Scalable generalized screening for high-order terms in the many-body expansion: Algorithm, open-source implementation, and demonstration

The many-body expansion lies at the heart of numerous fragment-based methods that are intended to sidestep the nonlinear scaling of ab initio quantum chemistry, making electronic structure calculations feasible in large systems. In principle, inclusion of higher-order n-body terms ought to improve the accuracy in a controllable way, but unfavorable combinatorics often defeats this in practice and applications with n ≥ 4 are rare. Here, we outline an algorithm to overcome this combinatorial bottleneck, based on a bottom-up approach to energy-based screening. This is implemented within a new open-source software application ("Fragme∩t"), which is integrated with a lightweight semi-empirical method that is used to cull subsystems, attenuating the combinatorial growth of higher-order terms in the graph that is used to manage the calculations. This facilitates applications of unprecedented size, and we report four-body calculations in (H2O)64 clusters that afford relative energies within 0.1 kcal/mol/monomer of the supersystem result using less than 10% of the unique subsystems. We also report n-body calculations in (H2O)20 clusters up to n = 8, at which point the expansion terminates naturally due to screening. These are the largest n-body calculations reported to date using ab initio electronic structure theory, and they confirm that high-order n-body terms are mostly artifacts of basis-set superposition error.

Accurate Calculation of Many-Body Energies in Water Clusters Using a Classical Geometry-Dependent Induction Model

We incorporate geometry-dependent distributed multipole and polarizability surfaces into an induction model that is used to describe the 3- and 4-body terms of the interaction between water molecules. The moment expansion is carried out up to the hexadecapole with the multipoles distributed on the atom sites. Dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole distributed polarizabilities are used to represent the response of the multipoles to an electric field. We compare the model against two large databases consisting of 43,844 3-body terms and 3,603 4-body terms obtained from high level ab initio calculations previously used to fit the MB-pol and q-AQUA classical interaction potentials for water. The classical induction model with no adjustable parameters reproduces the ab initio 3-/4-body terms contained in these two databases with a root-mean-square error (RMSE) of 0.104/0.058 and a mean-absolute error (MAE) of 0.054/0.026 kcal/mol, respectively. These results are on par with the ones obtained by fitting the same data using over 14,000 (for the 3-body) and 200 (for the 4-body) parameters via Permutationally Invariant Polynomials (PIPs). This demonstrates the accuracy of this physically motivated model in describing the 3- and 4-body terms in the interactions between water molecules with no adjustable parameters. The triple-dipole-dispersion energy, included in the calculation of the 3-body energy, was found to be small but not quite negligible. The model represents a practical, efficient, and transferable approach for obtaining accurate nonadditive interactions for multicomponent systems without the need to perform tens of thousands of high level electronic structure calculations and fitting them with PIPs.

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.

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.

Breaking covalent bonds in the context of the many-body expansion (MBE). I. The purported "first row anomaly" in XHn (X = C, Si, Ge, Sn; n = 1-4)

We present a new, novel implementation of the Many-Body Expansion (MBE) to account for the breaking of covalent bonds, thus extending the range of applications from its previous popular usage in the breaking of hydrogen bonds in clusters to molecules. A central concept of the new implementation is the in situ atomic electronic state of an atom in a molecule that casts the one-body term as the energy required to promote it to that state from its ground state. The rest of the terms correspond to the individual diatomic, triatomic, etc., fragments. Its application to the atomization energies of the XHn series, X = C, Si, Ge, Sn and n = 1-4, suggests that the (negative, stabilizing) 2-B is by far the largest term in the MBE with the higher order terms oscillating between positive and negative values and decreasing dramatically in size with increasing rank of the expansion. The analysis offers an alternative explanation for the purported "first row anomaly" in the incremental Hn-1X-H bond energies seen when these energies are evaluated with respect to the lowest energy among the states of the XHn molecules. Due to the "flipping" of the ground/first excited state between CH2 (3B1 ground state, 1A1 first excited state) and XH2, X = Si, Ge, Sn (1A1 ground state, 3B1 first excited state), the overall picture does not exhibit a "first row anomaly" when the incremental bond energies are evaluated with respect to the molecular states having the same in situ atomic states.

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.

A Classical Model for 3-body Interactions in Aqueous Ionic Systems

We present a classical electrostatic induction model to evaluate the 3-body Ion-Water-Water (I-W-W) and (W-W-W) interactions in aqueous ionic systems. The monatomic ions were described by a point charge and a dipole-dipole polarizability, while for the polyatomic ions distributed multipoles up to hexadecapole and distributed polarizabilities up to quadrupole-quadrupole were used. The accuracy of the classical model is benchmarked against an accurate dataset of 936 (I-W-W) and 2,184 (W-W-W) 3-body terms for 13 different monatomic and polyatomic cation and anion systems. The classical model shows excellent agreement with the reference MP2 and CCSD(T) 3-body energies. The Root-Mean-Square-Errors (RMSEs) for monatomic cations, monatomic anions, and polyatomic ions were 0.29 kcal/mol, 0.25 kcal/mol, and 0.12 kcal/mol, respectively. The corresponding RMSE for 1,744 CCSD(T)/aVTZ 3-body (W-W-W) energies, used to train MB-pol, was 0.12 kcal/mol. The accuracy of the classical model demonstrates that the 3-body term for aqueous ionic systems can be accurately modeled classically, without the need to fit to tens of thousands of high-level ab initio calculations. This approach provides a fast but accurate and efficient path towards modeling the 3-body effect in aqueous ionic systems that is fully transferable across systems with different ions.

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.

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.

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.

The many-body expansion for aqueous systems revisited: III. Hofmeister ion-water interactions

We report a Many Body Energy (MBE) analysis of aqueous ionic clusters containing anions and cations at the two opposite ends of the Hofmeister series, viz. the kosmotropes Ca²⁺ and SO₄^2- and the chaotropes NH₄⁺ and ClO₄^-, with 9 water molecules to quantify how these ions alter the interaction between the water molecules in their immediate surroundings. We specifically aim at quantifying how various ions (depending on their position in the Hofmeister series) affect the interaction between the surrounding water molecules and probe whether there is a qualitatively different behavior between kosmotropic vs. chaotropic ions. The current results when compared to the ones reported earlier for water clusters [J. P. Heindel and S. S. Xantheas, J. Chem. Theor. Comput., 2020, 16, 6843-6855] as well as for alkali metal and halide ion aqueous clusters of the same size [J. P. Heindel and S. S. Xantheas, J. Chem. Theor. Comput., 2021, 17, 2200-2216], which lie in the middle of the Hofmeister series, offer a complete account of the effect an ion across the Hofmeister series from "kosmotropes" to "chaotropes" has on the interaction between the neighboring water molecules. Through this analysis, noteworthy differences between the MBE of kosmotropes and chaotropes were identified. The MBE of kosmotropes is dominated by ion-water interactions that extend beyond the 4-body term, the rank at which the MBE of pure water converges. The percentage contribution of the 2-B term to the total cluster binding energy is noticeably larger. The disruption of the hydrogen bonded network due to the dominant ion-water interactions results in weak, unfavorable water-water interactions. The MBE for chaotropes, on the other hand, was found to converge more quickly as it more closely resembles that of pure water clusters. Chaotropes exhibit weaker overall binding energies and weaker ion-water interactions in favor of water-water interactions, somewhat recovering the pattern of the 2-4 body terms exemplified by pure water clusters. A remarkable anti-correlation between the 2-B ion-water (I-W) and water-water (W-W) interactions as well as between the 3-B (I-W-W) and (I-W) interactions was found for both kosmotropic and chaotropic ions. This anti-correlation is linear for both monatomic anions and monatomic cations, suggesting the existence of underlying physical mechanisms that were previously unexplored. The consideration of two different structural arrangements (ion inside and outside of a water cluster) suggests that fully solvated (ion inside) chaotropes disrupt the hydrogen bonding network in a similar manner to partially solvated (ion outside) kosmotropes and offers useful insights into the modeling requirements of bulk vs. interfacial ion solvation. It is noteworthy that the 2-B contribution to the total Basis Set Superposition Error (BSSE) correction for both kosmotropic and chaotropic ions follows the universal erf profile vs. intermolecular distance previously reported for pure water, halide ion-water and alkali metal ion-water clusters. When scaled for the corresponding dimer energies and distances, a single profile fits the current results together with all previously reported ones for pure water and halide water clusters. This finding lends further support to schemes for accurately estimating the 2-B BSSE correction in condensed environments.

The Many-Body Expansion for Aqueous Systems Revisited: II. Alkali Metal and Halide Ion-Water Interactions

We present a detailed study of the many-body expansion (MBE) for alkali metal and halide ion-water interactions and quantify the effect of these ions on the strength of the surrounding aqueous hydrogen bonding environment. Building on our previous work on neutral water clusters [J. P. Heindel and S. S. Xantheas, J. Chem. Theor. Comput. 16 (11), 6843-6855 (2020)], we carry out the MBE for the alkali metal and halide ion-water clusters, Z^+/-(H₂O)₉, where Z = Li⁺, Na⁺, K⁺, Rb⁺, Cs⁺, F^-, Cl^-, Br^-, and I^- and compare them with the results for a pure water cluster with the same number of "bodies", viz., (H₂O)₁₀. The 2-B ion-water (I-W) interaction accounts for a larger percentage of the total cluster binding energy compared to a pure water cluster of the same size, with the total 3-B term being smaller and of opposite sign (repulsive), whereas higher order terms are essentially negligible. The same oscillating behavior around zero for the MBE terms higher than the 5-B with a basis set that was reported for water clusters is also observed for the ion-water clusters considered here, with the basis set superposition error (BSSE) corrections amending this as in the water cluster case. A remarkable, linear anticorrelation between the total 2-B (I-W), the total 2-B (W-W), and also the 3-B (W-W-W) interactions is found, quantifying the effect of the different ions in disrupting and altering (weakening) the neighboring hydrogen bonded water network: stronger (I-W) interactions result in weaker (W-W) interactions. Additional linear correlations across the two series of alkali metals and halide ions were found between the 3-B (I-W-W) and the 2-B (I-W) as well as between the 3-B (I-W-W) and the 3-B (W-W-W) interactions, suggesting the existence of previously unrealized underlying physics governing these 2-B intermolecular and 3-B collective interactions. Our results further suggest a universal behavior of the two different families of ions (alkali metals and halides) for both the correlations of the various components of the total binding energies and the estimate of the 2-B BSSE correction, which is reported to follow a common profile for ion-water and water-water interactions when cast in terms of reduced distances and energies of the respective dimers. We expect the current results that quantify the interplay between ion-water and water-water interactions in aqueous clusters to impact the development of classical, ab initio-based force fields for monatomic ion solvation, whereas the insights into the nature of the BSSE to be critical in future ab initio-based, many-body molecular dynamics studies.

From Surface Tension to Molecular Distribution: Modeling Surfactant Adsorption at the Air-Water Interface

Surfactants are centrally important in many scientific and engineering fields and are used for many purposes such as foaming agents and detergents. However, many challenges remain in providing a comprehensive understanding of their behavior. Here, we provide a brief historical overview of the study of surfactant adsorption at the air-water interface, followed by a discussion of some recent advances in this area from our group. The main focus is on incorporating an accurate description of the adsorption layer thickness of surfactant at the air-water interface. Surfactants have a wide distribution at the air-water interface, which can have a significant effect on important properties such as the surface excess, surface tension, and surface potential. We have developed a modified Poisson-Boltzmann (MPB) model to describe this effect, which we outline here. We also address the remaining challenges and future research directions in this area. We believe that experimental techniques, modeling, and simulation should be combined to form a holistic picture of surfactant adsorption at the air-water interface.

Electrostatics, Charge Transfer, and the Nature of the Halide-Water Hydrogen Bond

Binary halide-water complexes X^-(H₂O) are examined by means of symmetry-adapted perturbation theory, using charge-constrained promolecular reference densities to extract a meaningful charge-transfer component from the induction energy. As is known, the X^-(H₂O) potential energy surface (for X = F, Cl, Br, or I) is characterized by symmetric left and right hydrogen bonds separated by a C_2v-symmetric saddle point, with a tunneling barrier height that is <2 kcal/mol except in the case of F^-(H₂O). Our analysis demonstrates that the charge-transfer energy is correspondingly small (<2 kcal/mol except for X = F), considerably smaller than the electrostatic interaction energy. Nevertheless, charge transfer plays a crucial role determining the conformational preferences of X^-(H₂O) and provides a driving force for the formation of quasi-linear X··· H-O hydrogen bonds. Charge-transfer energies correlate well with measured O-H vibrational redshifts for the halide-water complexes and also for OH^-(H₂O) and NO₂^-(H₂O), providing some indication of a general mechanism.

Isolating the Contributions of Specific Network Sites to the Diffuse Vibrational Spectrum of Interfacial Water with Isotopomer-Selective Spectroscopy of Cold Clusters

Decoding the structural information contained in the interfacial vibrational spectrum of water requires understanding how the spectral signatures of individual water molecules respond to their local hydrogen bonding environments. In this study, we isolated the contributions for the five classes of sites that differ according to the number of donor (D) and acceptor (A) hydrogen bonds that characterize each site. These patterns were measured by exploiting the unique properties of the water cluster cage structures formed in the gas phase upon hydration of a series of cations M⁺·(H₂O)_n (M = Li, Na, Cs, NH₄, CH₃NH₃, H₃O, and n = 5, 20-22). This selection of ions was chosen to systematically express the A, AD, AAD, ADD, and AADD hydrogen bonding motifs. The spectral signatures of each site were measured using two-color, IR-IR isotopomer-selective photofragmentation vibrational spectroscopy of the cryogenically cooled, mass selected cluster ions in which a single intact H₂O is introduced without isotopic scrambling, an important advantage afforded by the cluster regime. The resulting patterns provide an unprecedented picture of the intrinsic line shapes and spectral complexities associated with excitation of the individual OH groups, as well as the correlation between the frequencies of the two OH groups on the same water molecule, as a function of network site. The properties of the surrounding water network that govern this frequency map are evaluated by dissecting electronic structure calculations that explore how changes in the nearby network structures, both within and beyond the first hydration shell, affect the local frequency of an OH oscillator. The qualitative trends are recovered with a simple model that correlates the OH frequency with the network-modulated local electron density in the center of the OH bond.

Improved description of ligand polarization enhances transferability of ion-ligand interactions

The reliability of molecular mechanics (MM) simulations in describing biomolecular ion-driven processes depends on their ability to accurately model interactions of ions simultaneously with water and other biochemical groups. In these models, ion descriptors are calibrated against reference data on ion-water interactions, and it is then assumed that these descriptors will also satisfactorily describe interactions of ions with other biochemical ligands. The comparison against the experiment and high-level quantum mechanical data show that this transferability assumption can break down severely. One approach to improve transferability is to assign cross terms or separate sets of non-bonded descriptors for every distinct pair of ion type and its coordinating ligand. Here, we propose an alternative solution that targets an error-source directly and corrects misrepresented physics. In standard model development, ligand descriptors are never calibrated or benchmarked in the high electric fields present near ions. We demonstrate for a representative MM model that when the polarization descriptors of its ligands are improved to respond to both low and high fields, ligand interactions with ions also improve, and transferability errors reduce substantially. In our case, the overall transferability error reduces from 3.3 kcal/mol to 1.8 kcal/mol. These improvements are observed without compromising on the accuracy of low-field interactions of ligands in gas and condensed phases. Reference data for calibration and performance evaluation are taken from the experiment and also obtained systematically from "gold-standard" CCSD(T) in the complete basis set limit, followed by benchmarked vdW-inclusive density functional theory.

Excitation of hydrated Li+ and Na+ to their dissociative states: The effect of hydrogen bond on the dissociation of LiO and NaO bonds and the comparison of the TD-DFT and SAC-CI excited dissociative states

In this work, the ground and excited dissociative potential energy curves of hydrated Li⁺ and Na⁺ ions with different structures, containing two, three, and four M-O bonds (M = Li and Na), have been calculated. The vertical energies for the excitation of the clusters to their dissociative states were calculated. The scanning of the potential surfaces was performed in the direction of two normal vibrational modes related to the symmetric and asymmetric stretching of M-O bonds. The difference in the arrangement and number of water molecules around the alkali cation made it possible to study the effect of the hydrogen bond network on the dissociation of M-O bonds. Two different methods including the time-dependent density functional theory (TD-DFT) and direct-symmetry adapted cluster-configuration interaction (direct-SAC-CI) were used for calculating the potential energy curves, separately to compare the TD-DFT method with a correlative computational method such as SAC-CI. The TD-DFT method predicted the charge transfer from water molecules to alkali cation during the dissociation of clusters in the gas phase while the electrostatic field of water blocked this charge transfer. For some clusters, it was found that the change of the theoretical method from the TD-DFT to SAC-CI creates changes in the states of fragments obtained from the dissociation and charge transfer.

SN2 Reactions of N2O5 with Ions in Water: Microscopic Mechanisms, Intermediates, and Products

Reactions of dinitrogen pentoxide (N₂O₅) greatly affect the concentrations of NO₃, ozone, OH radicals, methane, and more. In this work, we employ ab initio molecular dynamics and other tools of computational chemistry to explore reactions of N₂O₅ with anions hydrated by 12 water molecules to shed light on this important class of reactions. The ions investigated are Cl^-, SO₄^2-, ClO₄^-, and RCOO^- (R = H, CH₃, C₂H₅). The following main results are obtained: (i) all the reactions take place by an S_N2-type mechanism, with a transition state that involves a contact ion pair (NO₂⁺NO₃^-) that interacts strongly with water molecules. (ii) Reactions of a solvent-separated nitronium ion (NO₂⁺) are not observed in any of the cases. (iii) An explanation is provided for the suppression of ClNO₂ formation from N₂O₅ reacting with salty water when sulfate or acetate ions are present, as found in recent experiments. (iv) Formation of novel intermediate species, such as (SO₄NO₂^-) and RCOONO₂, in these reactions is predicted. The results suggest atomistic-level mechanisms for the reactions studied and may be useful for the development of improved modeling of reaction kinetics in aerosol particles.

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.

Ion-Hydroxyl Interactions: From High-Level Quantum Benchmarks to Transferable Polarizable Force Fields

Ion descriptors in molecular mechanics models are calibrated against reference data on ion-water interactions. It is then typically assumed that these descriptors will also satisfactorily describe interactions of ions with other functional groups, such as those present in biomolecules. However, several studies now demonstrate that this transferability assumption produces, in many different cases, large errors. Here we address this issue in a representative polarizable model and focus on transferability of cationic interactions from water to a series of alcohols. Both water and alcohols use hydroxyls for ion-coordination, and, therefore, this set of molecules constitutes the simplest possible case of transferability. We obtain gas phase reference data systematically from "gold-standard" quantum Monte Carlo and CCSD(T) methods, followed by benchmarked vdW-corrected DFT. We learn that the original polarizable model yields large gas phase water → alcohol transferability errors - the RMS and maximum errors are 2.3 and 5.1 kcal/mol, respectively. These errors are, nevertheless, systematic in that ion-alcohol interactions are overstabilized, and systematic errors typically imply that some essential physics is either missing or misrepresented. A comprehensive analysis shows that when both low- and high-field responses of ligand dipole polarization are described accurately, then transferability improves significantly - the RMS and maximum errors in the gas phase reduce, respectively, to 0.9 and 2.5 kcal/mol. Additionally, predictions of condensed phase transfer free energies also improve. Nevertheless, within the limits of the extrathermodynamic assumptions necessary to separate experimental estimates of salt dissolution into constituent cationic and anionic contributions, we note that the error in the condensed phase is systematic, which we attribute, at least, partially to the parametrization in long-range electrostatics. Overall, this work demonstrates a rational approach to boosting transferability of ionic interactions that will be applicable broadly to improving other polarizable and nonpolarizable models.

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.

Single-Ion Thermodynamics from First Principles: Calculation of the Absolute Hydration Free Energy and Single-Electrode Potential of Aqueous Li+ Using ab Initio Quantum Mechanical/Molecular Mechanical Molecular Dynamics Simulations

A recently proposed thermodynamic integration (TI) approach formulated in the framework of quantum mechanical/molecular mechanical molecular dynamics (QM/MM MD) simulations is applied to study the structure, dynamics, and absolute intrinsic hydration free energy Δ_s G_M⁺,wat^◦ of the Li⁺ ion at a correlated ab initio level of theory. Based on the results, standard values (298.15 K, ideal gas at 1 bar, ideal solute at 1 molal) for the absolute intrinsic hydration free energy [Formula: see text] of the proton, the surface electric potential jump χ_wat^◦ upon entering bulk water, and the absolute single-electrode potential [Formula: see text] of the reference hydrogen electrode are calculated to be -1099.9 ± 4.2 kJ·mol^-1, 0.13 ± 0.08 V, and 4.28 ± 0.04 V, respectively, in excellent agreement with the standard values recommended by Hünenberger and Reif on the basis of an extensive evaluation of the available experimental data (-1100 ± 5 kJ·mol^-1, 0.13 ± 0.10 V, and 4.28 ± 0.13 V). The simulation results for Li⁺ are also compared to those for Na⁺ and K⁺ from a previous study in terms of relative hydration free energies ΔΔ_s G_M⁺,wat^◦ and relative electrode potentials [Formula: see text]. The calculated values are found to agree extremely well with the experimental differences in standard conventional hydration free energies ΔΔ_s G_M⁺,wat^• and redox potentials [Formula: see text]. The level of agreement between simulation and experiment, which is quantitative within error bars, underlines the substantial accuracy improvement achieved by applying a highly demanding QM/MM approach at the resolution-of-identity second-order Møller-Plesset perturbation (RIMP2) level over calculations relying on purely molecular mechanical or density functional theory (DFT) descriptions. A detailed analysis of the structural and dynamical properties of the Li⁺ hydrate indicates that a correct description of the solvation structure and dynamics is achieved as well at this level of theory. Consideration of the QM/MM potential-energy components also shows that the partitioning into QM and MM zones does not induce any significant energetic artifact for the system considered.

Assessing many-body contributions to intermolecular interactions of the AMOEBA force field using energy decomposition analysis of electronic structure calculations

In this work, we evaluate the accuracy of the classical AMOEBA model for representing many-body interactions, such as polarization, charge transfer, and Pauli repulsion and dispersion, through comparison against an energy decomposition method based on absolutely localized molecular orbitals (ALMO-EDA) for the water trimer and a variety of ion-water systems. When the 2- and 3-body contributions according to the many-body expansion are analyzed for the ion-water trimer systems examined here, the 3-body contributions to Pauli repulsion and dispersion are found to be negligible under ALMO-EDA, thereby supporting the validity of the pairwise-additive approximation in AMOEBA's 14-7 van der Waals term. However AMOEBA shows imperfect cancellation of errors for the missing effects of charge transfer and incorrectness in the distance dependence for polarization when compared with the corresponding ALMO-EDA terms. We trace the larger 2-body followed by 3-body polarization errors to the Thole damping scheme used in AMOEBA, and although the width parameter in Thole damping can be changed to improve agreement with the ALMO-EDA polarization for points about equilibrium, the correct profile of polarization as a function of intermolecular distance cannot be reproduced. The results suggest that there is a need for re-examining the damping and polarization model used in the AMOEBA force field and provide further insights into the formulations of polarizable force fields in general.

Real single ion solvation free energies with quantum mechanical simulation

Single ion solvation free energies are one of the most important properties of electrolyte solutions and yet there is ongoing debate about what these values are. Only the values for neutral ion pairs are known. Here, we use DFT interaction potentials with molecular dynamics simulation (DFT-MD) combined with a modified version of the quasi-chemical theory (QCT) to calculate these energies for the lithium and fluoride ions. A method to correct for the error in the DFT functional is developed and very good agreement with the experimental value for the lithium fluoride pair is obtained. Moreover, this method partitions the energies into physically intuitive terms such as surface potential, cavity and charging energies which are amenable to descriptions with reduced models. Our research suggests that lithium's solvation free energy is dominated by the free energetics of a charged hard sphere, whereas fluoride exhibits significant quantum mechanical behavior that cannot be simply described with a reduced model.

Metal Ion Modeling Using Classical Mechanics

Metal ions play significant roles in numerous fields including chemistry, geochemistry, biochemistry, and materials science. With computational tools increasingly becoming important in chemical research, methods have emerged to effectively face the challenge of modeling metal ions in the gas, aqueous, and solid phases. Herein, we review both quantum and classical modeling strategies for metal ion-containing systems that have been developed over the past few decades. This Review focuses on classical metal ion modeling based on unpolarized models (including the nonbonded, bonded, cationic dummy atom, and combined models), polarizable models (e.g., the fluctuating charge, Drude oscillator, and the induced dipole models), the angular overlap model, and valence bond-based models. Quantum mechanical studies of metal ion-containing systems at the semiempirical, ab initio, and density functional levels of theory are reviewed as well with a particular focus on how these methods inform classical modeling efforts. Finally, conclusions and future prospects and directions are offered that will further enhance the classical modeling of metal ion-containing systems.

Solvation of Li+ by argon: how important are three-body forces?

A global geometry search on a new potential energy surface for Li⁺Ar_n clusters revealed that three-body interactions must be included to reproduce ab initio structures and accurate energetic features.

Signatures of Size-Dependent Structural Patterns in Hydrated Copper(I) Clusters, Cu+(H2O)n=1-10

The isomers of a hydrated Cu(I) ion with n = 1-10 water molecules were investigated by using ab initio quantum chemistry and an automated isomer-search algorithm. The electronic structure and vibrational spectra of the hundreds of resulting isomers were used to analyze the source of the observed bonding patterns. A structural evolution from dominantly two-coordinate structures (n = 1-4) toward a mixture of two- and three-coordinate structures was observed at n = 5-6, where the stability provided by expanded hydrogen-bonding was competitive with the dominantly electrostatic interaction between the water ligand and remaining binding sites of the metal ion. Further hydration (n = 7-10) led to a mixture of three- and four-coordinate structures. The metal ion was found, through spectroscopic signatures, to appreciably perturb the O-H bonds of even third-shell water molecules, which highlighted the ability of this nominally simple ion to partially activate the surrounding water network.

Modeling Molecular Interactions in Water: From Pairwise to Many-Body Potential Energy Functions

Almost 50 years have passed from the first computer simulations of water, and a large number of molecular models have been proposed since then to elucidate the unique behavior of water across different phases. In this article, we review the recent progress in the development of analytical potential energy functions that aim at correctly representing many-body effects. Starting from the many-body expansion of the interaction energy, specific focus is on different classes of potential energy functions built upon a hierarchy of approximations and on their ability to accurately reproduce reference data obtained from state-of-the-art electronic structure calculations and experimental measurements. We show that most recent potential energy functions, which include explicit short-range representations of two-body and three-body effects along with a physically correct description of many-body effects at all distances, predict the properties of water from the gas to the condensed phase with unprecedented accuracy, thus opening the door to the long-sought "universal model" capable of describing the behavior of water under different conditions and in different environments.