Metrics for measuring distances in configuration spaces

Accelerating Structural Optimization through Fingerprinting Space Integration on the Potential Energy Surface

Structural optimization has been a crucial component in computational materials research, and structure predictions have relied heavily on this technique, in particular. In this study, we introduce a novel method that enhances the efficiency of local optimization by integrating extra fingerprint space into the optimization process. Our approach utilizes a mixed energy concept in the hyper potential energy surface (PES), combining real energy and a newly introduced fingerprint energy derived from the symmetry of the local atomic environment. This method strategically guides the optimization process toward high-symmetry, low-energy structures by leveraging the intrinsic symmetry of the atomic configurations. The effectiveness of our approach was demonstrated through structural optimizations of silicon, silicon carbide, and Lennard-Jones cluster systems. Our results show that the fingerprint space biasing technique significantly enhances the performance and probability of discovering energetically favorable, high-symmetry structures as compared to conventional optimizations. The proposed method is anticipated to streamline the search for new materials and facilitate the discovery of novel energetically favorable configurations.

Crystal-Structure Matches in Solid-Solid Phase Transitions

The exploration of solid-solid phase transition suffers from the uncertainty of how atoms in two crystal structures match. We devised a theoretical framework to describe and classify crystal-structure matches (CSM). Such description fully exploits the translational and rotational symmetries and is independent of the choice of supercells. This is enabled by the use of the Hermite normal form, an analog of reduced echelon form for integer matrices. With its help, exhausting all CSMs is made possible, which goes beyond the conventional optimization schemes. In an example study of the martensitic transformation of steel, our enumeration algorithm finds many candidate CSMs with lower strains than known mechanisms. Two long-sought CSMs accounting for the most commonly observed Kurdjumov-Sachs orientation relationship and the Nishiyama-Wassermann orientation relationship are unveiled. Given the comprehensiveness and efficiency, our enumeration scheme provide a promising strategy for solid-solid phase transition mechanism research.

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.

MAGUS: machine learning and graph theory assisted universal structure searcher

Crystal structure predictions based on first-principles calculations have gained great success in materials science and solid state physics. However, the remaining challenges still limit their applications in systems with a large number of atoms, especially the complexity of conformational space and the cost of local optimizations for big systems. Here, we introduce a crystal structure prediction method, MAGUS, based on the evolutionary algorithm, which addresses the above challenges with machine learning and graph theory. Techniques used in the program are summarized in detail and benchmark tests are provided. With intensive tests, we demonstrate that on-the-fly machine-learning potentials can be used to significantly reduce the number of expensive first-principles calculations, and the crystal decomposition based on graph theory can efficiently decrease the required configurations in order to find the target structures. We also summarized the representative applications of this method on several research topics, including unexpected compounds in the interior of planets and their exotic states at high pressure and high temperature (superionic, plastic, partially diffusive state, etc.); new functional materials (superhard, high-energy-density, superconducting, photoelectric materials), etc. These successful applications demonstrated that MAGUS code can help to accelerate the discovery of interesting materials and phenomena, as well as the significant value of crystal structure predictions in general.

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.

Electronic Descriptors for Supervised Spectroscopic Predictions

Spectroscopic properties of molecules hold great importance for the description of the molecular response under the effect of UV/vis electromagnetic radiation. Computationally expensive ab initio (e.g., MultiConfigurational SCF, Coupled Cluster) or TDDFT methods are commonly used by the quantum chemistry community to compute these properties. In this work, we propose a (supervised) Machine Learning approach to model the absorption spectra of organic molecules. Several supervised ML methods have been tested such as Kernel Ridge Regression (KRR), Multiperceptron Neural Networs (MLP), and Convolutional Neural Networks. [Ramakrishnan et al. J. Chem. Phys. 2015, 143, 084111. Ghosh et al. Adv. Sci. 2019, 6, 1801367.] The use of only geometrical-atomic number descriptors (e.g., Coulomb Matrix) proved to be insufficient for an accurate training. [Ramakrishnan et al. J. Chem. Phys. 2015, 143, 084111.] Inspired by the TDDFT theory, we propose to use a set of electronic descriptors obtained from low-cost DFT methods: orbital energy differences (Δϵ_ia = ϵ_a - ϵ_i), transition dipole moment between occupied and unoccupied Kohn-Sham orbitals (⟨ϕ_i|r|ϕ_a⟩), and when relevant, charge-transfer character of monoexcitations (R_ia). We demonstrate that with these electronic descriptors and the use of Neural Networks we can predict not only a density of excited states but also get a very good estimation of the absorption spectrum and charge-transfer character of the electronic excited states, reaching results close to chemical accuracy (∼2 kcal/mol or ∼0.1 eV).

A Practical Algorithm to Solve the Near-Congruence Problem for Rigid Molecules and Clusters

We present an improved algorithm to solve the near-congruence problem for rigid molecules and clusters based on the iterative application of assignment and alignment steps with biased Euclidean costs. The algorithm is formulated as a quasi-local optimization procedure with each optimization step involving a linear assignment (LAP) and a singular value decomposition (SVD). The efficiency of the algorithm is increased by up to 5 orders of magnitude with respect to the original unbiased noniterative method and can be applied to systems with hundreds or thousands of atoms, outperforming all state-of-the-art methods published so far in the literature. The Fortran implementation of the algorithm is available as an open source library (https://github.com/qcuaeh/molalignlib) and is suitable to be used in global optimization methods for the identification of local minima or basins.

Progressive alignment of crystals: reproducible and efficient assessment of crystal structure similarity

During in silico crystal structure prediction of organic molecules, millions of candidate structures are often generated. These candidates must be compared to remove duplicates prior to further analysis (e.g. optimization with electronic structure methods) and ultimately compared with structures determined experimentally. The agreement of predicted and experimental structures forms the basis of evaluating the results from the Cambridge Crystallographic Data Centre (CCDC) blind assessment of crystal structure prediction, which further motivates the pursuit of rigorous alignments. Evaluating crystal structure packings using coordinate root-mean-square deviation (RMSD) for N molecules (or N asymmetric units) in a reproducible manner requires metrics to describe the shape of the compared molecular clusters to account for alternative approaches used to prioritize selection of molecules. Described here is a flexible algorithm called Progressive Alignment of Crystals (PAC) to evaluate crystal packing similarity using coordinate RMSD and introducing the radius of gyration (R _g) as a metric to quantify the shape of the superimposed clusters. It is shown that the absence of metrics to describe cluster shape adds ambiguity to the results of the CCDC blind assessments because it is not possible to determine whether the superposition algorithm has prioritized tightly packed molecular clusters (i.e. to minimize R _g) or prioritized reduced RMSD (i.e. via possibly elongated clusters with relatively larger R _g). For example, it is shown that when the PAC algorithm described here uses single linkage to prioritize molecules for inclusion in the superimposed clusters, the results are nearly identical to those calculated by the widely used program COMPACK. However, the lower R _g values obtained by the use of average linkage are favored for molecule prioritization because the resulting RMSDs more equally reflect the importance of packing along each dimension. It is shown that the PAC algorithm is faster than COMPACK when using a single process and its utility for biomolecular crystals is demonstrated. Finally, parallel scaling up to 64 processes in the open-source code Force Field X is presented.

Five-fold symmetry in Au-Si metallic glass

The first metallic glass of Au-Si alloy for over half a century has been discovered, but its atomic structure is still puzzling. Herein, Au₈Si dodecahedrons with local five-fold symmetry are revealed as building blocks in Au-Si metallic glass, and the interconnection modes of Au₈Si dodecahedrons determine the medium-range order. With dimensionality reduction, the surface ordering is attributed to the motif transformation of Au₈Si dodecahedrons into planar Au₅Si pyramids with five-fold symmetry, and thus the self-assembly of Au₅Si pyramids leads to the formation of the ordered Au₂Si monolayer with the lowest energy. Furthermore, structural similarity analysis is performed to unveil the physical origin of the structural characteristics in different dimensions. The amorphism of Au-Si is due to the smooth energy landscape around the global minimum, while the ordered surface structure occurs due to the steep energy landscape.

Identifying molecular structural features by pattern recognition methods

Identification of molecular structural features is a central part of computational chemistry. It would be beneficial if pattern recognition techniques could be incorporated to facilitate the identification. Currently, the quantification of the structural dissimilarity is mainly carried out by root-mean-square-deviation (RMSD) calculations such as in molecular dynamics simulations. However, the RMSD calculation underperforms for large molecules, showing the so-called “curse of dimensionality” problem. Also, it requires consistent ordering of atoms in two comparing structures, which needs nontrivial effort to fulfill. In this work, we propose to take advantage of the point cloud recognition using convex hulls as the basis to recognize molecular structural features. Two advantages of the method can be highlighted. First, the dimension of the input data structure is largely reduced from the number of atoms of molecules to the number of atoms of convex hulls. Therefore, the dimensionality curse problem is avoided, and the atom ordering process is saved. Second, the construction of convex hulls can be used to define new molecular descriptors, such as the contact area of molecular interactions. These new molecular descriptors have different properties from existing ones, therefore they are expected to exhibit different behaviors for certain machine learning studies. Several illustrative applications have been carried out, which provide promising results for structure–activity studies.

Identification of molecular structural features by point clouds and convex hulls.

A DFT study of Se n Te n clusters

First principles calculations have been performed to study the characteristic properties of Se _n Te _n (n = 5-10) clusters. The present study reveals that the properties of these small clusters are consistent with the properties of Se-Te glassy systems. Several hundred equilibrium structures obtained from a genetic algorithm based USPEX code are relaxed to their minimum energy using the VASP code. Most of the lowest energy buckled ring-like structures are formed from Se-Te heteropolar bonds. Detailed structural analysis and distance energy plots unveil that many equilibrium structures are close in energy to their global minimum. The computed Raman and IR spectra show the dominance of Se-Te heteropolar bonds, consistent with earlier simulation and experimental findings in Se_1-x Te _x glass materials. Low frequency vibrational modes observed in small clusters are characteristic features of amorphous materials. Non-bonding orbitals (lone pair) are observed in the HOMO, whereas the LUMO is formed from purely antibonding orbitals. The dielectric functions corroborate the bonding mechanism and slightly polar nature of Se _n Te _n clusters. The energy loss and absorption coefficient indicate the presence of π-plasmons in the UV-visible region. Furthermore, it is ascertained that the use of a hybrid functional (B3LYP) does not affect the properties of small clusters appreciably, except causing a blue shift in the optical spectra. Hence, we find that the small clusters have bearing on the formation of glassy Se-Te systems.

Inclusion of More Physics Leads to Less Data: Learning the Interaction Energy as a Function of Electron Deformation Density with Limited Training Data

Machine learning (ML) approaches to predicting quantum mechanical (QM) properties have made great strides toward achieving the computational chemist's holy grail of structure-based property prediction. In contrast to direct ML methods, which encode a molecule with only structural information, in this work, we show that QM descriptors improve ML predictions of dimer interaction energy, both in terms of accuracy and data efficiency, by incorporating electronic information into the descriptor. We present the electron deformation density interaction energy machine learning (EDDIE-ML) model, which predicts the interaction energy as a function of Hartree-Fock electron deformation density. We compare its performance with leading direct ML schemes and modern DFT methods for the prediction of interaction energies for dimers of varying charge type, size, and intermolecular separation. Under a low-data regime, EDDIE-ML outperforms other direct ML schemes and is the only model readily transferrable to larger, more complex systems including base pair trimers and porous cages. The underlying physical connection between the density and interaction energy enables EDDIE-ML to reach an accuracy comparable to modern DFT functionals in fewer training data points compared to other ML methods.

IRA: A Shape Matching Approach for Recognition and Comparison of Generic Atomic Patterns

We propose a versatile, parameter-less approach for solving the shape matching problem, specifically in the context of atomic structures when atomic assignments are not known a priori. The algorithm Iteratively suggests Rotated atom-centered reference frames and Assignments (iterative rotations and assignments (IRA)). The frame for which a permutationally invariant set-set distance, namely, the Hausdorff distance, returns a minimal value is chosen as the solution of the matching problem. IRA is able to find rigid rotations, reflections, translations, and permutations between structures with different numbers of atoms, for any atomic arrangement and pattern, periodic or not. When distortions are present between the structures, optimal rotation and translation are found by further applying a standard singular value decomposition-based method. To compute the atomic assignments under the one-to-one assignment constraint, we develop our own algorithm, constrained shortest distance assignments (CShDA). The overall approach is extensively tested on several structures, including distorted structural fragments. The efficiency of the proposed algorithm is shown as a benchmark comparison against two other shape matching algorithms. We discuss the use of our approach for the identification and comparison of structures and structural fragments through two examples: a replica-exchange trajectory of a cyanine molecule, in which we show how our approach could aid the exploration of relevant collective coordinates for clustering the data, and a SiO₂ amorphous model, in which we compute distortion scores, and compare them with a classical strain-based potential. The source code and benchmark data are available at https://github.com/mammasmias/IterativeRotationsAssignments.

Visualization and Quantification of Geometric Diversity in Metal-Organic Frameworks

With ever-growing numbers of metal-organic framework (MOF) materials being reported, new computational approaches are required for a quantitative understanding of structure-property correlations in MOFs. Here, we show how structural coarse-graining and embedding ("unsupervised learning") schemes can together give new insights into the geometric diversity of MOF structures. Based on a curated data set of 1262 reported experimental structures, we automatically generate coarse-grained and rescaled representations which we couple to a kernel-based similarity metric and to widely used embedding schemes. This approach allows us to visualize the breadth of geometric diversity within individual topologies and to quantify the distributions of local and global similarities across the structural space of MOFs. The methodology is implemented in an openly available Python package and is expected to be useful in future high-throughput studies.

Molecular structure recognition by blob detection

Molecular structure recognition is fundamental in computational chemistry. The most common approach is to calculate the root mean square deviation (RMSD) between two sets of molecular coordinates. However, this method does not perform well for large molecules. In this work, a new method is proposed for structure comparison. Blob detection is used for recognizing structural features. Fragmentation of molecules is proposed as the pre-treatment. Mapping between blobs and atoms is developed as the post-treatment. A set of key parameters important for blob detections are determined. The dissimilarity is quantified by calculating the Euclidean metric of the blob vectors. The overall algorithm is found to be accurate to distinguish structural dissimilarity. The method has potential to be combined with other pattern recognition techniques for new chemistry discoveries.

Molecular structure recognition is fundamental in computational chemistry.

Combining Machine Learning and Computational Chemistry for Predictive Insights Into Chemical Systems

Machine learning models are poised to make a transformative impact on chemical sciences by dramatically accelerating computational algorithms and amplifying insights available from computational chemistry methods. However, achieving this requires a confluence and coaction of expertise in computer science and physical sciences. This Review is written for new and experienced researchers working at the intersection of both fields. We first provide concise tutorials of computational chemistry and machine learning methods, showing how insights involving both can be achieved. We follow with a critical review of noteworthy applications that demonstrate how computational chemistry and machine learning can be used together to provide insightful (and useful) predictions in molecular and materials modeling, retrosyntheses, catalysis, and drug design.

Physics-Inspired Structural Representations for Molecules and Materials

The first step in the construction of a regression model or a data-driven analysis, aiming to predict or elucidate the relationship between the atomic-scale structure of matter and its properties, involves transforming the Cartesian coordinates of the atoms into a suitable representation. The development of atomic-scale representations has played, and continues to play, a central role in the success of machine-learning methods for chemistry and materials science. This review summarizes the current understanding of the nature and characteristics of the most commonly used structural and chemical descriptions of atomistic structures, highlighting the deep underlying connections between different frameworks and the ideas that lead to computationally efficient and universally applicable models. It emphasizes the link between properties, structures, their physical chemistry, and their mathematical description, provides examples of recent applications to a diverse set of chemical and materials science problems, and outlines the open questions and the most promising research directions in the field.

Role of atomicity in the oxygen reduction reaction activity of platinum sub nanometer clusters: A global optimization study

Metal nanoclusters are an important class of materials for catalytic applications. Sub nanometer clusters are relatively less explored for their catalytic activity on account of undercoordinated surface structure. Taking this into account, we studied platinum-based sub nanometer clusters for their catalytic activity for oxygen reduction reaction (ORR). A comprehensive analysis with global optimization is carried out for structural prediction of the platinum clusters. The energetic and electronic properties of interactions of clusters with reaction intermediates are investigated. The role of structural sensitivity in the dynamics of clusters is unraveled, and unique intermediate specific interactions are identified. ORR energetics is examined, and exceptional activity for sub nanometer clusters are observed. An inverse size versus activity relationship is identified, challenging the conventional trends followed by larger nanoclusters. The principal role of atomicity in governing the catalytic activity of nanoclusters is illustrated. The structural norms governing the sub nanometer cluster activity are shown to be markedly different from larger nanoclusters.

How Many Isomers Do Metallic Clusters Have? Case of Magnesium Clusters of up to 55 Atoms

About 9000 structures of magnesium clusters Mg_n (n = 2-13) generated via different methods were optimized at the DFT levels in order to estimate the number of all possible stable structures that can exist for the given cluster size (∼820,000 PES points were explored in total). It was found that the number of possible cluster isomers N quickly grows with a number of atoms n; however, it is significantly lower than the number of possible nonisomorphic graph structures, which can be drawn for the given n. At the DFT potential energy surface, we found only 543 local minima corresponding to the isomers of Mg₂-Mg₁₃. The number of isomers obtained in the DFT optimizations grows with n approximately as n⁴, whereas the N values extrapolated to the infinite generation process grow as n⁸. The cluster geometries obtained from the global DFT optimization were then used to adjust two empirical potentials of Gupta type (GP) and modified Sutton-Chen type (SCG3) describing the interactions between the magnesium atoms. Using these potentials, the extensive sets of structures Mg₂-Mg₅₅ (up to 30,000 clusters for each n) were optimized to obtain the dependence of the cluster isomer count on n in the continuous range of n = 2-30 and for selected n up to n = 55. It was found that the SCG3 potential, which is closer to the DFT results, gives a number of possible isomers growing as approximately n^8.9, whereas GP potential results in the n^4.3 dependence.

A bin and hash method for analyzing reference data and descriptors in machine learning potentials

In recent years the development of machine learning potentials (MLPs) has become a very active field of research. Numerous approaches have been proposed, which allow one to perform extended simulations of large systems at a small fraction of the computational costs of electronic structure calculations. The key to the success of modern MLPs is the close-to first principles quality description of the atomic interactions. This accuracy is reached by using very flexible functional forms in combination with high-level reference data from electronic structure calculations. These data sets can include up to hundreds of thousands of structures covering millions of atomic environments to ensure that all relevant features of the potential energy surface are well represented. The handling of such large data sets is nowadays becoming one of the main challenges in the construction of MLPs. In this paper we present a method, the bin-and-hash (BAH) algorithm, to overcome this problem by enabling the efficient identification and comparison of large numbers of multidimensional vectors. Such vectors emerge in multiple contexts in the construction of MLPs. Examples are the comparison of local atomic environments to identify and avoid unnecessary redundant information in the reference data sets that is costly in terms of both the electronic structure calculations as well as the training process, the assessment of the quality of the descriptors used as structural fingerprints in many types of MLPs, and the detection of possibly unreliable data points. The BAH algorithm is illustrated for the example of high-dimensional neural network potentials using atom-centered symmetry functions for the geometrical description of the atomic environments, but the method is general and can be combined with any current type of MLP.

Machine learning-accelerated quantum mechanics-based atomistic simulations for industrial applications

Atomistic simulations have become an invaluable tool for industrial applications ranging from the optimization of protein-ligand interactions for drug discovery to the design of new materials for energy applications. Here we review recent advances in the use of machine learning (ML) methods for accelerated simulations based on a quantum mechanical (QM) description of the system. We show how recent progress in ML methods has dramatically extended the applicability range of conventional QM-based simulations, allowing to calculate industrially relevant properties with enhanced accuracy, at reduced computational cost, and for length and time scales that would have otherwise not been accessible. We illustrate the benefits of ML-accelerated atomistic simulations for industrial R&D processes by showcasing relevant applications from two very different areas, drug discovery (pharmaceuticals) and energy materials. Writing from the perspective of both a molecular and a materials modeling scientist, this review aims to provide a unified picture of the impact of ML-accelerated atomistic simulations on the pharmaceutical, chemical, and materials industries and gives an outlook on the exciting opportunities that could emerge in the future.

Fingerprint-Based Detection of Non-Local Effects in the Electronic Structure of a Simple Single Component Covalent System

Using fingerprints used mainly in machine learning schemes of the potential energy surface, we detect in a fully algorithmic way long range effects on local physical properties in a simple covalent system of carbon atoms. The fact that these long range effects exist for many configurations implies that atomistic simulation methods, such as force fields or modern machine learning schemes, that are based on locality assumptions, are limited in accuracy. We show that the basic driving mechanism for the long range effects is charge transfer. If the charge transfer is known, locality can be recovered for certain quantities such as the band structure energy.

Maximum volume simplex method for automatic selection and classification of atomic environments and environment descriptor compression

Fingerprint distances, which measure the similarity of atomic environments, are commonly calculated from atomic environment fingerprint vectors. In this work, we present the simplex method that can perform the inverse operation, i.e., calculating fingerprint vectors from fingerprint distances. The fingerprint vectors found in this way point to the corners of a simplex. For a large dataset of fingerprints, we can find a particular largest simplex, whose dimension gives the effective dimension of the fingerprint vector space. We show that the corners of this simplex correspond to landmark environments that can be used in a fully automatic way to analyze structures. In this way, we can, for instance, detect atoms in grain boundaries or on edges of carbon flakes without any human input about the expected environment. By projecting fingerprints on the largest simplex, we can also obtain fingerprint vectors that are considerably shorter than the original ones but whose information content is not significantly reduced.

Liquid water contains the building blocks of diverse ice phases

Water molecules can arrange into a liquid with complex hydrogen-bond networks and at least 17 experimentally confirmed ice phases with enormous structural diversity. It remains a puzzle how or whether this multitude of arrangements in different phases of water are related. Here we investigate the structural similarities between liquid water and a comprehensive set of 54 ice phases in simulations, by directly comparing their local environments using general atomic descriptors, and also by demonstrating that a machine-learning potential trained on liquid water alone can predict the densities, lattice energies, and vibrational properties of the ices. The finding that the local environments characterising the different ice phases are found in water sheds light on the phase behavior of water, and rationalizes the transferability of water models between different phases.

Molecular understanding of water is challenging due to the structural complexity of liquid water and the large number of ice phases. Here the authors use a machine-learning potential trained on liquid water to demonstrate the structural similarity of liquid water and that of 54 real and hypothetical ice phases.

Incompleteness of Atomic Structure Representations

Many-body descriptors are widely used to represent atomic environments in the construction of machine-learned interatomic potentials and more broadly for fitting, classification, and embedding tasks on atomic structures. There is a widespread belief in the community that three-body correlations are likely to provide an overcomplete description of the environment of an atom. We produce several counterexamples to this belief, with the consequence that any classifier, regression, or embedding model for atom-centered properties that uses three- (or four)-body features will incorrectly give identical results for different configurations. Writing global properties (such as total energies) as a sum of many atom-centered contributions mitigates the impact of this fundamental deficiency-explaining the success of current "machine-learning" force fields. We anticipate the issues that will arise as the desired accuracy increases, and suggest potential solutions.

Combining Machine Learning Potential and Structure Prediction for Accelerated Materials Design and Discovery

The theoretical structure prediction method via quantum mechanical atomistic simulations such as density functional theory (DFT), based solely on chemical composition, has already become a routine tool to determine the structures of physical and chemical systems, e.g., solids and clusters. However, the application of DFT to more realistic simulations, to a large extent, is impeded because of the unfavorable scaling of the computational cost with respect to the system size. During recent years, the machine learning potential (MLP) method has been rapidly rising as an accurate and efficient tool for atomistic simulations. In this Perspective, we provide an introduction to the basic principles and advantages of the combination of structure prediction and MLP, as well as the challenges and opportunities associated with this promising approach.

Mapping Materials and Molecules

The visualization of data is indispensable in scientific research, from the early stages when human insight forms to the final step of communicating results. In computational physics, chemistry and materials science, it can be as simple as making a scatter plot or as straightforward as looking through the snapshots of atomic positions manually. However, as a result of the "big data" revolution, these conventional approaches are often inadequate. The widespread adoption of high-throughput computation for materials discovery and the associated community-wide repositories have given rise to data sets that contain an enormous number of compounds and atomic configurations. A typical data set contains thousands to millions of atomic structures, along with a diverse range of properties such as formation energies, band gaps, or bioactivities.It would thus be desirable to have a data-driven and automated framework for visualizing and analyzing such structural data sets. The key idea is to construct a low-dimensional representation of the data, which facilitates navigation, reveals underlying patterns, and helps to identify data points with unusual attributes. Such data-intensive maps, often employing machine learning methods, are appearing more and more frequently in the literature. However, to the wider community, it is not always transparent how these maps are made and how they should be interpreted. Furthermore, while these maps undoubtedly serve a decorative purpose in academic publications, it is not always apparent what extra information can be garnered from reading or making them.This Account attempts to answer such questions. We start with a concise summary of the theory of representing chemical environments, followed by the introduction of a simple yet practical conceptual approach for generating structure maps in a generic and automated manner. Such analysis and mapping is made nearly effortless by employing the newly developed software tool ASAP. To showcase the applicability to a wide variety of systems in chemistry and materials science, we provide several illustrative examples, including crystalline and amorphous materials, interfaces, and organic molecules. In these examples, the maps not only help to sift through large data sets but also reveal hidden patterns that could be easily missed using conventional analyses.The explosion in the amount of computed information in chemistry and materials science has made visualization into a science in itself. Not only have we benefited from exploiting these visualization methods in previous works, we also believe that the automated mapping of data sets will in turn stimulate further creativity and exploration, as well as ultimately feed back into future advances in the respective fields.

Energy landscape of Au13: a global view of structure transformation

It has long been a challenge in physics and chemistry to acquire a global picture of the energy landscape of a specific material, as well as the kinetic transformation process between configurations of interest. Here we have presented a comprehensive approach to deal with the structure transformation problem, along with the illustration of the energy landscape, as exemplified with the case of Au13. A configuration space based on interatomic distances was proposed and demonstrated to have a strong correlation between structure and energy, with application in structure analysis to screen for trial transition pathways. As several representative configurations and their transition pathways ascertained and by projecting on a plane, a visual two-dimensional contour map was sketched revealing the unique energy landscape of Au13. It shows that the 2D and 3D clusters form two funnels in the high-dimensional configuration space, with a transition pathway with a 0.976 eV barrier bridging them.

Matching crystal structures atom-to-atom

Finding an optimal match between two different crystal structures underpins many important materials science problems, including describing solid-solid phase transitions and developing models for interface and grain boundary structures. In this work, we formulate the matching of crystals as an optimization problem where the goal is to find the alignment and the atom-to-atom map that minimize a given cost function such as the Euclidean distance between the atoms. We construct an algorithm that directly solves this problem for large finite portions of the crystals and retrieves the periodicity of the match subsequently. We demonstrate its capacity to describe transformation pathways between known polymorphs and to reproduce experimentally realized structures of semi-coherent interfaces. Additionally, from our findings, we define a rigorous metric for measuring distances between crystal structures that can be used to properly quantify their geometric (Euclidean) closeness.

Finding Reaction Pathways with Optimal Atomic Index Mappings

Finding complex reaction and transformation pathways involving many intermediate states is, in general, not possible on the density-functional theory level with existing simulation methods, due to the very large number of required energy and force evaluations. For complex reactions, it is not possible to determine which atom in the reactant is mapped onto which atom in the product. Trying out all possible atomic index mappings is not feasible because of the factorial increase in the number of possible mappings. We use a penalty function that is invariant under index permutations to bias the potential energy surface in such a way that it obtains the characteristics of a structure seeker, whose global minimum is the reaction product. By performing a minima-hopping-based global optimization on this biased potential energy surface, we rapidly find intermediate states that lead into the global minimum and allow us to then extract entire reaction pathways. We first demonstrate for a benchmark system, namely, the Lennard-Jones cluster LJ_{38}, that our method finds intermediate states relevant to the lowest energy reaction pathway, and hence we need to consider much fewer intermediate states than previous methods to find the lowest energy reaction pathway. Finally, we apply the method to two real systems, C_{60} and C_{20}H_{20}, and show that the reaction pathways found contain valuable information on how these molecules can be synthesized.

A universal algorithm for finding the shortest distance between systems of points

Three universal algorithms for geometrical comparison of abstract sets of n points in the Euclidean space R³ are proposed. It is proved that at an accuracy ε the efficiency of all the algorithms does not exceed O(n³/ε^3/2). The most effective algorithm combines the known Hungarian and Kabsch algorithms, but is free of their deficiencies and fast enough to match hundreds of points. The algorithm is applied to compare both finite (ligands) and periodic (nets) chemical objects.

A Global Optimizer for Nanoclusters

We have developed an algorithm to automatically build the global minimum and other low-energy minima of nanoclusters. This method is implemented in PyAR (https://github.com/anooplab/pyar) program. The global optimization in PyAR involves two parts, generation of several trial geometries and gradient-based local optimization of the trial geometries. While generating the trial geometries, a Tabu list is used for storing the information of the already used trial geometries to avoid using the similar trial geometries. In this recursive algorithm, an n-sized cluster is built from the geometries of n−1 clusters. The overall procedure automatically generates many unique minimum energy geometries of clusters with size from 2 up to n using this evolutionary growth strategy. We have used our strategy on some of the well-studied clusters such as Pd, Pt, Au, and Al homometallic clusters, Ru-Pt and Au-Pt binary clusters, and Ag-Au-Pt ternary cluster. We have analyzed some of the popular parameters to characterize the clusters, such as relative energy, singlet-triplet energy difference, binding energy, second-order energy difference, and mixing energy, and compared with the reported properties.

Phase Transition Pathway Sampling via Swarm Intelligence and Graph Theory

The prediction of reaction pathways for solid-solid transformations remains a key challenge. Here, we develop a pathway sampling method via swarm intelligence and graph theory and demonstrate that our pallas method is an effective tool to help understand phase transformations in solid-state systems. The method is capable of finding low-energy transition pathways between two minima without having to specify any details of the transition mechanism a priori. We benchmarked our pallas method against known phase transitions in cadmium selenide (CdSe) and silicon (Si). pallas readily identifies previously reported, low-energy phase transition pathways for the wurtzite to rock-salt transition in CdSe and reveals a novel lower-energy pathway that has not yet been observed. In addition, pallas provides detailed information that explains the complex phase transition sequence observed during the decompression of Si from high pressure. Given the efficiency to identify low-barrier-energy reaction pathways, the pallas methodology represents a promising tool for materials by design with valuable insights for novel synthesis.

Using symmetry to elucidate the importance of stoichiometry in colloidal crystal assembly

We demonstrate a method based on symmetry to predict the structure of self-assembling, multicomponent colloidal mixtures. This method allows us to feasibly enumerate candidate structures from all symmetry groups and is many orders of magnitude more computationally efficient than combinatorial enumeration of these candidates. In turn, this permits us to compute ground-state phase diagrams for multicomponent systems. While tuning the interparticle potentials to produce potentially complex interactions represents the conventional route to designing exotic lattices, we use this scheme to demonstrate that simple potentials can also give rise to such structures which are thermodynamically stable at moderate to low temperatures. Furthermore, for a model two-dimensional colloidal system, we illustrate that lattices forming a complete set of 2-, 3-, 4-, and 6-fold rotational symmetries can be rationally designed from certain systems by tuning the mixture composition alone, demonstrating that stoichiometric control can be a tool as powerful as directly tuning the interparticle potentials themselves.

Unsupervised machine learning in atomistic simulations, between predictions and understanding

Automated analyses of the outcome of a simulation have been an important part of atomistic modeling since the early days, addressing the need of linking the behavior of individual atoms and the collective properties that are usually the final quantity of interest. Methods such as clustering and dimensionality reduction have been used to provide a simplified, coarse-grained representation of the structure and dynamics of complex systems from proteins to nanoparticles. In recent years, the rise of machine learning has led to an even more widespread use of these algorithms in atomistic modeling and to consider different classification and inference techniques as part of a coherent toolbox of data-driven approaches. This perspective briefly reviews some of the unsupervised machine-learning methods-that are geared toward classification and coarse-graining of molecular simulations-seen in relation to the fundamental mathematical concepts that underlie all machine-learning techniques. It discusses the importance of using concise yet complete representations of atomic structures as the starting point of the analyses and highlights the risk of introducing preconceived biases when using machine learning to rationalize and understand structure-property relations. Supervised machine-learning techniques that explicitly attempt to predict the properties of a material given its structure are less susceptible to such biases. Current developments in the field suggest that using these two classes of approaches side-by-side and in a fully integrated mode, while keeping in mind the relations between the data analysis framework and the fundamental physical principles, will be key to realizing the full potential of machine learning to help understand the behavior of complex molecules and materials.

Neural network force fields for simple metals and semiconductors: construction and application to the calculation of phonons and melting temperatures

We present a practical procedure to obtain reliable and unbiased neural network based force fields for solids. Training and test sets are efficiently generated from global structural prediction runs, at the same time assuring the structural variety and importance of sampling the relevant regions of phase space. The neural networks are trained to yield not only good formation energies, but also accurate forces and stresses, which are the quantities of interest for molecular dynamics simulations. Finally, we construct, as an example, several force fields for both semiconducting and metallic elements, and prove their accuracy for a variety of structural and dynamical properties. These are then used to study the melting of bulk copper and gold.

Machine Learning Adaptive Basis Sets for Efficient Large Scale Density Functional Theory Simulation

It is chemically intuitive that an optimal atom centered basis set must adapt to its atomic environment, for example by polarizing toward nearby atoms. Adaptive basis sets of small size can be significantly more accurate than traditional atom centered basis sets of the same size. The small size and well conditioned nature of these basis sets leads to large saving in computational cost, in particular in a linear scaling framework. Here, it is shown that machine learning can be used to predict such adaptive basis sets using local geometrical information only. As a result, various properties of standard DFT calculations can be easily obtained at much lower costs, including nuclear gradients. In our approach, a rotationally invariant parametrization of the basis is obtained by employing a potential anchored on neighboring atoms to ultimately construct a rotation matrix that turns a traditional atom centered basis set into a suitable adaptive basis set. The method is demonstrated using MD simulations of liquid water, where it is shown that minimal basis sets yield structural properties in fair agreement with basis set converged results, while reducing the computational cost in the best case by a factor of 200 and the required flops by 4 orders of magnitude. Already a very small training set yields satisfactory results as the variational nature of the method provides robustness.

Force Field for Water Based on Neural Network

We developed a novel neural network-based force field for water based on training with high-level ab initio theory. The force field was built based on an electrostatically embedded many-body expansion method truncated at binary interactions. The many-body expansion method is a common strategy to partition the total Hamiltonian of large systems into a hierarchy of few-body terms. Neural networks were trained to represent electrostatically embedded one-body and two-body interactions, which require as input only one and two water molecule calculations at the level of ab initio electronic structure method CCSD/aug-cc-pVDZ embedded in the molecular mechanics water environment, making it efficient as a general force field construction approach. Structural and dynamic properties of liquid water calculated with our force field show good agreement with experimental results. We constructed two sets of neural network based force fields: nonpolarizable and polarizable force fields. Simulation results show that the nonpolarizable force field using fixed TIP3P charges has already behaved well, since polarization effects and many-body effects are implicitly included due to the electrostatic embedding scheme. Our results demonstrate that the electrostatically embedded many-body expansion combined with neural network provides a promising and systematic way to build next-generation force fields at high accuracy and low computational costs, especially for large systems.