Smooth Overlap Of Atomic Positions Research Articles

The rule of four: anomalous distributions in the stoichiometries of inorganic compounds

Why are materials with specific characteristics more abundant than others? This is a fundamental question in materials science and one that is traditionally difficult to tackle, given the vastness of compositional and configurational space. We highlight here the anomalous abundance of inorganic compounds whose primitive unit cell contains a number of atoms that is a multiple of four. This occurrence—named here the rule of four—has to our knowledge not previously been reported or studied. Here, we first highlight the rule’s existence, especially notable when restricting oneself to experimentally known compounds, and explore its possible relationship with established descriptors of crystal structures, from symmetries to energies. We then investigate this relative abundance by looking at structural descriptors, both of global (packing configurations) and local (the smooth overlap of atomic positions) nature. Contrary to intuition, the overabundance does not correlate with low-energy or high-symmetry structures; in fact, structures which obey the rule of four are characterized by low symmetries and loosely packed arrangements maximizing the free volume. We are able to correlate this abundance with local structural symmetries, and visualize the results using a hybrid supervised-unsupervised machine learning method.

Machine Learning Models for Predicting Zirconocene Properties and Barriers.

Zr metallocenes have significant potential to be highly tunable polyethylene catalysts through modification of the aromatic ligand framework. Here we report the development of multiple machine learning models using a large library (>700 systems) of DFT-calculated zirconocene properties and barriers for ethylene polymerization. We show that very accurate machine learning models are possible for HOMO-LUMO gaps of precatalysts but the performance significantly depends on the machine learning algorithm and type of featurization, such as fingerprints, Coulomb matrices, smooth overlap of atomic positions, or persistence images. Surprisingly, the description of the bonding hapticity, the number of direct connections between Zr and the ligand aromatic carbons, only has a moderate influence on the performance of most models. Despite robust models for HOMO-LUMO gaps, these types of machine learning models based on structure connectivity type features perform poorly in predicting ethylene migratory insertion barrier heights. Therefore, we developed several relatively robust and accurate machine learning models for barrier heights that are based on quantum-chemical descriptors (QCDs). The quantitative accuracy of these models depends on which potential energy surface structure QCDs were harvested from. This revealed a Hammett-type principle to naturally emerge showing that QCDs from the π-coordination complexes provide much better descriptions of the transition states than other potential-energy structures. Feature importance analysis of the QCDs provides several fundamental principles that influence zirconocene catalyst reactivity.

Gaussian processes for finite size extrapolation of many-body simulations.

Key to being able to accurately model the properties of realistic materials is being able to predict their properties in the thermodynamic limit. Nevertheless, because most many-body electronic structure methods scale as a high-order polynomial, or even exponentially, with system size, directly simulating large systems in their thermodynamic limit rapidly becomes computationally intractable. As a result, researchers typically estimate the properties of large systems that approach the thermodynamic limit by extrapolating the properties of smaller, computationally-accessible systems based on relatively simple scaling expressions. In this work, we employ Gaussian processes to more accurately and efficiently extrapolate many-body simulations to their thermodynamic limit. We train our Gaussian processes on Smooth Overlap of Atomic Positions (SOAP) descriptors to extrapolate the energies of one-dimensional hydrogen chains obtained using two high-accuracy many-body methods: coupled cluster theory and Auxiliary Field Quantum Monte Carlo (AFQMC). In so doing, we show that Gaussian processes trained on relatively short 10-30-atom chains can predict the energies of both homogeneous and inhomogeneous hydrogen chains in their thermodynamic limit with sub-milliHartree accuracy. Unlike standard scaling expressions, our GPR-based approach is highly generalizable given representative training data and is not dependent on systems' geometries or dimensionality. This work highlights the potential for machine learning to correct for the finite size effects that routinely complicate the interpretation of finite size many-body simulations.

Quantifying Local Atomic Distortions in UO2 Grain Boundaries: Correlation with Energetic and Electronic Properties.

UO2, as a key material in the nuclear industry, is composed of grains or crystallites in real applications. Their interfaces, known as grain boundaries (GBs), significantly impact thermal conductivity, corrosion resistance, and mechanical response. Here, utilizing Hubbard-corrected density functional theory, we systematically examine the local cluster structures, energetic stabilities, and electronic properties of five typical tilt UO2 GBs ranging from Σ 3 to Σ 11. We categorize all possible distorted U- and O-centered clusters at these GBs and identify their cluster morphologies and radial and angular distortions. Our results highlight the abundance of new U-O bonds stretching to "medium-range", a feature often overlooked in conventional coordination analysis. To quantitatively describe these distorted clusters, we use smooth overlap of atomic positions (SOAP) to represent the structural and chemical local environments, which takes into account both radial (2-body) and angular (3-body) distortions. We define a dissimilarity index by computing the inner product of SOAP descriptors between the distorted and the perfect motif in ideal UO2. Our findings show that the medium-range SOAP dissimilarity correlates well with the GB excess energy, outperforming metrics such as dangling bonds or bonding strain. Furthermore, it is found that the band gaps in sufficiently high-energy GBs are shortened, with excess states primarily contributed by the distorted U clusters. Our results present a comprehensive gallery of the local distorted clusters introduced by typical UO2 GBs and have implications on the structure-property relations of GBs and other interfaces.

Concurrent prediction of metallic glasses’ global energy and internal structural heterogeneity by interpretable machine learning

Predicting glasses’ properties from their structures is a formidable challenge because of the inherently disordered atomic configurations. Here we tackle the problem using a new two-stage (encoding/interpreting) machine learning pipeline. First, local environments are encoded by the Smooth Overlap of Atomic Positions (SOAP) descriptors, which are then fed into extreme gradient boosting tree algorithm to train/predict given samples’ configurational energy. 40 important unique local environments (ULEs) most responsible for the global energy of ZrCu-based glasses are identified. Markedly, we discover that the same short-range orders of Voronoi cells, when embedded in various ULEs, could impact the sample's global stability in qualitatively different manners. These new findings thus reveal a profound connection between short-range orders and medium-range orders. In the second stage, a designed interpreting stage is employed to decompose a sample's 3 N degrees-of-freedom configuration into a 40-dimension probability spectrum barcode via frequency mapping of those ULEs. We demonstrate that, in addition to the global energy prediction, by analyzing barcode-elements’ occupational fractions and fluctuations, one can simultaneously assess samples’ structural heterogeneity, which is known as a crucial quantity to dictate metallic glasses’ deformation behaviors. The implications of our findings to a barcode-mediated new strategy of inverse engineering design of metallic glasses are also discussed.

Persistent homology-based descriptor for machine-learning potential of amorphous structures.

High-accuracy prediction of the physical properties of amorphous materials is challenging in condensed-matter physics. A promising method to achieve this is machine-learning potentials, which is an alternative to computationally demanding abinitio calculations. When applying machine-learning potentials, the construction of descriptors to represent atomic configurations is crucial. These descriptors should be invariant to symmetry operations. Handcrafted representations using a smooth overlap of atomic positions and graph neural networks (GNN) are examples of methods used for constructing symmetry-invariant descriptors. In this study, we propose a novel descriptor based on a persistence diagram (PD), a two-dimensional representation of persistent homology (PH). First, we demonstrated that the normalized two-dimensional histogram obtained from PD could predict the average energy per atom of amorphous carbon at various densities, even when using a simple model. Second, an analysis of the dimensional reduction results of the descriptor spaces revealed that PH can be used to construct descriptors with characteristics similar to those of a latent space in a GNN. These results indicate that PH is a promising method for constructing descriptors suitable for machine-learning potentials without hyperparameter tuning and deep-learning techniques.

A fuzzy classification framework to identify equivalent atoms in complex materials and molecules.

The nature of an atom in a bonded structure-such as in molecules, in nanoparticles, or in solids, at surfaces or interfaces-depends on its local atomic environment. In atomic-scale modeling and simulation, identifying groups of atoms with equivalent environments is a frequent task, to gain an understanding of the material function, to interpret experimental results, or to simply restrict demanding first-principles calculations. However, while routine, this task can often be challenging for complex molecules or non-ideal materials with breaks in symmetries or long-range order. To automatize this task, we here present a general machine-learning framework to identify groups of (nearly) equivalent atoms. The initial classification rests on the representation of the local atomic environment through a high-dimensional smooth overlap of atomic positions (SOAP) vector. Recognizing that not least thermal vibrations may lead to deviations from ideal positions, we then achieve a fuzzy classification by mean-shift clustering within a low-dimensional embedded representation of the SOAP points as obtained through multidimensional scaling. The performance of this classification framework is demonstrated for simple aromatic molecules and crystalline Pd surface examples.

Updates to the DScribe library: New descriptors and derivatives.

We present an update of the DScribe package, a Python library for atomistic descriptors. The update extends DScribe's descriptor selection with the Valle-Oganov materials fingerprint and provides descriptor derivatives to enable more advanced machine learning tasks, such as force prediction and structure optimization. For all descriptors, numeric derivatives are now available in DScribe. For the many-body tensor representation (MBTR) and the Smooth Overlap of Atomic Positions (SOAP), we have also implemented analytic derivatives. We demonstrate the effectiveness of the descriptor derivatives for machine learning models of Cu clusters and perovskite alloys.

Machine Learning Electron Density Prediction Using Weighted Smooth Overlap of Atomic Positions.

Having access to accurate electron densities in chemical systems, especially for dynamical systems involving chemical reactions, ion transport, and other charge transfer processes, is crucial for numerous applications in materials chemistry. Traditional methods for computationally predicting electron density data for such systems include quantum mechanical (QM) techniques, such as density functional theory. However, poor scaling of these QM methods restricts their use to relatively small system sizes and short dynamic time scales. To overcome this limitation, we have developed a deep neural network machine learning formalism, which we call deep charge density prediction (DeepCDP), for predicting charge densities by only using atomic positions for molecules and condensed phase (periodic) systems. Our method uses the weighted smooth overlap of atomic positions to fingerprint environments on a grid-point basis and map it to electron density data generated from QM simulations. We trained models for bulk systems of copper, LiF, and silicon; for a molecular system, water; and for two-dimensional charged and uncharged systems, hydroxyl-functionalized graphane, with and without an added proton. We showed that DeepCDP achieves prediction R2 values greater than 0.99 and mean squared error values on the order of 10-5e2 Å-6 for most systems. DeepCDP scales linearly with system size, is highly parallelizable, and is capable of accurately predicting the excess charge in protonated hydroxyl-functionalized graphane. We demonstrate how DeepCDP can be used to accurately track the location of charges (protons) by computing electron densities at a few selected grid points in the materials, thus significantly reducing the computational cost. We also show that our models can be transferable, allowing prediction of electron densities for systems on which it has not been trained but that contain a subset of atomic species on which it has been trained. Our approach can be used to develop models that span different chemical systems and train them for the study of large-scale charge transport and chemical reactions.

Enabling automated high-throughput Density Functional Theory studies of amorphous material surface reactions

Amorphous materials are increasingly considered for electronic and energy applications. However, modeling site-specific processes on amorphous materials is prohibitively expensive due to the large configuration space, and the large simulation cell needed to properly capture material properties. In this work, we develop a high-throughput workflow, powered by semiempirical methods in combination with Density Functional Theory, to simulate amorphous surfaces and utilize statistical sampling methods to assess surface reaction kinetics. We employ the Smooth Overlap of Atomic Positions (SOAP) to featurize surface species and apply a Bayesian Gaussian Mixture clustering model to perform a site reduction analysis. Finally, we automate the generation of input images for Improved-Tangent Nudged Elastic Band (IT-NEB) to simulate an etching reaction on the reduced sites. The resulting etching barriers are found to follow experimental etching trends for both amorphous silicon and amorphous carbon. Notably, our method reproduces etching barriers using a significantly reduced amount of sites (6 sites) out of all potential sites on the amorphous material surfaces. The successful site reduction method described in this study opens amorphous materials to high-throughput explorations of interfacial chemistry and surface properties.

Do Machine-Learning Atomic Descriptors and Order Parameters Tell the Same Story? The Case of Liquid Water.

Machine-learning (ML) has become a key workhorse in molecular simulations. Building an ML model in this context involves encoding the information on chemical environments using local atomic descriptors. In this work, we focus on the Smooth Overlap of Atomic Positions (SOAP) and their application in studying the properties of liquid water both in the bulk and at the hydrophobic air-water interface. By using a statistical test aimed at assessing the relative information content of different distance measures defined on the same data space, we investigate if these descriptors provide the same information as some of the common order parameters that are used to characterize local water structure such as hydrogen bonding, density, or tetrahedrality to name a few. Our analysis suggests that the ML description and the standard order parameters of the local water structure are not equivalent. In particular, a combination of these order parameters probing local water environments can predict SOAP similarity only approximately, and vice versa, the environments that are similar according to SOAP are not necessarily similar according to the standard order parameters. We also elucidate the role of some of the metaparameters in the SOAP definition in encoding chemical information.

Leveraging genetic algorithms to maximise the predictive capabilities of the SOAP descriptor

The smooth overlap of atomic positions (SOAP) descriptor represents an increasingly common approach to encode local atomic environments in a form readily digestible to machine learning algorithms.

Spotting Local Environments in Self-Assembled Monolayer-Protected Gold Nanoparticles.

Organic-inorganic (O-I) nanomaterials are versatile platforms for an incredible high number of applications, ranging from heterogeneous catalysis to molecular sensing, cell targeting, imaging, and cancer diagnosis and therapy, just to name a few. Much of their potential stems from the unique control of organic environments around inorganic sites within a single O-I nanomaterial, which allows for new properties that were inaccessible using purely organic or inorganic materials. Structural and mechanistic characterization plays a key role in understanding and rationally designing such hybrid nanoconstructs. Here, we introduce a general methodology to identify and classify local (supra)molecular environments in an archetypal class of O-I nanomaterials, i.e., self-assembled monolayer-protected gold nanoparticles (SAM-AuNPs). By using an atomistic machine-learning guided workflow based on the Smooth Overlap of Atomic Positions (SOAP) descriptor, we analyze a collection of chemically different SAM-AuNPs and detect and compare local environments in a way that is agnostic and automated, i.e., with no need of a priori information and minimal user intervention. In addition, the computational results coupled with experimental electron spin resonance measurements prove that is possible to have more than one local environment inside SAMs, being the thickness of the organic shell and solvation primary factors in the determining number and nature of multiple coexisting environments. These indications are extended to complex mixed hydrophilic-hydrophobic SAMs. This work demonstrates that it is possible to spot and compare local molecular environments in SAM-AuNPs exploiting atomistic machine-learning approaches, establishes ground rules to control them, and holds the potential for the rational design of O-I nanomaterials instructed from data.

Machine-learning-based prediction of first-principles XANES spectra for amorphous materials

In this paper, a machine-learning-based method is proposed for predicting the x-ray absorption near-edge structure (XANES) for local configurations specific to amorphous materials. A combination of molecular dynamics and first-principles XANES simulations was adopted. The XANES spectrum was assumed to be accurately represented by linear regression of the local atomic descriptors. A comprehensive prediction of Si $K$-edge XANES spectra was performed based on an atom-centered symmetry function, smooth overlap of atomic positions, local many-body tensor representation, and spectral neighbor analysis potential. Furthermore, prediction accuracy was improved by compression of XANES spectral data and efficient sampling of training data.

Machine Learning Diffusion Monte Carlo Energies.

We present two machine learning methodologies that are capable of predicting diffusion Monte Carlo (DMC) energies with small data sets (≈60 DMC calculations in total). The first uses voxel deep neural networks (VDNNs) to predict DMC energy densities using Kohn-Sham density functional theory (DFT) electron densities as input. The second uses kernel ridge regression (KRR) to predict atomic contributions to the DMC total energy using atomic environment vectors as input (we used atom-centered symmetry functions, atomic environment vectors from the ANI models, and smooth overlap of atomic positions). We first compare the methodologies on pristine graphene lattices, where we find that the KRR methodology performs best in comparison to gradient boosted decision trees, random forest, Gaussian process regression, and multilayer perceptrons. In addition, KRR outperforms VDNNs by an order of magnitude. Afterward, we study the generalizability of KRR to predict the energy barrier associated with a Stone-Wales defect. Lastly, we move from 2D to 3D materials and use KRR to predict total energies of liquid water. In all cases, we find that the KRR models are more accurate than Kohn-Sham DFT and all mean absolute errors are less than chemical accuracy.

Quantum Chemical Roots of Machine-Learning Molecular Similarity Descriptors.

In this work, we explore the quantum chemical foundations of descriptors for molecular similarity. Such descriptors are key for traversing chemical compound space with machine learning. Our focus is on the Coulomb matrix and on the smooth overlap of atomic positions (SOAP). We adopt a basic framework that allows us to connect both descriptors to electronic structure theory. This framework enables us to then define two new descriptors that are more closely related to electronic structure theory, which we call Coulomb lists and smooth overlap of electron densities (SOED). By investigating their usefulness as molecular similarity descriptors, we gain new insights into how and why Coulomb matrix and SOAP work. Moreover, Coulomb lists avoid the somewhat mysterious diagonalization step of the Coulomb matrix and might provide a direct means to extract subsystem information that can be compared across Born-Oppenheimer surfaces of varying dimension. For the electron density, we derive the necessary formalism to create the SOED measure in close analogy to SOAP. Because this formalism is more involved than that of SOAP, we review the essential theory as well as introduce a set of approximations that eventually allow us to work with SOED in terms of the same implementation available for the evaluation of SOAP. We focus our analysis on elementary reaction steps, where transition state structures are more similar to either reactant or product structures than the latter two are with respect to one another. The prediction of electronic energies of transition state structures can, however, be more difficult than that of stable intermediates due to multi-configurational effects. The question arises to what extent molecular similarity descriptors rooted in electronic structure theory can resolve these intricate effects.

Effect of chemical disorder on the magnetic anisotropy in L10 FeNi from first-principles calculations

We use first-principles calculations to investigate how deviations from perfect chemical order affect the magnetocrystalline anisotropy energy (MAE) in $L{1}_{0}$ FeNi. We first analyze the local chemical environment of the Fe atoms in various partially ordered configurations, using the orbital magnetic moment anisotropy (OMA) as proxy for a local contribution to the MAE. We are able to identify a specific nearest neighbor configuration and use this ``favorable environment'' to successfully design various structures with MAE higher than the perfectly ordered system. However, a systematic analysis of the correlation between local environment and OMA using smooth overlap of atomic positions indicates only a partial correlation, which exists only if the deviation from full chemical order is not too large, whereas in general no such correlation can be identified even using up to third nearest neighbors. Guided by the observation that the identified favorable environment implies an Fe-rich composition, we investigate the effect of randomly inserting additional Fe into the nominal Ni planes of the perfectly ordered structure. We find that the MAE increases with Fe content, at least up to 62.5% Fe. Thus, our paper shows that the perfectly ordered case is not the one with highest MAE and that an increased MAE can be obtained for slightly Fe-rich compositions.

Inorganic Crystal Structure Prototype Database Based on Unsupervised Learning of Local Atomic Environments.

Recognition of structure prototypes from tremendous known inorganic crystal structures has been an important subject beneficial for materials science research and new materials design. The existing databases of inorganic crystal structure prototypes were mostly constructed by classifying materials in terms of the crystallographic space group information. Herein, we employed a distinct strategy to construct the inorganic crystal structure prototype database, relying on the classification of materials in terms of local atomic environments (LAEs) accompanied by unsupervised machine learning method. Specifically, we adopted a hierarchical clustering approach onto all experimentally known inorganic crystal structure data to identify structure prototypes. The criterion for hierarchical clustering is the LAE represented by the state-of-the-art structure fingerprints of the improved bond-orientational order parameters and the smooth overlap of atomic positions. This allows us to build up a LAE-based Inorganic Crystal Structure Prototype Database (LAE-ICSPD) containing 15,613 structure prototypes with defined stoichiometries. In addition, we have developed a Structure Prototype Generator Infrastructure (SPGI) package, which is a useful toolkit for structure prototype generation. Our developed SPGI toolkit and LAE-ICSPD are beneficial for investigating inorganic materials in a global way as well as accelerating the materials discovery process in the data-driven mode.

Atomic energy in grain boundaries studied by machine learning

Grain boundaries (GBs) have been studied for decades, but it remains a challenging task to describe characteristic GB properties by simple structural descriptors, especially at the local atomic level. In this paper, we use the atomic descriptor based on the smooth overlap of atomic positions (SOAP) to study the atomic energy at GBs by using machine learning and propose a route to simplify it. It is found that, compared with conventional local atomic descriptors such as the Voronoi index, excess volume, centrosymmetry, or local entropy, the SOAP vector shows excellent predictive performance for the atomic energy among the 172 Al and 388 Ni coincidence site lattice (CSL) GBs as well as general GBs in the nanocrystalline model. Additionally, we successfully used the datasets of GBs and amorphous models to predict the atomic energy of one another, which proves the similarity between the local atomic environments (LAEs) in GBs and the amorphous state. Furthermore, the distribution of local distortion factors based on the SOAP vector shows the transition in atomic pack ordering from special CSL GBs to general GBs and the amorphous structures. The simplified descriptor we propose can reduce the original SOAP vector from >1000 features to only a few yet still shows the superior predictive performance of the atomic energy at GBs in all cases than the conventional descriptors combined. It is expected that the simplification process can be adapted to study more complex GB behaviors. A simple and efficient descriptor of the LAEs should allow us to have a clearer picture of the structure-property correlation in GBs, which is essential for GB engineering.

Unraveling Crystallization Mechanisms and Electronic Structure of Phase‐Change Materials by Large‐Scale Ab Initio Simulations (Adv. Mater. 11/2022)

Phase-Change Memory Materials In article number 2109139, Wei Zhang, Riccardo Mazzarello, and co-workers study the crystallization and electronic properties of the GeSb2Te4 phase-change memory material, based on ab initio simulations with more than 1000 atoms. The smooth overlap of atomic positions (SOAP) similarity measure reveals the evolution of the crystallization process, and as-crystallized models are used to study the connection between structural and electronic properties, which is relevant to the transport behavior of Ge–Sb–Te alloys.