Perspective: Atomistic simulations of water and aqueous systems with machine learning potentials.

As the most important solvent, water has been at the center of interest since the advent of computer simulations. While early molecular dynamics and Monte Carlo simulations had to make use of simple model potentials to describe the atomic interactions, accurate ab initio molecular dynamics simulations relying on the first-principles calculation of the energies and forces have opened the way to predictive simulations of aqueous systems. Still, these simulations are very demanding, which prevents the study of complex systems and their properties. Modern machine learning potentials (MLPs) have now reached a mature state, allowing us to overcome these limitations by combining the high accuracy of electronic structure calculations with the efficiency of empirical force fields. In this Perspective, we give a concise overview about the progress made in the simulation of water and aqueous systems employing MLPs, starting from early work on free molecules and clusters via bulk liquid water to electrolyte solutions and solid-liquid interfaces.

Hydrogen atom scattering at the Al2O3(0001) surface: a combined experimental and theoretical study.

Investigating atom-surface interactions is the key to an in-depth understanding of chemical processes at interfaces, which are of central importance in many fields - from heterogeneous catalysis to corrosion. In this work, we present a joint experimental and theoretical effort to gain insights into the atomistic details of hydrogen atom scattering at the α-Al2O3(0001) surface. Surprisingly, this system has been hardly studied to date, although hydrogen atoms as well as α-Al2O3 are omnipresent in catalysis as reactive species and support oxide, respectively. We address this system by performing hydrogen atom beam scattering experiments and molecular dynamics (MD) simulations based on a high-dimensional machine learning potential trained to density functional theory data. Using this combination of methods we are able to probe the properties of the multidimensional potential energy surface governing the scattering process. Specifically, we compare the angular distribution and the kinetic energy loss of the scattered atoms obtained in experiment with a large number of MD trajectories, which, moreover, allow to identify the underlying impact sites at the surface.

How to train a neural network potential.

The introduction of modern Machine Learning Potentials (MLPs) has led to a paradigm change in the development of potential energy surfaces for atomistic simulations. By providing efficient access to energies and forces, they allow us to perform large-scale simulations of extended systems, which are not directly accessible by demanding first-principles methods. In these simulations, MLPs can reach the accuracy of electronic structure calculations, provided that they have been properly trained and validated using a suitable set of reference data. Due to their highly flexible functional form, the construction of MLPs has to be done with great care. In this Tutorial, we describe the necessary key steps for training reliable MLPs, from data generation via training to final validation. The procedure, which is illustrated for the example of a high-dimensional neural network potential, is general and applicable to many types of MLPs.

A new polymorph of white phosphorus at ambient conditions.

Phosphorus exists in several different allotropes: white, red, violet and black. For industrial and academic applications, white phosphorus is the most important. So far, three polymorphs of white phosphorus, all consisting of P4 tetrahedra, have been described. Among these, ß-P4 crystallizes in the space group P1 and γ-P4 in the space group C2/m. α-P4 forms soft plastic crystals with a proposed structure in the cubic space group I43m with the lattice constant a = 18.51 (3) Å, consisting of 58 rotationally disordered tetrahedra and thus is similar to the structure of α-Mn. Here we present a new polymorph, δ-P4. It crystallizes as a sixfold twin with the cell dimensions a = 18.302 (2), b = 18.302 (2), c = 36.441 (3) Šin the space group P212121 with 29 P4 tetrahedra in the asymmetric unit. The arrangement resembles the structure of α-Mn, but δ-P4 differs from α-P4. DFT calculations show δ-P4 to be metastable at a similar energy level to that of γ-P4.

High-Dimensional Neural Network Potentials for Accurate Prediction of Equation of State: A Case Study of Methane.

Machine learning-based interatomic potentials, such as those provided by neural networks, are increasingly important in molecular dynamics simulations. In the present work, we consider the applicability and robustness of machine learning molecular dynamics to predict the equation of state properties of methane by using high-dimensional neural network potentials (HDNNPs). We investigate two different strategies for generating training data: one strategy based upon bulk representations using periodic cells and another strategy based upon clusters of molecules. We assess the accuracy of the trained potentials by predicting the equilibrium mass density for a wide range of thermodynamic conditions to characterize the liquid phase, supercritical fluid, and gas phase, as well as the liquid-vapor coexistence curve. Our results show an excellent agreement with reference phase diagrams, with an average error below ∼2% for all studied phases. Moreover, we confirm the applicability of models trained on cluster data sets for producing accurate and reliable results.

Accurate Fourth-Generation Machine Learning Potentials by Electrostatic Embedding.

In recent years, significant progress has been made in the development of machine learning potentials (MLPs) for atomistic simulations with applications in many fields from chemistry to materials science. While most current MLPs are based on environment-dependent atomic energies, the limitations of this locality approximation can be overcome, e.g., in fourth-generation MLPs, which incorporate long-range electrostatic interactions based on an equilibrated global charge distribution. Apart from the considered interactions, the quality of MLPs crucially depends on the information available about the system, i.e., the descriptors. In this work we show that including─in addition to structural information─the electrostatic potential arising from the charge distribution in the atomic environments significantly improves the quality and transferability of the potentials. Moreover, the extended descriptor allows current limitations of two- and three-body based feature vectors to be overcome regarding artificially degenerate atomic environments. The capabilities of such an electrostatically embedded fourth-generation high-dimensional neural network potential (ee4G-HDNNP), which is further augmented by pairwise interactions, are demonstrated for NaCl as a benchmark system. Employing a data set containing only neutral and negatively charged NaCl clusters, even small energy differences between different cluster geometries can be resolved, and the potential shows an impressive transferability to positively charged clusters as well as the melt.

Machine learning transferable atomic forces for large systems from underconverged molecular fragments.

Machine learning potentials (MLP) enable atomistic simulations with first-principles accuracy at a small fraction of the costs of electronic structure calculations. Most modern MLPs rely on constructing the potential energy, or a major part of it, as a sum of atomic energies, which are given as a function of the local chemical environments up to a cutoff radius. Since analytic forces are readily available, nowadays it is common practice to make use of both, reference energies and forces, for training these MLPs. This can be computationally demanding since often large systems are required to obtain structurally converged reference forces experienced by atoms in realistic condensed phase environments. In this work we show how density-functional theory calculations of molecular fragments, which are too small to provide such structurally converged forces, can be used to learn forces exhibiting excellent transferability to extended systems. The general procedure and the accuracy of the method are illustrated for metal-organic frameworks using second-generation high-dimensional neural network potentials.

High-dimensional neural network potentials for accurate vibrational frequencies: the formic acid dimer benchmark.

In recent years, machine learning potentials (MLP) for atomistic simulations have attracted a lot of attention in chemistry and materials science. Many new approaches have been developed with the primary aim to transfer the accuracy of electronic structure calculations to large condensed systems containing thousands of atoms. In spite of these advances, the reliability of modern MLPs in reproducing the subtle details of the multi-dimensional potential-energy surface is still difficult to assess for such systems. On the other hand, moderately sized systems enabling the application of tools for thorough and systematic quality-control are nowadays rarely investigated. In this work we use benchmark-quality harmonic and anharmonic vibrational frequencies as a sensitive probe for the validation of high-dimensional neural network potentials. For the case of the formic acid dimer, a frequently studied model system for which stringent spectroscopic data became recently available, we show that high-quality frequencies can be obtained from state-of-the-art calculations in excellent agreement with coupled cluster theory and experimental data.

Coupled Cluster Molecular Dynamics of Condensed Phase Systems Enabled by Machine Learning Potentials: Liquid Water Benchmark.

Coupled cluster theory is a general and systematic electronic structure method, but in particular the highly accurate "gold standard" coupled cluster singles, doubles and perturbative triples, CCSD(T), can only be applied to small systems. To overcome this limitation, we introduce a framework to transfer CCSD(T) accuracy of finite molecular clusters to extended condensed phase systems using a high-dimensional neural network potential. This approach, which is automated, allows one to perform high-quality coupled cluster molecular dynamics, CCMD, as we demonstrate for liquid water including nuclear quantum effects. The machine learning strategy is very efficient, generic, can be systematically improved, and is applicable to a variety of complex systems.

A Hessian-based assessment of atomic forces for training machine learning interatomic potentials.

In recent years, many types of machine learning potentials (MLPs) have been introduced, which are able to represent high-dimensional potential-energy surfaces (PESs) with close to first-principles accuracy. Most current MLPs rely on atomic energy contributions given as a function of the local chemical environments. Frequently, in addition to total energies, atomic forces are also used to construct the potentials, as they provide detailed local information about the PES. Since many systems are too large for electronic structure calculations, obtaining reliable reference forces from smaller subsystems, such as molecular fragments or clusters, can substantially simplify the construction of the training sets. Here, we propose a method to determine structurally converged molecular fragments, providing reliable atomic forces based on an analysis of the Hessian. The method, which serves as a locality test and allows us to estimate the importance of long-range interactions, is illustrated for a series of molecular model systems and the metal-organic framework MOF-5 as an example for a complex organic-inorganic hybrid material.

Neural Network Potentials: A Concise Overview of Methods.

In the past two decades, machine learning potentials (MLPs) have reached a level of maturity that now enables applications to large-scale atomistic simulations of a wide range of systems in chemistry, physics, and materials science. Different machine learning algorithms have been used with great success in the construction of these MLPs. In this review, we discuss an important group of MLPs relying on artificial neural networks to establish a mapping from the atomic structure to the potential energy. In spite of this common feature, there are important conceptual differences among MLPs, which concern the dimensionality of the systems, the inclusion of long-range electrostatic interactions, global phenomena like nonlocal charge transfer, and the type of descriptor used to represent the atomic structure, which can be either predefined or learnable. A concise overview is given along with a discussion of the open challenges in the field.

Mechanism of amorphous phase stabilization in ultrathin films of monoatomic phase change material.

Elemental antimony has been recently proposed as a promising material for phase change memories with improved performances with respect to the most used ternary chalcogenide alloys. The compositional simplification prevents reliability problems due to demixing of the alloy during memory operation. This is made possible by the dramatic stabilization of the amorphous phase once Sb is confined in an ultrathin film 3-5 nm thick. In this work, we shed light on the microscopic origin of this effect by means of large scale molecular dynamics simulations based on an interatomic potential generated with a machine learning technique. The simulations suggest that the dramatic reduction of the crystal growth velocity in the film with respect to the bulk is due to the effect of nanoconfinement on the fast ß relaxation dynamics while the slow α relaxation is essentially unaffected.

Four Generations of High-Dimensional Neural Network Potentials.

Since their introduction about 25 years ago, machine learning (ML) potentials have become an important tool in the field of atomistic simulations. After the initial decade, in which neural networks were successfully used to construct potentials for rather small molecular systems, the development of high-dimensional neural network potentials (HDNNPs) in 2007 opened the way for the application of ML potentials in simulations of large systems containing thousands of atoms. To date, many other types of ML potentials have been proposed continuously increasing the range of problems that can be studied. In this review, the methodology of the family of HDNNPs including new recent developments will be discussed using a classification scheme into four generations of potentials, which is also applicable to many other types of ML potentials. The first generation is formed by early neural network potentials designed for low-dimensional systems. High-dimensional neural network potentials established the second generation and are based on three key steps: first, the expression of the total energy as a sum of environment-dependent atomic energy contributions; second, the description of the atomic environments by atom-centered symmetry functions as descriptors fulfilling the requirements of rotational, translational, and permutation invariance; and third, the iterative construction of the reference electronic structure data sets by active learning. In third-generation HDNNPs, in addition, long-range interactions are included employing environment-dependent partial charges expressed by atomic neural networks. In fourth-generation HDNNPs, which are just emerging, in addition, nonlocal phenomena such as long-range charge transfer can be included. The applicability and remaining limitations of HDNNPs are discussed along with an outlook at possible future developments.

A fourth-generation high-dimensional neural network potential with accurate electrostatics including non-local charge transfer.

Machine learning potentials have become an important tool for atomistic simulations in many fields, from chemistry via molecular biology to materials science. Most of the established methods, however, rely on local properties and are thus unable to take global changes in the electronic structure into account, which result from long-range charge transfer or different charge states. In this work we overcome this limitation by introducing a fourth-generation high-dimensional neural network potential that combines a charge equilibration scheme employing environment-dependent atomic electronegativities with accurate atomic energies. The method, which is able to correctly describe global charge distributions in arbitrary systems, yields much improved energies and substantially extends the applicability of modern machine learning potentials. This is demonstrated for a series of systems representing typical scenarios in chemistry and materials science that are incorrectly described by current methods, while the fourth-generation neural network potential is in excellent agreement with electronic structure calculations.

General-Purpose Machine Learning Potentials Capturing Nonlocal Charge Transfer.

The development of first-principles-quality machine learning potentials (MLP) has seen tremendous progress, now enabling computer simulations of complex systems for which sufficiently accurate interatomic potentials have not been available. These advances and the increasing use of MLPs for more and more diverse systems gave rise to new questions regarding their applicability and limitations, which has constantly driven new developments. The resulting MLPs can be classified into several generations depending on the types of systems they are able to describe. First-generation MLPs, as introduced 25 years ago, have been applicable to low-dimensional systems such as small molecules. MLPs became a practical tool for complex systems in chemistry and materials science with the introduction of high-dimensional neural network potentials (HDNNP) in 2007, which represented the first MLP of the second generation. Second-generation MLPs are based on the concept of locality and express the total energy as a sum of environment-dependent atomic energies, which allows applications to very large systems containing thousands of atoms with linearly scaling computational costs. Since second-generation MLPs do not consider interactions beyond the local chemical environments, a natural extension has been the inclusion of long-range interactions without truncation, mainly electrostatics, employing environment-dependent charges establishing the third MLP generation. A variety of second- and, to some extent, also third-generation MLPs are currently the standard methods in ML-based atomistic simulations.In spite of countless successful applications, in recent years it has been recognized that the accuracy of MLPs relying on local atomic energies and charges is still insufficient for systems with long-ranged dependencies in the electronic structure. These can, for instance, result from nonlocal charge transfer or ionization and are omnipresent in many important types of systems and chemical processes such as the protonation and deprotonation of organic and biomolecules, redox reactions, and defects and doping in materials. In all of these situations, small local modifications can change the system globally, resulting in different equilibrium structures, charge distributions, and reactivity. These phenomena cannot be captured by second- and third-generation MLPs. Consequently, the inclusion of nonlocal phenomena has been identified as a next key step in the development of a new fourth generation of MLPs. While a first fourth-generation MLP, the charge equilibration neural network technique (CENT), was introduced in 2015, only very recently have a range of new general-purpose methods applicable to a broad range of physical scenarios emerged. In this Account, we show how fourth-generation HDNNPs can be obtained by combining the concepts of CENT and second-generation HDNNPs. These new MLPs allow for a highly accurate description of systems where nonlocal charge transfer is important.

Insights into lithium manganese oxide-water interfaces using machine learning potentials.

Unraveling the atomistic and the electronic structure of solid-liquid interfaces is the key to the design of new materials for many important applications, from heterogeneous catalysis to battery technology. Density functional theory (DFT) calculations can, in principle, provide a reliable description of such interfaces, but the high computational costs severely restrict the accessible time and length scales. Here, we report machine learning-driven simulations of various interfaces between water and lithium manganese oxide (LixMn2O4), an important electrode material in lithium ion batteries and a catalyst for the oxygen evolution reaction. We employ a high-dimensional neural network potential to compute the energies and forces several orders of magnitude faster than DFT without loss in accuracy. In addition, a high-dimensional neural network for spin prediction is utilized to analyze the electronic structure of the manganese ions. Combining these methods, a series of interfaces is investigated by large-scale molecular dynamics. The simulations allow us to gain insights into a variety of properties, such as the dissociation of water molecules, proton transfer processes, and hydrogen bonds, as well as the geometric and electronic structure of the solid surfaces, including the manganese oxidation state distribution, Jahn-Teller distortions, and electron hopping.

Correction: A critical comparison of neural network potentials for molecular reaction dynamics with exact permutation symmetry.

Correction for 'A critical comparison of neural network potentials for molecular reaction dynamics with exact permutation symmetry' by Jun Li et al., Phys. Chem. Chem. Phys., 2019, 21, 9672-9682, DOI: .

Predicting oxidation and spin states by high-dimensional neural networks: Applications to lithium manganese oxide spinels.

Lithium ion batteries often contain transition metal oxides such as LixMn2O4 (0 ≤ x ≤ 2). Depending on the Li content, different ratios of MnIII to MnIV ions are present. In combination with electron hopping, the Jahn-Teller distortions of the MnIIIO6 octahedra can give rise to complex phenomena such as structural transitions and conductance. While for small model systems oxidation and spin states can be determined using density functional theory (DFT), the investigation of dynamical phenomena by DFT is too demanding. Previously, we have shown that a high-dimensional neural network potential can extend molecular dynamics (MD) simulations of LixMn2O4 to nanosecond time scales, but these simulations did not provide information about the electronic structure. Here, we extend the use of neural networks to the prediction of atomic oxidation and spin states. The resulting high-dimensional neural network is able to predict the spins of the Mn ions with an error of only 0.03 ℏ. We find that the Mn eg electrons are correctly conserved and that the number of Jahn-Teller distorted MnIIIO6 octahedra is predicted precisely for different Li loadings. A charge ordering transition is observed between 280 K and 300 K, which matches resistivity measurements. Moreover, the activation energy of the electron hopping conduction above the phase transition is predicted to be 0.18 eV, deviating only 0.02 eV from experiment. This work demonstrates that machine learning is able to provide an accurate representation of both the geometric and the electronic structure dynamics of LixMn2O4 on time and length scales that are not accessible by ab initio MD.

An experimentally validated neural-network potential energy surface for H-atom on free-standing graphene in full dimensionality.

We present a first principles-quality potential energy surface (PES) describing the inter-atomic forces for hydrogen atoms interacting with free-standing graphene. The PES is a high-dimensional neural network potential that has been parameterized to 75 945 data points computed with density-functional theory employing the PBE-D2 functional. Improving over a previously published PES [Jiang et al., Science, 2019, 364, 379], this neural network exhibits a realistic physisorption well and achieves a 10-fold reduction in the RMS fitting error, which is 0.6 meV per atom. The chemisorption barrier is 172 meV, which is lower than that of the REBO-EMFT PES (260 meV). We used this PES to calculate about 1.5 million classical trajectories with carefully selected initial conditions to allow for direct comparison to results of H- and D-atom scattering experiments performed at incidence translational energy of 1.9 eV and a surface temperature of 300 K. The theoretically predicted scattering angular and energy loss distributions are in good agreement with experiment, despite the fact that the experiments employed graphene grown on Pt(111). Compared to previous calculations, the agreement with experiments is improved. The remaining discrepancies between experiment and theory are likely due to the influence of the Pt substrate only present in the experiment.

Insights into Water Permeation through hBN Nanocapillaries by Ab Initio Machine Learning Molecular Dynamics Simulations.

Water permeation between stacked layers of hBN sheets forming 2D nanochannels is investigated using large-scale ab initio-quality molecular dynamics simulations. A high-dimensional neural network potential trained on density-functional theory calculations is employed. We simulate water in van der Waals nanocapillaries and study the impact of nanometric confinement on the structure and dynamics of water using both equilibrium and nonequilibrium methods. At an interlayer distance of 10.2 Å confinement induces a first-order phase transition resulting in a well-defined AA-stacked bilayer of hexagonal ice. In contrast, for h < 9 Å, the 2D water monolayer consists of a mixture of different locally ordered patterns of squares, pentagons, and hexagons. We found a significant change in the transport properties of confined water, particularly for monolayer water where the water-solid friction coefficient decreases to half and the diffusion coefficient increases by a factor of 4 as compared to bulk water. Accordingly, the slip-velocity is found to increase under confinement and we found that the overall permeation is dominated by monolayer water adjacent to the hBN membranes at extreme confinements. We conclude that monolayer water in addition to bilayer ice has a major contribution to water transport through 2D nanochannels.