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

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

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

Nuclear Quantum Effects on the Electronic Structure of Water and Ice

The electronic properties and optical response of ice and water are intricately shaped by their molecular structure, including the quantum mechanical nature of the hydrogen atoms. Despite numerous previous studies, a comprehensive understanding of the nuclear quantum effects (NQEs) on the electronic structure of water and ice at finite temperatures remains elusive. Here, we utilize molecular simulations that harness efficient machine-learning potentials and many-body perturbation theory to assess how NQEs impact the electronic bands of water and hexagonal ice. By comparing path-integral and classical simulations, we find that NQEs lead to a larger renormalization of the fundamental gap of ice, compared to that of water, ultimately yielding similar bandgaps in the two systems, consistent with experimental estimates. Our calculations suggest that the increased quantum mechanical delocalization of protons in ice, relative to water, is a key factor leading to the enhancement of NQEs on the electronic structure of ice.

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.

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.

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.

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.

Balance between Physical Interpretability and Energetic Predictability in Widely Used Dispersion-Corrected Density Functionals

We assess the performance of different dispersion models for several popular density functionals across a diverse set of noncovalent systems, ranging from the benzene dimer to molecular crystals. By analyzing the interaction energies and their individual components, we demonstrate that there exists variability across different systems for empirical dispersion models, which are calibrated for reproducing the interaction energies of specific systems. Thus, parameter fitting may undermine the underlying physics, as dispersion models rely on error compensation among the different components of the interaction energy. Energy decomposition analyses reveal that, the accuracy of revPBE-D3 for some aqueous systems originates from significant compensation between dispersion and charge transfer energies. However, revPBE-D3 is less accurate in describing systems where error compensation is incomplete, such as the benzene dimer. Such cases highlight the propensity for unpredictable behavior in various dispersion-corrected density functionals across a wide range of molecular systems, akin to the behavior of force fields. On the other hand, we find that SCAN-rVV10, a targeted-dispersion approach, affords significant reductions in errors associated with the lattice energies of molecular crystals, while it has limited accuracy in reproducing structural properties. Given the ubiquitous nature of noncovalent interactions and the key role of density functional theory in computational sciences, the future development of dispersion models should prioritize the faithful description of the dispersion energy, a shift that promises greater accuracy in capturing the underlying physics across diverse molecular and extended systems.

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.

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.

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.

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.

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.

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.