nrg potential energy functions (PEFs) to achieve nearly quantitative agreement with the experimental EXAFS spectra, thus enabling the development of a molecular-level picture of the hydration structure of both Na⁺ and K⁺ in solution. Since the MB-nrg PEFs have already been shown to accurately describe isomeric equilibria and vibrational spectra of small ion-water clusters in the gas phase, the present study demonstrates that the MB-nrg PEFs effectively represent the long-sought-after models able to correctly predict the properties of ionic aqueous systems from the gas to the liquid phase, which has so far remained elusive.

Promoted Activity of Surface Hydroxyls on γ-Al2O3 Mineral Dust with the Coexistence of SO2 and NH3

Sulfate and ammonium formed on mineral dust can be mutually accelerated through the heterogeneous reactions of coexisting SO₂ and NH₃. However, little is known about the underlying mechanism, especially the pivotal reactive sites. Using combined Born-Oppenheimer molecular dynamics simulations and density functional theory calculations, the results show that, compared to that of SO₂ or NH₃ alone on the γ-Al₂O₃ surface, the increased level of formation of sulfate and ammonium can be attributed to the promoted activity of the surface-bridged hydroxyl with the coexistence of SO₂ and NH₃. In the specific mechanism, the O and H of the surface-bridged hydroxyl group are attacked by the adjacent SO₂ and NH₃, respectively, which directly enhances the formation of absorbed sulfite and ammonium, and indirectly facilitates the production of sulfate by oxidation of atmospheric O₂. The proposed mechanisms can be broadly applied to other aluminum-based suspended dust particles, such as kaolinite, montmorillonite, and clay dust.

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.

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.

Data-Driven Many-Body Potential Energy Functions for Generic Molecules: Linear Alkanes as a Proof-of-Concept Application

We present a generalization of the many-body energy (MB-nrg) theoretical/computational framework that enables the development of data-driven potential energy functions (PEFs) for generic covalently bonded molecules, with arbitrary quantum mechanical accuracy. The "nearsightedness of electronic matter" is exploited to define monomers as "natural building blocks" on the basis of their distinct chemical identity. The energy of generic molecules is then expressed as a sum of individual many-body energies of incrementally larger subsystems. The MB-nrg PEFs represent the low-order n-body energies, with n = 1-4, using permutationally invariant polynomials derived from electronic structure data carried out at an arbitrary quantum mechanical level of theory, while all higher-order n-body terms (n > 4) are represented by a classical many-body polarization term. As a proof-of-concept application of the general MB-nrg framework, we present MB-nrg PEFs for linear alkanes. The MB-nrg PEFs are shown to accurately reproduce reference energies, harmonic frequencies, and potential energy scans of alkanes, independently of their length. Since, by construction, the MB-nrg framework introduced here can be applied to generic covalently bonded molecules, we envision future computer simulations of complex molecular systems using data-driven MB-nrg PEFs, with arbitrary quantum mechanical accuracy.

Self-Consistent Implementation of Kohn-Sham Adiabatic Connection Models with Improved Treatment of the Strong-Interaction Limit

Adiabatic connection models (ACMs), which interpolate between the limits of weak and strong interaction, are powerful tools to build accurate exchange–correlation functionals. If the exact weak-interaction expansion from the second-order perturbation theory is included, a self-consistent implementation of these functionals is challenging and still absent in the literature. In this work, we fill this gap by presenting a fully self-consistent-field (SCF) implementation of some popular ACM functionals. While using second-order perturbation theory at weak interactions, we have also introduced new generalized gradient approximations (GGAs), beyond the usual point-charge-plus-continuum model, for the first two leading terms at strong interactions, which are crucial to ensure robustness and reliability. We then assess the SCF–ACM functionals for molecular systems and for prototypical strong-correlation problems. We find that they perform well for both the total energy and the electronic density and that the impact of SCF orbitals is directly connected to the accuracy of the ACM functional form. For the H₂ dissociation, the SCF–ACM functionals yield significant improvements with respect to standard functionals also thanks to the use of the new GGAs for the strong-coupling functionals.

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.