Aqueous Solution Chemistry In Silico and the Role of Data Driven Approaches

1 arXiv:2403.06236v1 [physics.chem-ph] 10 Mar 2024 Aqueous Solution Chemistry In Silico and the Role of Data Driven Approaches Debarshi Banerjee,†,‡Khatereh Azizi,¶,†Colin K. Egan,†Edward Danquah Donkor,†,‡Cesare Malosso,§Solana Di Pino,§Gonzalo D´ıaz Mir´on,†Martina Stella,†Giulia Sormani,†Germaine Neza Hozana,†,∥Marta Monti,†Uriel N. Morzan,⊥Alex Rodriguez,#,†Giuseppe Cassone,@Asja Jelic,†Damian Scherlis,§ and Ali Hassanali† †International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy ‡Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265, 34136 Trieste, Italy ¶School of Nano Science, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran §Departamento de Quimica Inorganica, Analitica y Quimica Fisica/INQUIMAE. Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires. Pabell´on II Ciudad Universitaria 1428 Buenos Aires - Argentina ∥Dipartimento di Fisica, Universit´a degli Studi di Trieste, Via Alfonso Valerio 2, 34127, Trieste, Italy ⊥Instituto de Fisica de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales Pabell´on I Ciudad Universitaria 1428 Buenos Aires - Argentina #Dipartimento di Matematica e Geoscienze, Universit´a degli Studi di Trieste, via Alfonso Valerio 12/1, 34127, Trieste, Italy @Institute for Chemical-Physical Processes, National Research Council (IPCF-CNR), Via S. d’Alcontres 37, 98158 Messina, Italy E-mail: 2 Abstract The use of computer simulations to study the properties of aqueous systems is, today more than ever, an active area of research. In this context, during the last decade there has been a tremendous growth in the use of data-driven approaches to develop more accurate potentials for water as well as to characterize its complexity in chemical and biological contexts. We highlight the progress, giving a historical context, on the path to the development of many-body and reactive potentials to model aqueous chemistry, including the role of machine learning strategies. We focus specifically on conceptual and methodological challenges along the way in performing simulations that seek to tackle problems in modeling the chemistry of aqueous solutions. In conclusion, we summarize our perspectives on the use and integration of advanced data-science techniques to provide chemical insights in physical chemistry and how this will influence computer simulations of aqueous systems in the future. Introduction Liquid water is one of the key ingredients for life.1It forms a central lubricant for biological materials and is perhaps the most ubiquitous solvent in physical, chemical, engineering and technological applications.2–4 Many of the unique properties of water arise from its hydrogenbond network5–8 and how it changes in different thermodynamic conditions. Dissecting the microscopic structure of water and aqueous solutions in terms of both the static and dynamical properties of hydrogen bonding has been the subject of numerous experimental and theoretical studies and rather lively controversies.9–11 Computer simulations of varying levels of complexity have played an important role in providing molecular-level insights into the structural, dynamical and electronic properties of both bulk water and solutes in aqueous solutions.11–13 Over the last five decades, the typical models that are used to simulate aqueous systems can be broadly separated into two categories namely molecular mechanics (MM)14 empirical potentials and first-principles 3 ab-initio (AIMD) approaches.15 While MM based water models provide a powerful way to explore, for example, the phase diagram of water6,9 and perform simulations of complex biological systems,16–18 they typically do not allow for chemistry to occur. Although being more computationally prohibitive, electronic structure-based AIMD simulations overcome this limitation allowing for modeling chemical reactions where bond breaking or formation occurs. One of the most popular electronic structure approaches for modeling the properties of water has been Density Functional Theory (DFT),12,13,19 although over the last decade more advanced approaches have also been employed, using, for example, quantum Monte-Carlo20 and quantum chemistry techniques such as Møller–Plesset perturbation theory (MP2).21 In both the MM and AIMD approaches to study aqueous systems, there are several challenges that have emerged if one is interested in producing meaningful simulations that can be interpreted and compared with experiments. The first is the quality of the electronic structure theory which ultimately controls the underlying potential energy surface associated with the hydrogen bonds needed to reproduce structural, dynamical and spectroscopic properties of water.19,22,23 Secondly, there is the problem of sampling arising from the fact that it takes a long time for the system to hop over free-energy barriers.24,25 This is particularly true for modeling chemical reactions such as water ionization in different environments26–28 with AIMD, as well as exploring the complex phase diagram of water, for example, under super cooled conditions.29 Finally, in many particle systems involving the coupling of both the solute and its aqueous environment, identifying the relevant and important degrees of freedom (commonly referred to as order parameters, reaction coordinates or collective variables) along which physical, chemical or biological processes occur, is far from trivial.30,31 Herein, we will provide an overview of the most recent advances in the field aimed at overcoming these challenges. Figure 1 provides a schematic outline of our review. In the first part, we begin with a brief historical perspective on the challenges in modeling liquid water from first principles electronic structure simulations. We thereafter discuss the advances made in the development of reactive, many-body and machine-learning (ML) based 4 potentials which are opening new turf in studying the physical chemistry of liquid water (Figure 1a,b). We focus specifically on highlighting important advances that have been made in the applications of these techniques using them to also underscore the challenges in modeling aqueous chemistry in the bulk, at interfaces and under confinement. Furthermore, a critical aspect associated with understanding aqueous chemistry is the determination and exploration of relevant order parameters of reaction coordinates. Due to the collective nature associated with processes involving chemistry in water, identifying these coordinates requires going beyond chemical imagination; we highlight in this context the importance of data-driven approaches (Figure 1c). Finally, we conclude with some perspectives on the future of data-driven approaches in empowering conceptual advances in the chemical physics and physical chemistry of aqueous systems. Towards Advanced Data Science Techniques The present section is essentially divided into three parts, as illustrated in Figure 1. We begin by providing an overview of the main approaches that have been used to simulate aqueous chemistry focusing on DFT, Many-Body and reactive potentials. As indicated in the Introduction, our focus is to highlight key applications related to aqueous chemistry that underscore the challenges. This subsequently serves as motivation for the second part where we focus on recent developments in the use of machine-learning potentials for modeling water and its constituent ions in different contexts. In the last part of this section, we discuss advances and challenges in data-mining for water-chemistry. 5 (a) DFT/AIMD/ … initial structure {r3N} V(r3N) Larger box simulation (b) CV PES [kBT] (c) Atomic descriptors High-dim data Low-dim data Free energy surface DFT AIMD MB-pol Reactive Neural Network Potentials Collective Variables Unsupervised Learning Figure 1: Schematic summary outlining the content of our review. Panel (a) illustrates the main potential energy functions used to simulate aqueous systems discussed in this review. Panel (b) highlights recent developments in the training of neural-network based potentials facilitating larger system sizes and longer simulation times. Panel (c) shows the use of datadriven approaches to enable the interpretation and analysis of simulations where aqueous chemistry occurs. DFT Water, Many-Body and Reactive Potentials DFT Water DFT has become one of the most popular electronic structure methods for simulating complex chemical and biological systems from first principles32,33 due to its favourable compromise between computational efficiency and accuracy. The standard way in which one can couple DFT with finite temperature molecular dynamics is using Born-Oppenheimer (BO) or Car-Parrinello (CP) MD. For more details on the theory of DFT, BOMD and CPMD, the interested reader is referred to specific books and other reviews on this subject and references therein.13,34,35 6 DFT-based ab initio molecular dynamics simulations (AIMD) of bulk liquid water and its constituent ions, the proton and hydroxide, offered a first glimpse into the coupling of nuclear and electronic degrees of freedom in water.36–39 In particular, such techniques have opened up a microscopic window into the celebrated Grotthuss mechanism40 involving the inter-conversion of covalent and hydrogen bonds in the water network. These simulations have since guided the interpretation of many IR and Raman-based spectroscopy experiments of acidic and basic water.41–44 Although DFT is an exact theory, within the Kohn-Sham formalism, the functional form for the exchange-correlation (XC) energy is not known and various approximations need to be made to treat this term. As a result, this poses limits on the accuracy that DFT can achieve for the prediction of electronic structure properties. Numerous studies over the last decades have highlighted how the choice of the XC functional can, in fact, hugely impact the description of the structural and dynamical properties of water. For a detailed review on the effect of DFT on water in different conditions, the reader is referred to a perspective by Michaelides and co-workers.19 Panel (a) in Figure 2 shows the radial distribution functions (RDF) obtained for two different popular functionals with and without the inclusion of dispersion interaction corrections (e.g., Grimme’s semiempirical D345), compared to experimental measurements.46,47 In summary, it can be seen that accounting for dispersion interactions plays an important role in correcting for approximations made in standard generalized-gradient approximation (GGA) functionals, an aspect that has been reinforced in numerous studies.36,48–50 Specifically, standard GGA functionals yield a much more glassy liquid. Besides the inclusion of dispersion corrections,19,51–53 the inclusion of exact Hartree-Fock exchange through the use of hybrid functionals has also been shown to give more accurate estimates of water polarizability.54 However, despite the generally favourable scaling of DFT and especially in cases where sophisticated functionals and extra dispersion corrections are employed, the computational cost of the simulations poses severe limits on both the size of the system investigated (relatively small box sizes (<2 nm)) and the length 7 of the dynamics (short simulation times ( ∼100 ps)). Besides the challenges of dealing with the quality of the electronic structure, another issue in simulating the physical and chemical behavior of water is the high zero-point energy (ZPE) of the O-H covalent bonds which is almost about an order of magnitude larger than thermal energy at room temperature. This is reflected in the O-H pair correlation function of water which tends to be overlocalized in simulations where the nuclei are treated classically. Car and co-workers were the first to elucidate the importance of nuclear quantum effects (NQEs) on the structural properties of the hydrogen bonds in water by using path-integral molecular dynamics (PIMD) together with CPMD simulations.55 Examining the O-H RDF, one observes that in the PIMD simulations, the proton is much more delocalized, leading to a broader first peak consistent with the experiments. In this context, path-integral approaches coupled with BOMD or CPMD simulations have become somewhat routine, especially in combination with generalized-Langevin based thermostats.56–58 These and other types of simulations that include NQEs have shown that these effects are important for understanding the structural, dynamical and even electronic properties of bulk water as well as chemical and biologically relevant solutes.59–67 The broader O-H bond length distribution found in PIMD was associated with enhanced transient autoionization events where charged-pairs of waters’ constituent ions apparently form due to extreme events involving protons delocalizing along hydrogen bonds.68 The extent of these proton fluctuations are however, very sensitive to the quality in the underlying electronic structure. Specifically, the details of the DFT functional along with dispersion corrections compete with NQEs in highly non-trivial ways, for example, in affecting spectroscopic properties of hydrogen bonds. Figure 2b nicely demonstrates this effect by illustrating how the high frequency modes of the O-H vibrations change as a function of combining hybrid DFT functionals with nuclear quantum dynamics. Marselek and Markland show that standard GGA functionals, even when including dispersion corrections, exaggerate the fraction of these transient autoionization events leading to larger red shifts in the O-H vibrational 8 stretch frequencies.69 Upon using hybrid functionals together with dispersion corrections, the inclusion of quantum effects almost perfectly reproduces the experimental spectra. The transient autoionization events previously discussed are, of course, the initial seeds for water dissociation that leads to the creation of hydronium (H3O)+and hydroxide (OH)− ions. The preceding issues regarding the choice of the quality in the electronic structure description within the framework of DFT have been shown to play a critical role in affecting structural and dynamical properties of these ionic topological defects. In Figure 2c, Voth and co-workers perform a systematic analysis on the factors affecting the diffusion constant of the excess proton in water.70 The reported values of the diffusion constant are extremely sensitive to the inclusion of dispersion corrections, the choice of the basis set, density functional employed and, finally, the initial conditions. Furthermore, Car and co-workers recently studied the role of including exact exchange on the diffusion mechanism of the excess proton and hydroxide in liquid water.71 Previous studies from some of us, had suggested that the Grotthuss mechanism involves concerted proton hopping events for both the proton and hydroxide ion.72 Specifically, it was found that protons and proton-holes (hydroxide ions) diffuse along water wires involving concerted double jumps. In Figure 2d, Car and coworkers demonstrate that while the use of GGA functionals display consistent results for the excess proton compared to the hybrids, the story is much more complicated for the hydroxide ion. In particular, they show that the migration of hydroxide is dominated by a stepwise mechanism rather than concerted hopping. The qualitative difference between the effect of the DFT electronic structure on the proton and hydroxide is rooted in the fact that the use of hybrid functionals along with ab initio dispersion corrections sensitively affects the solvent environment of the hydroxide, favoring a hyper-coordinated versus a three-coordinated solvation structure. 9 rately captures the experimentally measured disorder in the hydrogen bonding network (see Figure 3d). It is important to note that condensed phase structure and thermodynamics were not directly included in the development of MB-pol, so accuracy in reproducing these properties reflects the importance of close-range low-order many-body effects in aqueous systems. As alluded to earlier in the review, water is characterized by different types of anomalies which are not found in simple liquids. These include for example, the density maximum at 4◦C134 and the minimum in isothermal compressibility at 46.5◦C.135 The MB-pol model has been successful at reproducing many of these anomalies where standard DFT-based methods at the GGA level of accuracy typically fail136 and effective MM potentials have experienced significant challenges. In part, this originates from the lack of transferability of effective MM potentials as they are often parameterized to work at a specific region of the phase diagram.137 In Figure 3e we show the isothermal compressibility of liquid water from ambient temperatures to the supercooled regime, comparing MB-pol to experimental measurements. The agreement is extremely impressive. In addition to the well-established minimum at 46.5◦C, MB-pol also reproduces the value of the measured compressibility at the tentative maximum at around -44oC (albeit with large error bars due to long relaxation times near the Widom line).121,138 The quality of the potential energy surface sampled by MB-pol is clearly superior to the vast majority of MM models and commonly used XC functionals on the market. Among the many interesting properties of water, perhaps the most challenging is determining spectroscopic properties both in the bulk and at interfaces. Specifically, vibrational spectra in liquid water are sensitive to both the details of microscopic structure, including properties of the hydrogen bonding network, as well as the potential energy surface along specific vibrational modes. Early studies validating the MB-pol potential demonstrated that the model successfully reproduces experimentally measured features of the bulk vibrational (infrared and Raman) spectra, within the limits of the approximate quantum dynamics 16 method used, demonstrating that liquid water simulations with MB-pol produce accurate hydrogen bond dynamics.139,140 Additionally, the ability of MB-pol to reproduce the features of the experimental SFG spectrum of the air-water interface (within the limits of the approximate quantum dynamics used) is a noteworthy achievement. In particular, the MB-pol calculations predicted the lack of a positive feature around 3000 cm−1prior to the correction of experimental artifacts between 2011 and 2015.141 These examples further show that the bottom up approach involving converging the MBE expansion ensures the transferability of the potential. Finally, we would like to showcase recent extensions of MB-pol (MB-nrg) to go beyond bulk water and allow for modeling solutions such as ions in water.93,122,125–131 An instructive example is the dilute aqueous Na+solution, for which pairwise classical potentials tend to over-structure the solvation shells around Na+, while classical polarizable models understructure the Na+solvation, predicting an almost continuous transition between the first and second shells.122 The accurate description of close-range quantum mechanical effects, such as charge transfer and charge penetration, at the 2B and 3B levels provided by a full many-body potential93 allows MB-nrg to predict the correct solvation structure, and thus reproduce the experimentally measured extended X-ray absorption fine structure (EXAFS) spectrum with unparalleled accuracy.122 Figure 3f compares the K-edge EXAFS spectra for water solvating the Na+ion obtained from MB-nrg and experiments. Interestingly, in the case of Na+ solvation, it was found that the MB-nrg potential without 3B corrections produced a solvation structure similar to the classical polarizable potential. EXAFS spectra computed for solutions of halides and other alkali ions using the MB-nrg potentials are presented in several references, demonstrating the applicability of the approach to each system.122,129–131 An MB-nrg model for treating solvated hydrocarbons such as methane, has also been recently reported.142,143 Inspired by both experiments144 and previous DFT-based AIMD simulations145 it was shown that under pressure, methane picks up a dipole moment which leads to its enhanced solubility. In this case, the use of MB-nrg potentials allows for running larger 17 and longer simulations than the far more expensive AIMD approaches permit, offering the possibility of obtaining accurate predictions on the corresponding thermodynamics.143 Reactive Empirical Potentials Although many-body polarizable potentials, such as MB-pol, allow for an extremely accurate characterization of the structural, dynamical and spectroscopic properties of bulk water, they do not allow for bond-breaking and bond-formation. One of the most fundamental processes in aqueous chemistry that requires chemistry to occur in water is the dissociation of water into its constituent ions, proton and hydroxide.146,147 This equilibrium determines the pH of water and has numerous implications for the biochemistry of biomolecules in solution. A lot of our understanding of water dissociation and the diffusion of protons and hydroxide ions in solution has come from molecular simulations that allow for reactive chemistry to occur. In this regard, DFT based AIMD simulations of liquid water and its ionic products have been instrumental over the last three decades in providing a molecular lens into the structure of the excess proton in water. These simulations have guided and motivated state-of-the-art spectroscopy experiments.41,148 As alluded to earlier, for chemical reactions such as those involving the breaking of covalent bonds, the standard DFT simulations are hindered by short timescales and small system sizes. Besides the ML-based force fields designed to overcome these challenges that will be discussed in the next section, there are also empirical potentials that have been developed to study water dissociation. Attempts to construct dissociative water potentials date back to work almost four decades ago by Stillinger and Rahman.149–152 These initial studies laid the groundwork for one of the most popular dissociative water potentials, that describes both intramolecular and intermolecular interactions associated with water and its constituent ions, namely the OSS family.153 The functional forms of these potentials include 2-body radial, 3body angular contributions, and polarizable oxygen ions. The first form of the OSS potential was shown to be rather promising at characterizing potential energy surfaces in protonated 18 clusters. Over the last two decades it has been systematically improved to allow for modeling proton and hydroxide transfer in the bulk.154–156 The left panel of Figure 4a illustrates the mean square displacement (MSD) and, thus, the inferred diffusion constants for the excess proton, hydroxide, and neutral water, obtained from the OSS2 potential by Rasaiah and coworkers.155 It is first important to note that the potential successfully predicts the relative differences in the diffusion constants for the three species, namely that DH+>DOH−>DH2O, consistent with experiments. Instead, the right panel of Figure 4a shows the magnitude of the diffusion constants for the proton and hydroxide, at different temperatures again compared with experimental results. While the trends in the change of the diffusion constants as a function of temperature are consistently reproduced by the model, the differences appear to get more pronounced at larger temperatures. Being able to quantitatively predict these types of dynamical properties of water’s constituent ions at different thermodynamic conditions remains an open challenge. Another family of dissociative potentials that have caught some momentum in the literature are those developed by Garofalini and co-workers to deal with water dissociation in the bulk and near inorganic metal oxide interfaces such as silica.157 Similar to the OSS potential, water dissociation is facilitated by incorporating intramolecular interactions, which encompass 2 and 3-body terms. This approach has been employed in recent studies to investigate the mechanisms involved in water dissociation. Garofalini and co-workers show that liquid water is characterized by a larger fraction of transient autoionization events, similar to previous studies where this was observed upon inclusion of nuclear quantum effects.158 They also point out the challenge in identifying the relevant collective variables that ultimately lead to separation of the hydronium and hydroxide, a topic that will be discussed at greater length later in the review. Over the last decade, Wiedemair and co-workers have extended the original potential proposed by Garofalini, in an effort to improve various dynamical properties by replacing the non-Coulombic Morse-like potential with either a Lennard-Jones or Buckhingham potential, allowing for it to be of greater practical use in other common MD 19 softwares.159 In the last two decades, the Voth group has pioneered the development of reactive potentials based on the multi-state empirical valence bond formalism (MS-EVB). The EVBtype potentials have been typically trained with different levels of DFT simulations and have allowed for examining problems such as proton transfer in bulk liquid water with impressive detail and accuracy.160–165 Since this has already been tackled in several previous reviews,163,166 here we focus on highlighting a specific problem that has been the subject of raging debate and regards the propensity of waters constituent ions for the air-water interface and the apparent negative charge near hydrophobic interfaces.167,168 Figure 4b illustrates potential of mean force (PMF) calculations performed by the Voth group, which show that the excess proton has a slight propensity for the surface of water while the hydroxide ion is repelled.169,170 On the other hand, previous studies using DFT-based AIMD simulations reached different conclusions regarding the propensity of the hydronium and hydroxide ions for the surface of water – the latter, hydroxide ion, is instead found to be weakly attracted while the proton has no affinity for the air-water interface.171,172 Such a dual – acidic or basic – behavior of the surface of water has also recently been the subject of a large-scale QM/MM study, attempting at resolving the controversial finding that the excess proton presents a higher affinity for the surface compared to the hydroxide.173 The origin of this behavior is attributed to differences in the local solvation structures which are challenging to sample with standard DFT based simulations. The Hertzfeld group have also developed LEWIS based potentials, which include valence electrons as semi-classical particles interacting with each other through pairwise potentials.174,175 The LEWIS model tends to predict a much higher affinity of hydroxide ions for the surface of water compared to both MS-EVB and DFT based simulations (see Figure 4c).169,176 Besides, potentials from the Netz group, by including several extra bond-stretching and angle potentials, as well as altering the point charges, have been interfaced with the SPC/E water model. While this does not allow for studying the Grotthuss mechanism, 20 thermodynamic properties such as solvation energies and surface activities are accurately reproduced.177 This is illustrated in Figure 4d which compares the surface activities extracted from experiments, simulations and through PMF calculations, for HCl, NaOH and NaCl. One observes that the proton is surface active, while the hydroxide is not, consistent with the predictions of the PMFs by Voth shown in Figure 4c. Needless to say, it is clear that this is an extremely challenging problem and, depending on the flavor of the underlying potential that is used, can lead to very different qualitative and quantitative conclusions. Finally, another dissociative potential of relevance is the ReaxFF force-field developed by van Duin, Goddard III and co-workers.178–180 This is a bond-order based force field, typically employed for molecular dynamics simulations involving chemical reactions. Conventional force fields face limitations in representing chemical reactions due to their explicit bond definition requirements. In contrast, ReaxFF employs bond orders rather than explicit bonds, enabling continuous bond formation and breaking, making it more suitable for modeling chemical reactions. The first-generation ReaxFF water force field (water-2010) underestimated the bulk density of liquid water by ∼8% at ambient conditions181 (an issue also occurring with many DFT XC functionals), but also incorrectly predicted the order of the diffusion coefficients as: D(H2O) <D(H3O+)<D(OH−).182 The second-generation of ReaxFF force field183(water-2017) was developed by fitting energies and structures to QM data (as in water-2010) and, in addition, by explicitly including both the experimental density of water and the correct diffusion coefficients of H2O, H3O+, OH−in the training procedure. This resolved the aforementioned issues and resulted in a more accurate density of bulk water, as well as the correct order of the diffusion coefficients: D(H2O) <D(OH−)<D(H3O+). This force field has also been shown to describe water dissociation and the Grotthuss mechanism underlying the propagation of protons. However, in comparison to water-2010, the water-2017 force field slightly misjudges the location and intensity of the first peak in the O-O and H-O RDFs, even though they are comparable to what is predicted by common classical water models, such as SPC/E, TIP3P, and TIP4P21 2005.183 It was hypothesized that this was caused by including the density of bulk water in the training data, which might have resulted in an overestimation of the intermolecular interactions. The reactive nature of this force field has allowed it to be used to simulate water dissociation in liquid-solid interfaces.184 (c) (d) (a) (b) Figure 4: Panel (a) shows the mean square displacement (MSD) of water and its constituent ions obtained with the OSS2 potential in the left panel and the estimates of the diffusion constant of the proton and hydroxide at a range of temperatures, in the right panel. This figure is reproduced from Reference 155. Panel (b) shows the potentials of mean force for the excess proton and hydroxide ion to bind to the surface of water obtained using MS-EVB simulations. This figure is reproduced from Reference 169. Panel (c) shows probability distributions of the water (violet), the proton (blue) and hydroxide ion (red) obtained from MD simulations using LEWIS based potentials. This figure is reproduced from Reference 174. Panel (d) compares the surface activities of different salt solutions obtained with three different methods: experiments, equilibrium molecular simulations and free energy calculations (pmf). This figure is reproduced from Reference 177. 22 Aqueous Chemistry with Machine-Learning Force Fields One of the challenges with developing accurate potential energy surfaces (PES) for use in empirical potentials like the ones described earlier is the construction of an appropriate functional form for the interacting particles. The reactive potentials discussed in the previous section can often have very complicated functional forms. This has triggered the use of models for which the knowledge of the functional form is not needed, with Neural-Networks (NN) being a natural choice, as they are universal approximators of any mathematical function.185 Therefore, in principle, by properly optimizing the internal weights of the NN, one can fit any PES as a function of the atomic positions without having to explicitly separate out the reactive or bonded/non-bonded nature of the interactions. The vast majority of NN potentials for water used in molecular dynamics applications builds on the seminal work by Behler and Parrinello186 who developed a generalized NN representation of high-dimensional PES that was trained on DFT data. The reader is referred to several other detailed reviews on the topic.187–189 Here, we outline the essential principles of the method. The idea is to express the total energy of a system as a sum of individual atomic contributions Eiwhich depend on the local chemical environment. Each atom is dressed with its own NN so that the total energy of the system is given by: E=PN i=1 Ei. In order to be physically meaningful, each Eimust be invariant under the permutation of identical atoms in the local environment and under the rotation/translation of their coordinates. However, learning these symmetries (although theoretically possible) would be extremely costly since the algorithm would need a lot of data points during the training. Therefore, it is a common practice to employ some kind of symmetry functions (Gi) that encode the local atomic structure and satisfy these symmetry conditions, providing a direct link between the atomic energies Eiand the local atomic environments. Specifically, in the Behler-Parrinello NN (BPNN) framework, the atomic coordinates are mapped onto a set of twoand three-body symmetry functions. The structure of the atomic NN shown in Figure 1b is inspired by the architecture of 23 artificial neurons which consists of nodes organized in various layers. The symmetry function coordinates described earlier are used as input for the first layer, while the last layer produces as output the atomic energy. The weights of the connecting nodes are optimized to reduce the error of these energies with respect to electronic structure calculations.189 Several versions of neural network potentials of water have been developed over the last decade which began by originally fitting the potential energy surface of the water dimer based on environment-dependent atomic energies and charges.190 Over time, this potential was refined to handle water clusters highlighting the importance of dispersion interactions.191 These NN potentials clearly allow for the simulation of systems over much longer timescales. Naturally, however, the quality of the potential depends on where exactly the training data set comes from. For example, training an NN potential using DFT-based molecular dynamics without dispersion corrections leads to overstructured and glassy water dynamics.192 Figure 5a illustrates a powerful example of using different types of NN potentials to study the behavior of the density maximum of liquid water, as well as the thermodynamics associated with the ice-water equilibrium. Behler and Dellago developed a series of NN potentials with which they demonstrated that the inclusion of dispersion corrections with different DFT functionals, reflects the correct existence of a density maximum.192 In this work, they also computed melting temperatures for systems like the one shown in Figure 5a, consisting of over 2000 water molecules and with numerous simulations on the nanosecond timescale. Table C in Figure 5a shows that the melting temperatures are significantly improved upon the inclusion of dispersion corrections, in comparison with the experiments. Again, these conclusions are based on the possibility of conducting large-scale and long-time (several nanoseconds) simulations with the NN potentials. The use of NN-potentials for water has opened up exciting applications in which they are coupled with path integral molecular dynamics (PIMD). A nice example of this synergy is a recent application involving Ceriotti and Behler that uses a NN-potential of water combined with path integral techniques to predict the thermodynamic properties of liquid water, as 24 well as hexagonal and cubic ice193 By training a NN with a hybrid functional, these simulations rigorously take into account nuclear quantum effects, the disorder of the protons, and anharmonic fluctuations. The upper panel of Figure 5b shows the density isobars computed with the NN for liquid water, hexagonal, and cubic ice. The temperature of maximum density is in perfect agreement with the experiments, while the density isobars for all studied systems are within 3% of the experiments. The lower panel of Figure 5b, instead, shows the coupling of the NN with path integral simulations being used to determine the role of NQEs in controlling the structure of liquid water. The PIMD simulations coupled with the NN show excellent agreement with experiments for all three pair-correlation functions. Furthermore, as pointed out earlier, quantum effects are essential for capturing the delocalization of the proton along the hydrogen bond as seen in the bottom two panels of the RDFs. In recent years, Car and others have extended the scope of standard NN potentials with the development of deep neural network potentials (DNN).194–197 DNN potentials have emerged in order to tackle some critical limitations with the classical NN methodology, including the somewhat ad-hoc choice of symmetry functions. Furthermore, DNNs have demonstrated a substantial enhancement in the efficiency of the learning phase for potential functions. This improvement is achieved through the incorporation of optimal loss functions and local decomposition methods, which enable training DNNs on relatively small systems while maintaining their applicability to much larger systems. It is noteworthy that the computational cost of employing DNNs scales linearly with system size, primarily because they are highly parallelizable due to the earlier mentioned local decomposition approach. Their adoption has expanded the scope of applications in simulating various aqueous systems. Notably, DNNs exhibit remarkable accuracy in reproducing average energy, density, and RDFs across a range of water and ice systems when compared to AIMD simulations.194 Car and co-workers have recently also used the DNN framework to not only learn structural properties, but also the environment-dependent polarizability tensor of water molecules.198 This, in turn, has allowed determining the Raman spectrum of liquid wa25 and products. Mining simulation data Another very active and growing area of research currently is the use of unsupervised learning approaches to characterize and understand patterns that arise in molecular simulations without prior imposition of knowledge often introduced by chemical bias.228 In this context, a wide range of local atomic descriptors are now being used to encode information about atomic environments because they preserve important symmetries. A particularly popular one, that is used in the study of liquid water for example, is the smooth overlap of atomic positions (SOAP) which essentially expands the atomic neighbor density of a chemical species onto a basis of radial basis functions and spherical harmonics.235,236 These types of atomic descriptors are high-dimensional vectors encoding details of the local environment. In order to extract useful and interpretable information from them, one needs to perform some form of dimensionality reduction and subsequently project along the relevant degrees of freedom using clustering. Figure 6d illustrates an example of this approach by Cheng and co-workers: SOAP descriptors of bulk liquid water are built and later used for a principal component analysis.237 These authors also try to relate the fluctuations in liquid water to milestone structures involving different phases of ice. In particular, they demonstrate that one can think about fluctuations in liquid water as transiently forming different local structures resembling phases of ice. Pavan and co-workers have also demonstrated similar ideas, again by taking advantage of the generality of the SOAP descriptors, comparing similarity measures of different empirical potentials of liquid water to phases of ice.238 Recently, our group has also taken important steps in this direction in an effort to understand the thermodynamic landscape of liquid water and how it changes around model hydrophobic polymers, amino acids, peptide groups, and at interfaces.239–244 Specifically, using the SOAP descriptors we have investigated the number of independent degrees of freedom (often referred to as the intrinsic dimension (ID) of a data set) needed to characterize 32 the fluctuations in liquid water at room temperature and, thereafter, we extracted the high dimensional free energies.245 One important observation is that even at the local structure level, for example, between the first and second solvation shells, the ID of the system is quite large. This implies that a correct description of the underlying thermodynamics and dynamics of the system requires looking at many orthogonal degrees of freedom concertedly. Contrary to current descriptions of water in terms of two-state liquid, the left panel of Figure 7 shows that liquid water at room temperature is characterized by a single broad minimum. Furthermore, we do not find any evidence for two states or stable local structures corresponding to low-density (LDL) and high-density liquid (HDL). More recently, we have applied similar techniques to address the question regarding the structure of the proton in liquid water, typically discussed in the literature in terms of a competition between idealized limiting states, namely the Eigen and Zundel.246 Our agnostic approach, instead, shows that the Eigen and Zundel are neither limiting nor stable thermodynamic states. Contrasting the two panels of Figure 7, shows that the simulations with the excess proton (concentrated HCl to be specific) lead to the creation of two additional minima on the free energy landscape. The excess proton in water is best seen as a charged topological defect that strongly perturbs its local environment, leading to an enhancement of the concentration of neutral water defects. 33 (a) (b) (d) (c) Figure 6: Panel (a) shows a water wire connecting the hydronium and hydroxide ion and an example of the water wire compression that leads to concerted proton transfer. This figure was reproduced from Reference 224. Panel (b) shows the decision tree obtained by using a machine learning procedure on ionization trajectories generated from AIMD simulations. As seen the parameter w4corresponds to the water wire distance which is identified as the most important parameter. Others include the tetrahedrality and the local hydrogen bond patterns involving the number of accepting/donating hydrogen bonds of water molecules along the wire. This figure is reproduced from Reference 230. Panel (c) (top panel) shows an illustration of the graph chemical network used to quantify different molecules. The bottom panels compare the free energy surface associated with the pre-biotic chemistry of formamide in the gas phase (left) and in solution (right panel). This figure is reproduced from Reference 232. Panel (d) shows a two-dimensional scatter plot of SOAP descriptors of liquid water using principal component analysis. Also shown are the projections of the points associated with different phases of Ice. This figure is reproduced from Reference 237. 34 Figure 7: The 2-projections using UMAP of the SOAP variables of liquid water at room temperature on the left, and of 2M HCl on the right. This figure is reproduced from Reference 246. Future Perspectives The goal of this review is to give a broad overview of the historical context of studying aqueous chemistry in solution, both from the perspective of the use of DFT-based AIMD simulations as well as modern empirical potentials with a focus on dissociative schemes. This background motivated our discussion on the development of machine-learning potentials and, more generally, data-driven approaches to both model aqueous systems with greater accuracy and sophistication as well as to drive the learning of complex phenomena in these systems. Where possible, we have made an attempt to elucidate critical challenges that have arisen in different types of problems relevant to the physical chemistry and chemical physics of solvation. The growth of machine-learning approaches to study aqueous chemistry, in particular the dissociation of water, offers enormous potential for future applications. Specifically, how these models can be extended to tackle other problems involving salt solutions,247 organic molecules such as amino acids, proteins, and DNA,18 and, finally, processes out of equilibrium such as in external electric fields,66,248–255 remains an open question. The challenge 35 here involves both the generation of accurate data sets for training as well as assessing the transferability of existing models. For example, the exchange of protons in solution is a critical step underlying isotope fractionation of biomolecules in water, which is thought to be catalyzed by negatively charged hydroxide ions.256 However, the underlying mechanisms by which this happens still remain uncharted territory. In the context of externally driven aqueous solutions, large external electric fields, for example, have been shown to enhance autoprotolysis in solution.66,249,250 It is clear that the breadth and depth of computer simulations that have been achieved with standard methods such as AIMD will play a critical role in helping push forward the development and application of machine-learning-based approaches, as recently reported in the literature.257 Another very interesting and challenging area that will certainly benefit from data-driven approaches is excited-state chemistry in solution, which presents many other methodological issues at the moment that are typically absent in the ground-state such as the near-sightedness of the electronic degrees of freedom. The reader is referred to the following review and references therein for more details.258 Besides the use of ML-based potentials to simulate aqueous chemistry with higher accuracy and on larger system sizes, another active area that we believe will continue to grow is the use of both supervised and unsupervised learning to unravel the complexity of chemical processes in aqueous solutions. While these approaches allow an agnostic inference of models based on the underlying structure of the data, chemical and physical interpretability is often a critical missing link. Therefore, attempts to provide physical interpretability, for example, to atomic descriptors used in neural networks (NN), is an area that requires more attention.244 This understanding has important implications on how to translate physics-based interaction models into the structure of NNs. While the focus of this review has been on motivating data-driven potentials relevant for first-principles simulations, there are also other examples of ML being used to improve coarse-grained potentials. The mW model developed by the Molinero group259 despite being 36 a coarse-grained model, outperforms many of the other empirical potentials such as SPC and TIP in predicting thermodynamic properties such as surface-tension. While this model does not capture realistically the dynamics of water, it has been successful at studying phenomena such ice-nucleation and thermodynamics of water under confinement.260–266 More recently, it has also been interfaced to combining it with ions, molecules and polymers to tackle more complex aqueous solutions as well as systems involving interfaces such as membranes.267–271 ML approaches have the potential to also enable coarse-grained models such as mW. 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