Toward chemical accuracy in the description of ion–water interactions through many-body representations. Alkali-water dimer potential energy surfaces

Current Status of the MB-pol Data-Driven Many-Body Potential for Predictive Simulations of Water Across Different Phases

Developing a molecular-level understanding of the properties of water is central to numerous scientific and technological applications. However, accurately modeling water through computer simulations has been a significant challenge due to the complex nature of the hydrogen-bonding network that water molecules form under different thermodynamic conditions. This complexity has led to over five decades of research and many modeling attempts. The introduction of the MB-pol data-driven many-body potential energy function marked a significant advancement toward a universal molecular model capable of predicting the structural, thermodynamic, dynamical, and spectroscopic properties of water across all phases. By integrating physics-based and data-driven (i.e., machine-learned) components, which correctly capture the delicate balance among different many-body interactions, MB-pol achieves chemical and spectroscopic accuracy, enabling realistic molecular simulations of water, from gas-phase clusters to liquid water and ice. In this review, we present a comprehensive overview of the data-driven many-body formalism adopted by MB-pol, highlight the main results and predictions made from computer simulations with MB-pol to date, and discuss the prospects for future extensions to data-driven many-body potentials of generic and reactive molecular systems.

Entropy of Liquid Water as Predicted by the Two-Phase Thermodynamic Model and Data-Driven Many-Body Potentials

We investigate the entropy of liquid water at ambient conditions using the two-phase thermodynamic (2PT) model, applied to both common pairwise-additive water models and the MB-pol and MB-pol(2023) data-driven many-body potentials. Our simulations demonstrate that the 2PT model yields entropy values in semiquantitative agreement with experimental data when using MB-pol and MB-pol(2023). Additionally, our analyses indicate that the entropy values predicted by pairwise-additive water models may benefit from error compensation between the inherent approximations of the 2PT model and the known limitations of these water models in describing many-body interactions. Despite its approximate nature, the simplicity of the 2PT model makes it a valuable tool for estimating relative entropy changes of liquid water across various environments, especially when combined with water models that provide a consistently robust representation of the structural, thermodynamic, and dynamical properties of liquid water.

To Pair or not to Pair? Machine-Learned Explicitly-Correlated Electronic Structure for NaCl in Water

The extent of ion pairing in solution is an important phenomenon to rationalize transport and thermodynamic properties of electrolytes. A fundamental measure of this pairing is the potential of mean force (PMF) between solvated ions. The relative stabilities of the paired and solvent shared states in the PMF and the barrier between them are highly sensitive to the underlying potential energy surface. However, direct application of accurate electronic structure methods is challenging, since long simulations are required. We develop wave function based machine learning potentials with the random phase approximation (RPA) and second order Møller-Plesset (MP2) perturbation theory for the prototypical system of Na and Cl ions in water. We show both methods in agreement, predicting the paired and solvent shared states to have similar energies (within 0.2 kcal/mol). We also provide the same benchmarks for different DFT functionals as well as insight into the PMF based on simple analyses of the interactions in the system.

Molecular Insights into Chemical Reactions at Aqueous Aerosol Interfaces

Atmospheric aerosols facilitate reactions between ambient gases and dissolved species. Here, we review our efforts to interrogate the uptake of these gases and the mechanisms of their reactions both theoretically and experimentally. We highlight the fascinating behavior of N2O5 in solutions ranging from pure water to complex mixtures, chosen because its aerosol-mediated reactions significantly impact global ozone, hydroxyl, and methane concentrations. As a hydrophobic, weakly soluble, and highly reactive species, N2O5 is a sensitive probe of the chemical and physical properties of aerosol interfaces. We employ contemporary theory to disentangle the fate of N2O5 as it approaches pure and salty water, starting with adsorption and ending with hydrolysis to HNO3, chlorination to ClNO2, or evaporation. Flow reactor and gas-liquid scattering experiments probe even greater complexity as added ions, organic molecules, and surfactants alter the interfacial composition and reaction rates. Together, we reveal a new perspective on multiphase chemistry in the atmosphere.

A Many-Body Perspective of Nuclear Quantum Effects in Aqueous Clusters

Nuclear quantum effects play an important role in the structure and thermodynamics of aqueous systems. By performing a many-body expansion with nuclear-electronic orbital (NEO) theory, we show that proton quantization can give rise to significant energetic contributions for many-body interactions spanning several molecules in single-point energy calculations of water clusters. Although zero-point motion produces a large increase in energy at the one-body level, nuclear quantum effects serve to stabilize higher-order molecular interactions. These results are significant because they demonstrate that nuclear quantum effects play a nontrivial role in many-body interactions of aqueous systems. Our approach also provides a pathway for incorporating nuclear quantum effects into water potential energy surfaces. The NEO approach is advantageous for many-body expansion analyses because it includes nuclear quantum effects directly in the energies.

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 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.

MBX: A many-body energy and force calculator for data-driven many-body simulations

Many-Body eXpansion (MBX) is a C++ library that implements many-body potential energy functions (PEFs) within the "many-body energy" (MB-nrg) formalism. MB-nrg PEFs integrate an underlying polarizable model with explicit machine-learned representations of many-body interactions to achieve chemical accuracy from the gas to the condensed phases. MBX can be employed either as a stand-alone package or as an energy/force engine that can be integrated with generic software for molecular dynamics and Monte Carlo simulations. MBX is parallelized internally using Open Multi-Processing and can utilize Message Passing Interface when available in interfaced molecular simulation software. MBX enables classical and quantum molecular simulations with MB-nrg PEFs, as well as hybrid simulations that combine conventional force fields and MB-nrg PEFs, for diverse systems ranging from small gas-phase clusters to aqueous solutions and molecular fluids to biomolecular systems and metal-organic frameworks.

Many-Body Effects in Aqueous Systems: Synergies Between Interaction Analysis Techniques and Force Field Development

Interaction analysis techniques, including the many-body expansion (MBE), symmetry-adapted perturbation theory, and energy decomposition analysis, allow for an intuitive understanding of complex molecular interactions. We review these methods by first providing a historical context for the study of many-body interactions and discussing how nonadditivities emerge from Hamiltonians containing strictly pairwise-additive interactions. We then elaborate on the synergy between these interaction analysis techniques and the development of advanced force fields aimed at accurately reproducing the Born-Oppenheimer potential energy surface. In particular, we focus on ab initio-based force fields that aim to explicitly reproduce many-body terms and are fitted to high-level electronic structure results. These force fields generally incorporate many-body effects through (a) parameterization of distributed multipoles, (b) explicit fitting of the MBE, (c) inclusion of many-atom features in a neural network, and (d) coarse-graining of many-body terms into an effective two-body term. We also discuss the emerging use of the MBE to improve the accuracy and speed of ab initio molecular dynamics.

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.

A scheme for rapid evaluation of the intermolecular three-body polarization effect in water clusters

The ability to accurately and rapidly evaluate the intermolecular many-body polarization effect of the water system is very important for computer simulations of biomolecule in aqueous. In this paper, a scheme is proposed based on the polarizable dipole-dipole interaction model and used to rapidly estimate the intermolecular many-body polarization effect in water clusters. We use a bond-dipole-based polarization function to evaluate the polarization energy. We regard two OH bonds of a water molecule as two bond-dipoles and set the permanent OH bond-dipole moment of a water molecule to be 1.51 Debye. We estimate the induced OH bond-dipole moment via a simple formula in which only one correction factor is needed. This scheme is then applied to tens of water clusters to calculate the three- and four-body interaction energies. The three-body interaction energies of 93 water clusters produced by our scheme are compared with those produced by the counterpoise-corrected CCSD(T)/aug-cc-pVDZ, MP2/aug-cc-pVDZ, M06-2X/jul-cc-pVTZ methods, by the AMOEBApro13, iAMOEBA, AMOEBA+, AMOEBA+(CF) methods, and by the MB-pol method. The four-body interaction energies of 47 water clusters yielded by our scheme are compared with those yielded by the counterpoise-corrected MP2/aug-cc-pVDZ and M06-2X/ jul-cc-pVTZ methods, by the AMOEBApro13, AMOEBA+, AMOEBA+(CF) methods, and by the MB-pol method. The comparison results show that the scheme proposed in this paper can reproduce the counterpoise-corrected CCSD(T)/aug-cc-pVDZ three-body interaction energies and reproduce the counterpoise-corrected MP2/aug-cc-pVDZ four-body interaction energies both accurately and efficiently. We anticipate the scheme proposed here can be useful for computer simulations of liquid water and aqueous 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.

Accurate Modeling of Bromide and Iodide Hydration with Data-Driven Many-Body Potentials

Ion-water interactions play a central role in determining the properties of aqueous systems in a wide range of environments. However, a quantitative understanding of how the hydration properties of ions evolve from small aqueous clusters to bulk solutions and interfaces remains elusive. Here, we introduce the second generation of data-driven many-body energy (MB-nrg) potential energy functions (PEFs) representing bromide-water and iodide-water interactions. The MB-nrg PEFs use permutationally invariant polynomials to reproduce two-body and three-body energies calculated at the coupled cluster level of theory, and implicitly represent all higher-body energies using classical many-body polarization. A systematic analysis of the hydration structure of small Br^-(H₂O)_n and I^-(H₂O)_n clusters demonstrates that the MB-nrg PEFs predict interaction energies in quantitative agreement with "gold standard" coupled cluster reference values. Importantly, when used in molecular dynamics simulations carried out in the isothermal-isobaric ensemble for single bromide and iodide ions in liquid water, the MB-nrg PEFs predict extended X-ray absorption fine structure (EXAFS) spectra that accurately reproduce the experimental spectra, which thus allows for characterizing the hydration structure of the two ions with a high level of confidence.

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.

Toward High-level Machine Learning Potential for Water Based on Quantum Fragmentation and Neural Networks

Accurate and efficient simulation of liquids, such as water and salt solutions, using high-level wave function theories is still a formidable task for computational chemists owing to the high computational costs. In this study, we develop a deep machine learning potential based on fragment-based second-order Møller-Plesset perturbation theory (DP-MP2) for water through neural networks. We show that the DP-MP2 potential predicts the structural, dynamical, and thermodynamic properties of liquid water in better agreement with the experimental data than previous studies based on density functional theory (DFT). The nuclear quantum effects (NQEs) on the properties of liquid water are also examined, which are noticeable in affecting the structural and dynamical properties of liquid water under ambient conditions. This work provides a general framework for quantitative predictions of the properties of condensed-phase systems with the accuracy of high-level wave function theory while achieving significant computational savings compared to ab initio simulations.

The behavior of methane-water mixtures under elevated pressures from simulations using many-body potentials

Non-polarizable empirical potentials have been proven to be incapable of capturing the mixing of methane-water mixtures at elevated pressures. Although density functional theory-based ab initio simulations may circumvent this discrepancy, they are limited in terms of the relevant time and length scales associated with mixing phenomena. Here, we show that the many-body MB-nrg potential, designed to reproduce methane-water interactions with coupled cluster accuracy, successfully captures this phenomenon up to 3 GPa and 500 K with varying methane concentrations. Two-phase simulations and long time scales that are required to fully capture the mixing, affordable due to the speed and accuracy of the MBX software, are assessed. Constructing the methane-water equation of state across the phase diagram shows that the stable mixtures are denser than the sum of their parts at a given pressure and temperature. We find that many-body polarization plays a central role, enhancing the induced dipole moments of methane by 0.20 D during mixing under pressure. Overall, the mixed system adopts a denser state, which involves a significant enthalpic driving force as elucidated by a systematic many-body energy decomposition analysis.

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.

Density Functional Theory of Water with the Machine-Learned DM21 Functional

The delicate interplay between functional-driven and density-driven errors in density functional theory (DFT) has hindered traditional density functional approximations (DFAs) from providing an accurate description of water for over 30 years. Recently, the deep-learned DeepMind 21 (DM21) functional has been shown to overcome the limitations of traditional DFAs as it is free of delocalization error. To determine if DM21 can enable a molecular-level description of the physical properties of aqueous systems within Kohn-Sham DFT, we assess the accuracy of the DM21 functional for neutral, protonated, and deprotonated water clusters. We find that the ability of DM21 to accurately predict the energetics of aqueous clusters varies significantly with cluster size. Additionally, we introduce the many-body MB-DM21 potential derived from DM21 data within the many-body expansion of the energy and use it in simulations of liquid water as a function of temperature at ambient pressure. We find that size-dependent functional-driven errors identified in the analysis of the energetics of small clusters calculated with the DM21 functional result in the MB-DM21 potential systematically overestimating the hydrogen-bond strength and, consequently, predicting a more ice-like local structure of water at room temperature.

Uptake of N2O5 by aqueous aerosol unveiled using chemically accurate many-body potentials

The reactive uptake of N2O5 to aqueous aerosol is a major loss channel for nitrogen oxides in the troposphere. Despite its importance, a quantitative picture of the uptake mechanism is missing. Here we use molecular dynamics simulations with a data-driven many-body model of coupled-cluster accuracy to quantify thermodynamics and kinetics of solvation and adsorption of N2O5 in water. The free energy profile highlights that N2O5 is selectively adsorbed to the liquid-vapor interface and weakly solvated. Accommodation into bulk water occurs slowly, competing with evaporation upon adsorption from gas phase. Leveraging the quantitative accuracy of the model, we parameterize and solve a reaction-diffusion equation to determine hydrolysis rates consistent with experimental observations. We find a short reaction-diffusion length, indicating that the uptake is dominated by interfacial features. The parameters deduced here, including solubility, accommodation coefficient, and hydrolysis rate, afford a foundation for which to consider the reactive loss of N2O5 in more complex solutions.

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.

MB-Fit: Software infrastructure for data-driven many-body potential energy functions

Many-body potential energy functions (MB-PEFs), which integrate data-driven representations of many-body short-range quantum mechanical interactions with physics-based representations of many-body polarization and long-range interactions, have recently been shown to provide high accuracy in the description of molecular interactions from the gas to the condensed phase. Here, we present MB-Fit, a software infrastructure for the automated development of MB-PEFs for generic molecules within the TTM-nrg (Thole-type model energy) and MB-nrg (many-body energy) theoretical frameworks. Besides providing all the necessary computational tools for generating TTM-nrg and MB-nrg PEFs, MB-Fit provides a seamless interface with the MBX software, a many-body energy and force calculator for computer simulations. Given the demonstrated accuracy of the MB-PEFs, particularly within the MB-nrg framework, we believe that MB-Fit will enable routine predictive computer simulations of generic (small) molecules in the gas, liquid, and solid phases, including, but not limited to, the modeling of quantum isomeric equilibria in molecular clusters, solvation processes, molecular crystals, and phase diagrams.

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.

Data-driven many-body models enable a quantitative description of chloride hydration from clusters to bulk

We present a new data-driven potential energy function (PEF) describing chloride-water interactions, which is developed within the many-body-energy (MB-nrg) theoretical framework. Besides quantitatively reproducing low-order many-body energy contributions, the new MB-nrg PEF is able to correctly predict the interaction energies of small chloride-water clusters calculated at the coupled cluster level of theory. Importantly, classical and quantum molecular dynamics simulations of a single chloride ion in water demonstrate that the new MB-nrg PEF predicts x-ray spectra in close agreement with the experimental results. Comparisons with an popular empirical model and a polarizable PEF emphasize the importance of an accurate representation of short-range many-body effect while demonstrating that pairwise additive representations of chloride-water and water-water interactions are inadequate for correctly representing the hydration structure of chloride in both gas-phase clusters and solution. We believe that the analyses presented in this study provide additional evidence for the accuracy and predictive ability of the MB-nrg PEFs, which can then enable more realistic simulations of ionic aqueous systems in different environments.

Structures of ions accommodated in salty ice Ih crystals

Frozen aqueous electrolytes are ubiquitous and involved in various phenomena occurring in the natural environment. Although salts are expelled from ice during freezing of aqueous solutions, minor amounts of the constituent ions are accommodated in the crystal lattice of ice. This phenomenon was associated with the generation of the Workman-Reynolds freezing potential. Molecular simulations also confirmed the ion incorporation in the crystal lattice of ice Ih upon freezing of aqueous electrolytes and identified possible local structures of the ions. However, no experimental information is available on the structure of ions accommodated in the crystal lattice of ice Ih. In this work, we use X-ray absorption fine structure (XAFS) to study the local structures of K⁺ and Cl^- accommodated in ice Ih single crystals. Previous molecular simulations predicted that ions are trapped in the hexagonal cavities of the ice structure or replace two water molecules in the crystal lattice. Four possible configurations are considered and optimized by the calculations using ONIOM (QM/QM/QM). The results are evaluated in terms of the agreement between the experimental XAFS spectra and those simulated from the optimized structures. The spectra are most reasonably interpreted by assuming that K⁺ replaces one water molecule in the ice crystal lattice and is accommodated in a tetrahedral coordination cage. Similarly, Cl^- probably adopts the same configuration, because it explains the coordination number better than other structures, such as that assuming the replacement of two water molecules belonging to the same hexagonal planes.

Toward a First-Principles Framework for Predicting Collective Properties of Electrolytes

Given the universal importance of electrolyte solutions, it is natural to expect that we have a nearly complete understanding of the fundamental properties of these solutions (e.g., the chemical potential) and that we can therefore explain, predict, and control the phenomena occurring in them. In fact, reality falls short of these expectations. But, recent advances in the simulation and modeling of electrolyte solutions indicate that it should soon be possible to make progress toward these goals. In this Account, we will discuss the use of first-principles interaction potentials based in quantum mechanics (QM) to enhance our understanding of electrolyte solutions. Specifically, we will focus on the use of quantum density functional theory (DFT) combined with molecular dynamics simulation (DFT-MD) as the foundation for our approach. The overarching concept is to understand and accurately reproduce the balance between local or short-ranged (SR) structural details and long-range (LR) correlations, allowing the prediction of the thermodynamics of both single ions in solution as well as the collective interactions characterized by activity/osmotic coefficients. In doing so, relevant collective motions and driving forces characterized by chemical potentials can be determined.In this Account, we will make the case that understanding electrolyte solutions requires a faithful QM representation of the SR nature of the ion-ion, ion-water, and water-water interactions. However, the number of molecules that is required for collective behavior makes the direct application of high-level QM methods that contain the best SR physics untenable, making methods that balance accuracy and efficiency a practical goal. Alternatives such as continuum solvent models (CSMs) and empirically based classical molecular dynamics have been extensively employed to resolve this problem but without yet overcoming the fundamental issue of SR accuracy. We will demonstrate that accurately describing the SR interaction is imperative for predicting both intrinsic properties, namely, at infinite dilution, and collective properties of electrolyte solutions.DFT has played an important role in our understanding of condensed phase systems, e.g., bulk liquid water, the air-water interface, ions in bulk, and at the air-water interface. This approach holds huge promise to provide benchmark calculations of electrolyte solution properties that will allow for the development and improvement of more efficient methods, as well as an enhanced understanding of fundamental phenomena. However, the standard protocol using the generalized gradient approximation with van der Waals (vdW) correction requires improvement in order to achieve a high level of quantitative accuracy. Simply simulating with higher level DFT functionals may not be the best route considering the significant computational cost. Alternative methods of incorporating information from higher levels of QM should be explored; e.g., using force matching techniques on small clusters, where high level benchmark calculations are possible, to develop ideal correction terms to the DFT functional is a promising possibility. We argue that DFT with statistical mechanics is becoming an increasingly useful framework enabling the prediction of collective electrolyte properties.

Highly Accurate Many-Body Potentials for Simulations of N2O5 in Water: Benchmarks, Development, and Validation

Dinitrogen pentoxide (N₂O₅) is an important intermediate in the atmospheric chemistry of nitrogen oxides. Although there has been much research, the processes that govern the physical interactions between N₂O₅ and water are still not fully understood at a molecular level. Gaining a quantitative insight from computer simulations requires going beyond the accuracy of classical force fields while accessing length scales and time scales that are out of reach for high-level quantum-chemical approaches. To this end, we present the development of MB-nrg many-body potential energy functions for nonreactive simulations of N₂O₅ in water. This MB-nrg model is based on electronic structure calculations at the coupled cluster level of theory and is compatible with the successful MB-pol model for water. It provides a physically correct description of long-range many-body interactions in combination with an explicit representation of up to three-body short-range interactions in terms of multidimensional permutationally invariant polynomials. In order to further investigate the importance of the underlying interactions in the model, a TTM-nrg model was also devised. TTM-nrg is a more simplistic representation that contains only two-body short-range interactions represented through Born-Mayer functions. In this work, an active learning approach was employed to efficiently build representative training sets of monomer, dimer, and trimer structures, and benchmarks are presented to determine the accuracy of our new models in comparison to a range of density functional theory methods. By assessing the binding curves, distortion energies of N₂O₅, and interaction energies in clusters of N₂O₅ and water, we evaluate the importance of two-body and three-body short-range potentials. The results demonstrate that our MB-nrg model has high accuracy with respect to the coupled cluster reference, outperforms current density functional theory models, and thus enables highly accurate simulations of N₂O₅ in aqueous environments.

Support Vector Regression-Based Monte Carlo Simulation of Flexible Water Clusters

Molecular simulations based on classical force fields are computationally efficient but lack accuracy due to the empirical formulation of non-bonded interactions. Quantum mechanical (QM) methods, albeit accurate, have inhibitory computational costs for large molecules and clusters. Hence, to overcome the bottleneck, machine learning (ML)-based methods have been employed in the recent years. We had earlier reported a combined scheme of many-body expansion (MBE) and ML to predict the interaction energies of rigid water clusters. In this work, we proceed toward building a flexible water model using the ML-MBE scheme. This ML-MBE scheme has an error of <1% for interaction energy prediction in comparison to the parent QM method for flexible water decamers. Machine learning-based Monte Carlo simulations (MLMC) are performed with this water model, and the structural properties of these configurations are compared with those obtained from ab initio molecular dynamics (AIMD) and the TIP3P classical force field. The radial distribution functions, tetrahedral order parameters, and number of hydrogen bonds in AIMD and MLMC have a similar qualitative and quantitative trend, whereas the classical force fields show a significant deviation.

Quantifying the hydration structure of sodium and potassium ions: taking additional steps on Jacob's Ladder

The ability to reproduce the experimental structure of water around the sodium and potassium ions is a key test of the quality of interaction potentials due to the central importance of these ions in a wide range of important phenomena.

From ab initio data to high-dimensional potential energy surfaces: A critical overview and assessment of the development of permutationally invariant polynomial potential energy surfaces for single molecules

The representation of high-dimensional potential energy surfaces by way of the many-body expansion and permutationally invariant polynomials has become a well-established tool for improving the resolution and extending the scope of molecular simulations. The high level of accuracy that can be attained by these potential energy functions (PEFs) is due in large part to their specificity: for each term in the many-body expansion, a species-specific training set must be generated at the desired level of theory and a number of fits attempted in order to obtain a robust and reliable PEF. In this work, we attempt to characterize the numerical aspects of the fitting problem, addressing questions which are of simultaneous practical and fundamental importance. These include concrete illustrations of the nonconvexity of the problem, the ill-conditionedness of the linear system to be solved and possible need for regularization, the sensitivity of the solutions to the characteristics of the training set, and limitations of the approach with respect to accuracy and the types of molecules that can be treated. In addition, we introduce a general approach to the generation of training set configurations based on the familiar harmonic approximation and evaluate the possible benefits to the use of quasirandom sequences for sampling configuration space in this context. Using sulfate as a case study, the findings are largely generalizable and expected to ultimately facilitate the efficient development of PIP-based many-body PEFs for general systems via automation.

Improving the description of solvent pairwise interactions using local solute/solvent three-body functions. The case of halides and carboxylates in aqueous environment

We propose a general strategy to remediate force-field artifacts in describing pairwise interactions among similar molecules M in the vicinity of another chemical species, C, like water molecules interacting at short distance from a monoatomic ion. This strategy is based on introducing a three-body potential energy term that alters the pairwise interactions among M-type molecules when they lie at short range from the species C. In other words the species C is the center of a space domain where the pairwise interactions among the molecules M is altered. Here, we apply it to improve the description of the water interactions provided by the polarizable water model TCPE/2013 in the vicinity of halides, from F^- to At^- , and of the prototypical carboxylate anion CH₃ COO^- . We show the accuracy and the transferability of such an approach to investigate not only the hydration process of single anions but also of a salt solution NH 4 + / Cl - in aqueous phase. This strategy can be used to remediate the drawbacks of any kind of force fields. © 2019 Wiley Periodicals, Inc.

Many-Body Effects Determine the Local Hydration Structure of Cs+ in Solution

A systematic analysis of the hydration structure of Cs⁺ ions in solution is derived from simulations carried out using a series of molecular models built upon a hierarchy of approximate representations of many-body effects in ion-water interactions. It is found that a pairwise-additive model, commonly used in biomolecular simulations, provides poor agreement with experimental X-ray spectra, indicating an incorrect description of the underlying hydration structure. Although the agreement with experiment improves in simulations with a polarizable model, the predicted hydration structure is found to lack the correct sequence of water shells. Progressive inclusion of explicit many-body effects in the representation of Cs⁺-water interactions as well as accounting for nuclear quantum effects is shown to be necessary for quantitatively reproducing the experimental X-ray spectra. Besides emphasizing the importance of many-body effects, these results suggest that molecular models rigorously derived from many-body expansions hold promise for realistic simulations of aqueous solutions.

Machine learning prediction of interaction energies in rigid water clusters

Classical force fields form a computationally efficient avenue for calculating the energetics of large systems. However, due to the constraints of the underlying analytical form, it is sometimes not accurate enough. Quantum mechanical (QM) methods, although accurate, are computationally prohibitive for large systems. In order to circumvent the bottle-neck of interaction energy estimation of large systems, data driven approaches based on machine learning (ML) have been employed in recent years. In most of these studies, the method of choice is artificial neural networks (ANN). In this work, we have shown an alternative ML method, support vector regression (SVR), that provides comparable accuracy with better computational efficiency. We have further used many body expansion (MBE) along with SVR to predict interaction energies in water clusters (decamers). In the case of dimer and trimer interaction energies, the root mean square errors (RMSEs) of the SVR based scheme are 0.12 kcal mol-1 and 0.34 kcal mol-1, respectively. We show that the SVR and MBE based scheme has a RMSE of 2.78% in the estimation of decamer interaction energy against the parent QM method in a computationally efficient way.