Accelerating Fourth-Generation Machine Learning Potentials Using Quasi-Linear Scaling Particle Mesh Charge Equilibration.

Machine learning potentials (MLPs) have revolutionized the field of atomistic simulations by describing atomic interactions with the accuracy of electronic structure methods at a small fraction of the cost. Most current MLPs construct the energy of a system as a sum of atomic energies, which depend on information about the atomic environments provided in the form of predefined or learnable feature vectors. If, in addition, nonlocal phenomena like long-range charge transfer are important, fourth-generation MLPs need to be used, which include a charge equilibration (Qeq) step to take the global structure of the system into account. This Qeq can significantly increase the computational cost and thus can become a computational bottleneck for large systems. In this Article, we present a highly efficient formulation of Qeq that does not require the explicit computation of the Coulomb matrix elements, resulting in a quasi-linear scaling method. Moreover, our approach also allows for the efficient calculation of energy derivatives, which explicitly consider the global structure-dependence of the atomic charges as obtained from Qeq. Due to its generality, the method is not restricted to MLPs and can also be applied within a variety of other force fields.

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.

Principles of isomer stability in small clusters.

In this work we study isomers of several representative small clusters to find principles for their stability. Our conclusions about the principles underlying the structure of clusters are based on a huge database of 44 000 isomers generated for 58 different clusters on the density functional theory level by Minima Hopping. We explore the potential energy surface of small neutral, anionic and cationic isomers, moving left to right across the third period of the periodic table and varying the number of atoms n and the cluster charge state q (X q n , with X = {Na, Mg, Al, Si, Ge}, q = -1, 0, 1, 2). We use structural descriptors such as bond lengths and atomic coordination numbers, the surface to volume ratios and the shape factor as well as electronic descriptors such as shell filling and hardness to detect correlations with the stability of clusters. The isomers of metallic clusters are found to be structure seekers with a strong tendency to adopt compact shapes. However certain numbers of atoms can suppress the formation of nearly spherical metallic clusters. Small non-metallic clusters typically also do not adopt compact spherical shapes for their lowest energy structures. In both cases spherical jellium models are not any more applicable. Nevertheless for many structures, that frequently have a high degree of symmetry, the Kohn-Sham eigenvalues are bunched into shells and if the available electrons can completely fill such shells, a particularly stable structure can result. We call such a cluster whose shape gives rise to shells that can be completely filled by the number of available electrons an optimally matched cluster, since both the structure and the number of electrons must be special and match. In this way we can also explain the stability trends for covalent silicon and germanium cluster isomers, whose stability was previously explained by the presence of certain structural motifs. Thus we propose a unified framework to explain trends in the stability of isomers and to predict their structure for a wide range of small clusters.

Experimental absence of the non-perovskite ground state phases of MaPbI3 explained by a Funnel Hopping Monte Carlo study based on a neural network potential.

Methylammonium lead iodide is a material known for its exceptional opto-electronic properties that make it a promising candidate for many high performance applications, such as light emitting diodes or solar cells. A recent computational structure search revealed two previously unknown non-perovskite polymorphs, that are lower in energy than the experimentally observed perovskite phases. To investigate the elusiveness of the non-perovskite phases in experimental studies, we extended our Funnel Hopping Monte Carlo (FHMC) method to periodic systems and performed extensive MC simulations driven by a machine learned potential. FHMC simulations that also include these newly discovered non-perovskite phases show that above temperatures of 200 K the perovskite phases are thermodynamically preferred. A comparison with the quasi-harmonic approximation highlights the importance of anharmonic effects captured by FHMC.

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.

Funnel hopping Monte Carlo: An efficient method to overcome broken ergodicity.

Monte Carlo simulations are a powerful tool to investigate the thermodynamic properties of atomic systems. In practice, however, sampling of the complete configuration space is often hindered by high energy barriers between different regions of configuration space, which can make ergodic sampling completely infeasible within accessible simulation times. Although several extensions to the conventional Monte Carlo scheme have been developed, which enable the treatment of such systems, these extensions often entail substantial computational cost or rely on the harmonic approximation. In this work, we propose an exact method called Funnel Hopping Monte Carlo (FHMC) that is inspired by the ideas of smart darting but is more efficient. Gaussian mixtures are used to approximate the Boltzmann distribution around local energy minima, which are then used to propose high quality Monte Carlo moves that enable the Monte Carlo simulation to directly jump between different funnels. We demonstrate the method's performance on the example of the 38 as well as the 75 atom Lennard-Jones clusters, which are well known for their double funnel energy landscapes that prevent ergodic sampling with conventional Monte Carlo simulations. By integrating FHMC into the parallel tempering scheme, we were able to reduce the number of steps required significantly until convergence of the simulation.