Whereas the interactions between water molecules are dominated by strongly directional hydrogen bonds (HBs), it was recently proposed that relatively weak, isotropic van der Waals (vdW) forces are essential for understanding the properties of liquid water and ice. This insight was derived from ab initio computer simulations, which provide an unbiased description of water at the atomic level and yield information on the underlying molecular forces. However, the high computational cost of such simulations prevents the systematic investigation of the influence of vdW forces on the thermodynamic anomalies of water. Here, we develop efficient ab initio-quality neural network potentials and use them to demonstrate that vdW interactions are crucial for the formation of water's density maximum and its negative volume of melting. Both phenomena can be explained by the flexibility of the HB network, which is the result of a delicate balance of weak vdW forces, causing, e.g., a pronounced expansion of the second solvation shell upon cooling that induces the density maximum.water structure | van der Waals interactions | neural network potentials | ab initio liquid water | density-functional theory W ater is an exceptional liquid exhibiting several anomalies, of which the density maximum at 4°C is the most prominent (1). Together with the negative volume of melting, it is responsible for the fact that water freezes from the top down and ice floats on water. The unusual behavior of water can be directly related to its ability to form hydrogen bonds (HBs) which are of strongly directional nature and determine the microscopic structure of water (2, 3). To investigate the anomalies of water at the molecular level, atomistic computer simulations have become an essential tool. Important contributions have been made by simulations using simple empirical water models (3-8).Simulations based on ab initio molecular dynamics (AIMD) (9-11) allow determination of the properties of water with high predictive power and enable a detailed analysis of their underlying microscopic mechanisms. In contrast to empirical water models (5), which depend on experimental data resulting in a limited transferability, in AIMD the atomic forces that govern the molecular dynamics are obtained directly from quantum mechanics. Although this approach is in principle exact [in combination with methods that account for the quantum nature of the nuclei (12, 13)], ab initio simulations of condensed matter systems are feasible only if approximate but efficient methods such as density-functional theory (DFT) are used. Even then, however, simulations are restricted to short times and small systems. AIMD simulations have been used to a limited extent to investigate the phase behavior of water, for instance by estimating melting temperatures (14, 15) and vapor-liquid coexistence curves (16, 17). However, many fundamental thermodynamic properties of water have not been evaluated to date. To circumvent the limitations of on-the-fly AIMD, various efficient water potentia...

Neural networks and other machine learning approaches have been successfully used to accurately represent atomic interaction potentials derived from computationally demanding electronic structure calculations. Due to their low computational cost, such representations open the possibility for large scale reactive molecular dynamics simulations of processes with bonding situations that cannot be described accurately with traditional empirical force fields. Here, we present a library of functions developed for the implementation of neural network potentials. Written in C++, this library incorporates several strategies resulting in a very high efficiency of neural network potential-energy and force evaluations. Based on this library, we have developed an implementation of the neural network potential within the molecular dynamics package LAMMPS and demonstrate its performance using liquid water as a test system.

Over the past years high-dimensional neural network potentials (HDNNPs), fitted to accurately reproduce ab initio potential energy surfaces, have become a powerful tool in chemistry, physics and materials science. Here, we focus on the training of the neural networks that lies at the heart of the HDNNP method. We present an efficient approach for optimizing the weight parameters of the neural network via multistream Kalman filtering, using potential energies and forces as reference data. In this procedure, the choice of the free parameters of the Kalman filter can have a significant impact on the fit quality. Carrying out a large parameter study, we determine optimal settings and demonstrate how to optimize training results of HDNNPs. Moreover, we illustrate our HDNNP training approach by revisiting previously presented fits for water and developing a new potential for copper sulfide. This material, accessible in computer simulations so far only via first-principles methods, forms a particularly complex solid structure at low temperatures and undergoes a phase transition to a superionic state upon heating. Analyzing MD simulations carried out with the Cu2S HDNNP, we confirm that the underlying ab initio reference method indeed reproduces this behavior.

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The prediction of chemical properties using machine learning techniques calls for a set of appropriate descriptors that accurately describe atomic and, on a larger scale, molecular environments. A mapping of conformational information on a space spanned by atom-centred symmetry functions (SF) has become a standard technique for energy and force predictions using high-dimensional neural network potentials (HDNNP). An appropriate choice of SFs is particularly crucial for accurate force predictions. Established atom-centred SFs, however, are limited in their flexibility, since their functional form restricts the angular domain that can be sampled without introducing problematic derivative discontinuities. Here, we introduce a class of atom-centred SFs based on polynomials with compact support called polynomial symmetry functions (PSF), which enable a free choice of both, the angular and the radial domain covered. We demonstrate that the accuracy of PSFs is either on par or considerably better than that of conventional, atom-centred SFs. In particular, a generic set of PSFs with an intuitive choice of the angular domain inspired by organic chemistry considerably improves prediction accuracy for organic molecules in the gaseous and liquid phase, with reductions in force prediction errors over a test set approaching 50% for certain systems. Contrary to established atom-centred SFs, computation of PSF does not involve any exponentials, and their intrinsic compact support supersedes use of separate cutoff functions, facilitating the choice of their free parameters. Most importantly, the number of floating point operations required to compute polynomial SFs introduced here is considerably lower than that of other state-of-the-art SFs, enabling their efficient implementation without the need of highly optimised code structures or caching, with speedups with respect to other state-of-the-art SFs reaching a factor of 4.5 to 5. This low-effort performance benefit substantially simplifies their use in new programs and emerging platforms such as graphical processing units. Overall, polynomial SFs with compact support improve accuracy of both, energy and force predictions with HDNNPs while enabling significant speedups compared to their well-established counterparts.