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.

Transferability of data-driven, many-body models for CO2 simulations in the vapor and liquid phases.

Extending on the previous work by Riera et al. [J. Chem. Theory Comput. 16, 2246-2257 (2020)], we introduce a second generation family of data-driven many-body MB-nrg models for CO2 and systematically assess how the strength and anisotropy of the CO2-CO2 interactions affect the models' ability to predict vapor, liquid, and vapor-liquid equilibrium properties. Building upon the many-body expansion formalism, we construct a series of MB-nrg models by fitting one-body and two-body reference energies calculated at the coupled cluster level of theory for large monomer and dimer training sets. Advancing from the first generation models, we employ the charge model 5 scheme to determine the atomic charges and systematically scale the two-body energies to obtain more accurate descriptions of vapor, liquid, and vapor-liquid equilibrium properties. Challenges in model construction arise due to the anisotropic nature and small magnitude of the interaction energies in CO2, calling for the necessity of highly accurate descriptions of the multidimensional energy landscape of liquid CO2. These findings emphasize the key role played by the training set quality in the development of transferable, data-driven models, which, accurately representing high-dimensional many-body effects, can enable predictive computer simulations of molecular fluids across the entire phase diagram.

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.

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.

Vapor-liquid equilibrium of water with the MB-pol many-body potential.

Among the many existing molecular models of water, the MB-pol many-body potential has emerged as a remarkably accurate model, capable of reproducing thermodynamic, structural, and dynamic properties across water's solid, liquid, and vapor phases. In this work, we assessed the performance of MB-pol with respect to an important set of properties related to vapor-liquid coexistence and interfacial behavior. Through direct coexistence classical molecular dynamics simulations at temperatures of 400 K < T < 600 K, we calculated properties such as equilibrium coexistence densities, vapor-liquid interfacial tension, vapor pressure, and enthalpy of vaporization and compared the MB-pol results to experimental data. We also compared rigid vs fully flexible variants of the MB-pol model and evaluated system size effects for the properties studied. We found that the MB-pol model predictions are in good agreement with experimental data, even for temperatures approaching the vapor-liquid critical point; this agreement was largely insensitive to system sizes or the rigid vs flexible treatment of the intramolecular degrees of freedom. These results attest to the chemical accuracy of MB-pol and its high degree of transferability, thus enabling MB-pol's application across a large swath of water's phase diagram.

Infrared signatures of isomer selectivity and symmetry breaking in the Cs+(H2O)3 complex using many-body potential energy functions.

A quantitative description of the interactions between ions and water is key to characterizing the role played by ions in mediating fundamental processes that take place in aqueous environments. At the molecular level, vibrational spectroscopy provides a unique means to probe the multidimensional potential energy surface of small ion-water clusters. In this study, we combine the MB-nrg potential energy functions recently developed for ion-water interactions with perturbative corrections to vibrational self-consistent field theory and the local-monomer approximation to disentangle many-body effects on the stability and vibrational structure of the Cs+(H2O)3 cluster. Since several low-energy, thermodynamically accessible isomers exist for Cs+(H2O)3, even small changes in the description of the underlying potential energy surface can result in large differences in the relative stability of the various isomers. Our analysis demonstrates that a quantitative account for three-body energies and explicit treatment of cross-monomer vibrational couplings are required to reproduce the experimental spectrum.

Halide Ion Microhydration: Structure, Energetics, and Spectroscopy of Small Halide-Water Clusters.

Replica exchange molecular dynamics simulations and vibrational spectroscopy calculations are performed using halide-water many-body potential energy functions to provide a bottom-up analysis of the structures, energetics, and hydrogen-bonding arrangements in X-(H2O) n ( n = 3-6) clusters, with X = F, Cl, Br, and I. Independently of the cluster size, it is found that all four halides prefer surface-type structures in which they occupy one of the vertices in the underlying three-dimensional hydrogen-bond networks. For fluoride-water clusters, this is in contrast to previous reports suggesting that fluoride prefers interior-type arrangements, where the ion is fully hydrated. These differences can be ascribed to the variability in how various molecular models are capable of reproducing the subtle interplay between halide-water and water-water interactions. Our results thus emphasize the importance of a correct representation of individual many-body contributions to the molecular interactions for a quantitative description of halide ion hydration.

Isomeric Equilibria, Nuclear Quantum Effects, and Vibrational Spectra of M+(H2O) n=1-3 Clusters, with M = Li, Na, K, Rb, and Cs, through Many-Body Representations.

A quantitative characterization of the molecular mechanisms that regulate ion solvation is key to the microscopic understanding of fundamental processes taking place in aqueous environments with major implications for different fields, from atmospheric chemistry to materials research and biochemistry. This study presents a systematic analysis of isomeric equilibria for small M+(H2O) n clusters, with M = Li, Na, K, Rb, and Cs, from 0 to 200 K. To determine the relative stability of different isomers of each M+(H2O) n cluster as a function of temperature, replica exchange simulations are carried out at both classical and quantum levels with the recently developed many-body MB-nrg potential energy functions, which have been shown to exhibit chemical accuracy. Anharmonic vibrational spectra are then calculated within the local monomer approximation and found to be in agreement with the available experimental data, providing further support for the accuracy of the MB-nrg potential energy functions. The present analysis indicates that nuclear quantum effects become increasingly important for larger M+(H2O) n clusters containing the heavier alkali metal ions, which is explained in terms of competing ion-water and water-water interactions along with the interplay between energetic and entropic effects. By directly connecting experimental measurements with molecular properties calculated at the quantum mechanical level, this study represents a further step toward the development of a consistent picture of ion hydration from the gas to the condensed phase.

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

This study presents the extension of the MB-nrg (Many-Body energy) theoretical/computational framework of transferable potential energy functions (PEFs) for molecular simulations of alkali metal ion-water systems. The MB-nrg PEFs are built upon the many-body expansion of the total energy and include the explicit treatment of one-body, two-body, and three-body interactions, with all higher-order contributions described by classical induction. This study focuses on the MB-nrg two-body terms describing the full-dimensional potential energy surfaces of the M+(H2O) dimers, where M+ = Li+, Na+, K+, Rb+, and Cs+. The MB-nrg PEFs are derived entirely from "first principles" calculations carried out at the explicitly correlated coupled-cluster level including single, double, and perturbative triple excitations [CCSD(T)-F12b] for Li+ and Na+ and at the CCSD(T) level for K+, Rb+, and Cs+. The accuracy of the MB-nrg PEFs is systematically assessed through an extensive analysis of interaction energies, structures, and harmonic frequencies for all five M+(H2O) dimers. In all cases, the MB-nrg PEFs are shown to be superior to both polarizable force fields and ab initio models based on density functional theory. As previously demonstrated for halide-water dimers, the MB-nrg PEFs achieve higher accuracy by correctly describing short-range quantum-mechanical effects associated with electron density overlap as well as long-range electrostatic many-body interactions.

The i-TTM model for ab initio-based ion-water interaction potentials. II. Alkali metal ion-water potential energy functions.

A new set of i-TTM potential energy functions describing the interactions between alkali metal ions and water molecules is reported. Following our previous study on halide ion-water interactions [J. Phys. Chem. B, 2016, 120, 1822], the new i-TTM potentials are derived from fits to CCSD(T) reference energies and, by construction, are compatible with the MB-pol many-body potential, which has been shown to accurately predict the properties of water from the gas to the condensed phase. Within the i-TTM formalism, two-body repulsion, electrostatic, and dispersion energies are treated explicitly, while many-body effects are represented by classical induction. The accuracy of the new i-TTM potentials is assessed through extensive comparisons with results obtained from different ab initio methods, including CCSD(T), CCSD(T)-F12b, DF-MP2, and several DFT models, as well as from polarizable force fields for M+(H2O)n clusters with M+ = Li+, Na+, K+, Rb+, and Cs+, and n = 1-4.

On the accuracy of the MB-pol many-body potential for water: Interaction energies, vibrational frequencies, and classical thermodynamic and dynamical properties from clusters to liquid water and ice.

The MB-pol many-body potential has recently emerged as an accurate molecular model for water simulations from the gas to the condensed phase. In this study, the accuracy of MB-pol is systematically assessed across the three phases of water through extensive comparisons with experimental data and high-level ab initio calculations. Individual many-body contributions to the interaction energies as well as vibrational spectra of water clusters calculated with MB-pol are in excellent agreement with reference data obtained at the coupled cluster level. Several structural, thermodynamic, and dynamical properties of the liquid phase at atmospheric pressure are investigated through classical molecular dynamics simulations as a function of temperature. The structural properties of the liquid phase are in nearly quantitative agreement with X-ray diffraction data available over the temperature range from 268 to 368 K. The analysis of other thermodynamic and dynamical quantities emphasizes the importance of explicitly including nuclear quantum effects in the simulations, especially at low temperature, for a physically correct description of the properties of liquid water. Furthermore, both densities and lattice energies of several ice phases are also correctly reproduced by MB-pol. Following a recent study of DFT models for water, a score is assigned to each computed property, which demonstrates the high and, in many respects, unprecedented accuracy of MB-pol in representing all three phases of water.

Toward Data-Driven Many-Body Simulations of Biomolecules in Solution: N-Methyl Acetamide as a Proxy for the Protein Backbone.

The development of molecular models with quantum-mechanical accuracy for predictive simulations of biomolecular systems has been a long-standing goal in the field of computational biophysics and biochemistry. As a first step toward a transferable force field for biomolecules entirely derived from "first-principles", we introduce a data-driven many-body energy (MB-nrg) potential energy function (PEF) for N-methylacetamide (NMA), a peptide bond capped by two methyl groups that is commonly used as a proxy for the protein backbone. The MB-nrg PEF is shown to accurately describe the energetics and structural properties of an isolated NMA molecule, including the normal modes of both cis and trans isomers and the energy variation along the isomerization path, as well as the multidimensional potential energy landscape of the NMA-H2O dimer in the gas phase. Importantly, we show that the MB-nrg PEF is fully transferable, enabling molecular dynamics simulations of NMA in solution with quantum-mechanical accuracy. Comparisons with results obtained with a popular pairwise-additive force field for biomolecules and a classical polarizable PEF demonstrate the ability of the MB-nrg PEF to accurately represent many-body effects in NMA-H2O interactions at both short and long distances, which is key to guaranteeing full transferability from the gas phase to the liquid phase.

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.

MB-pol(2023): Sub-chemical Accuracy for Water Simulations from the Gas to the Liquid Phase.

We use the MB-pol theoretical/computational framework to introduce a new family of data-driven many-body potential energy functions (PEFs) for water, named MB-pol(2023). By employing larger 2-body and 3-body training sets, including an explicit machine-learned representation of 4-body energies, and adopting more sophisticated machine-learned representations of 2-body and 3-body energies, we demonstrate that the MB-pol(2023) PEFs achieve sub-chemical accuracy in modeling the energetics of the hexamer isomers, outperforming both the original MB-pol and q-AQUA PEFs, which currently provide the most accurate description of water clusters in the gas phase. Importantly, the MB-pol(2023) PEFs provide remarkable agreement with the experimental results for various properties of liquid water, improving upon the original MB-pol PEF and effectively closing the gap with experimental measurements.

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.

Anomalies and Local Structure of Liquid Water from Boiling to the Supercooled Regime as Predicted by the Many-Body MB-pol Model.

For the past 50 years, researchers have sought molecular models that can accurately reproduce water's microscopic structure and thermophysical properties across broad ranges of its complex phase diagram. Herein, molecular dynamics simulations with the many-body MB-pol model are performed to monitor the thermodynamic response functions and local structure of liquid water from the boiling point down to deeply supercooled temperatures at ambient pressure. The isothermal compressibility and isobaric heat capacity show maxima near 223 K, in excellent agreement with recent experiments, and the liquid density exhibits a minimum at ∼208 K. A local tetrahedral arrangement, where each water molecule accepts and donates two hydrogen bonds, is found to be the most probable hydrogen-bonding topology at all temperatures. This work suggests that MB-pol may provide predictive capability for studies of liquid water's physical properties across broad ranges of thermodynamic states, including the so-called water's "no man's land" which is difficult to probe experimentally.

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.

Data-Driven Many-Body Models with Chemical Accuracy for CH4/H2O Mixtures.

Many-body potential energy functions (PEFs) based on the TTM-nrg and MB-nrg theoretical/computational frameworks are developed from coupled cluster reference data for neat methane and mixed methane/water systems. It is shown that the MB-nrg PEFs achieve subchemical accuracy in the representation of individual many-body effects in small clusters and enables predictive simulations from the gas to the liquid phase. Analysis of structural properties calculated from molecular dynamics simulations of liquid methane and methane/water mixtures using both TTM-nrg and MB-nrg PEFs indicates that, while accounting for polarization effects, is important for a correct description of many-body interactions in the liquid phase, an accurate representation of short-range interactions, as provided by the MB-nrg PEFs, is necessary for a quantitative description of the local solvation structure in liquid mixtures.

Data-Driven Many-Body Models for Molecular Fluids: CO2/H2O Mixtures as a Case Study.

In this study, we extend the scope of the many-body TTM-nrg and MB-nrg potential energy functions (PEFs), originally introduced for halide ion-water and alkali-metal ion-water interactions, to the modeling of carbon dioxide (CO2) and water (H2O) mixtures as prototypical examples of molecular fluids. Both TTM-nrg and MB-nrg PEFs are derived entirely from electronic structure data obtained at the coupled cluster level of theory and are, by construction, compatible with MB-pol, a many-body PEF that has been shown to accurately reproduce the properties of water. Although both TTM-nrg and MB-nrg PEFs adopt the same functional forms for describing permanent electrostatics, polarization, and dispersion, they differ in the representation of short-range contributions, with the TTM-nrg PEFs relying on conventional Born-Mayer expressions and the MB-nrg PEFs employing multidimensional permutationally invariant polynomials. By providing a physically correct description of many-body effects at both short and long ranges, the MB-nrg PEFs are shown to quantitatively represent the global potential energy surfaces of the CO2-CO2 and CO2-H2O dimers and the energetics of small clusters, as well as to correctly reproduce various properties in both gas and liquid phases. Building upon previous studies of aqueous systems, our analysis provides further evidence for the accuracy and efficiency of the MB-nrg framework in representing molecular interactions in fluid mixtures at different temperature and pressure conditions.

Nature of Alkali Ion-Water Interactions: Insights from Many-Body Representations and Density Functional Theory. II.

Interaction energies of alkali ion-water dimers, M+(H2O), and trimers, M+(H2O)2, with M = Li, Na, K, Rb, and Cs, are investigated using various many-body potential energy functions and exchange-correlation functionals selected across the hierarchy of density functional theory approximations. Analysis of interaction energy decompositions indicates that close-range interactions such as Pauli repulsion, charge transfer, and charge penetration must be captured in order to reproduce accurate interaction energies. In particular, it is found that simple classical polarizable models must be supplemented with dedicated terms which account for these close-range interactions in order to achieve chemical accuracy. It is also found that the exchange-correlation functionals mostly differ from each other in their Pauli repulsion + dispersion energies and, hence, benefit from the inclusion of nonlocal terms such as Hartree-Fock exchange and dependence on the electronic kinetic energy density in order to reproduce the interactions that contribute to this term, namely, Pauli repulsion and intermediate-range dispersion. As a continuation of the analysis performed in J. Chem. Theory Comput. 2019, 15, 2983, 10.1021/acs.jctc.9b00064, we make comparisons between findings for alkali ion-water interactions with those for halide-water interactions.