Evolutionary approach for determining first-principles hamiltonians

Machine Learning Force Field-Aided Cluster Expansion Approach to Phase Diagram of Alloyed Materials

First-principles approaches based on density functional theory (DFT) have played important roles in the theoretical study of multicomponent alloyed materials. Considering the highly demanding computational cost of direct DFT-based sampling of the configurational space, it is crucial to build efficient and low-cost surrogate Hamiltonian models with DFT accuracy for efficient simulation of alloyed systems with configurational disorder. Recently, the machine learning force field (MLFF) method has been proposed to tackle complicated multicomponent disordered systems. However, the importance of integrating significant physical considerations, including, in particular, convex hull preservation, which is the prerequisite for the accurate prediction of phase diagrams, into the training process of the MLFF remains rarely addressed. In this work, a workflow is proposed to train a convex-hull-preserved (CHP) MLFF for binary alloy systems, based on which the order-disorder phase boundary is predicted by using the Wang-Landau Monte Carlo (WLMC) technique. The predicted values for order-disorder phase transition temperatures agree well with the experiment. The CHP-MLFF is further used to build CE models with the same accuracy as the MLFF and higher efficiency in sampling configurational space. Using the results obtained from the MLFF-based WLMC simulation as a reference, the performances of different schemes for constructing CE models were evaluated in a transparent manner, which revealed the close correlation between the prediction accuracy of ground-state configurations and that of the order-disorder phase transition temperature. This work clearly indicates the great importance of reproducing the convex hull and energetics of ground-state configurations when constructing surrogate Hamiltonians for the statistical modeling of alloyed systems.

A Data and DFT-Driven Framework for Predicting the Microstructure of Submonolayer Inverse Metal Oxide on Metal Catalysts

Metal oxides on metal (inverse) catalysts can selectively drive many important reactions. However, understanding the active site under experimentally relevant conditions is lacking. Herein, we introduce a computational framework for predicting atomic models of stable inverse catalysts and demonstrate it for WOx on Pt(553) and a Pt79 nanoparticle at variable WOx coverages. An evolutionary algorithm identifies a small (5%) subset of promising atomic configurations on which DFT simulations are performed. We predict a maximum coverage of ∼50% WOx on Pt(553), consisting of small clusters (tetramers and pentamers), which preferentially reside on the terrace, with their oxygen atoms interacting with the Pt step sites. Consistently, WOx does not lie on curved and undercoordinated metal sites of Pt nanoparticles. The oxide clusters prefer a partially reduced oxidation state. Theoretical EXAFS spectra for select configurations provide insights into interpreting experimental spectra of inverse catalysts. The framework applies to other catalysts.

Mixtures Recomposition by Neural Nets: A Multidisciplinary Overview

Artificial Neural Networks (ANNs) are transforming how we understand chemical mixtures, providing an expressive view of the chemical space and multiscale processes. Their hybridization with physical knowledge can bridge the gap between predictivity and understanding of the underlying processes. This overview explores recent progress in ANNs, particularly their potential in the 'recomposition' of chemical mixtures. Graph-based representations reveal patterns among mixture components, and deep learning models excel in capturing complexity and symmetries when compared to traditional Quantitative Structure-Property Relationship models. Key components, such as Hamiltonian networks and convolution operations, play a central role in representing multiscale mixtures. The integration of ANNs with Chemical Reaction Networks and Physics-Informed Neural Networks for inverse chemical kinetic problems is also examined. The combination of sensors with ANNs shows promise in optical and biomimetic applications. A common ground is identified in the context of statistical physics, where ANN-based methods iteratively adapt their models by blending their initial states with training data. The concept of mixture recomposition unveils a reciprocal inspiration between ANNs and reactive mixtures, highlighting learning behaviors influenced by the training environment.

High-throughput computational materials screening of transition metal peroxides

Semiconductor materials of abnormal stoichiometric ratio often exhibit unique properties, yet it is still a challenge to determine the structures of such materials in an efficient way. Herein, we propose a method for structurally biased screening according to the coordination numbers and the numbers of Wyckoff positions, balancing the atom local environment and the global symmetry of structures. Based on first-principles calculations, we have predicted two metastable peroxides P21/c-ScO2 and Pmmn-TiO3 with more than six coordination points. For these two structures, the most stable intrinsic defect is the oxygen vacancy (VO) at the peroxide anion (O2-2), which induces the absence of antibonding orbital formed by O2-2 near the valence band maximum. With the introduction of VO, the decrease of coordination numbers leads to charge recombination, and results in the appearance of an ordered phase TiO2.5 with stronger Ti-O orbital hybridization. The proposed method presents a promising and feasible approach for the screening of novel compounds.

Block sparsity promoting algorithm for efficient construction of cluster expansion models for multicomponent alloys

Multicomponent alloys are gaining significance as drivers of technological breakthroughs especially in structural and energy storage materials. The vast configuration space of these materials prohibit computational modeling using first-principles based methods alone. The cluster expansion (CE) method is the most widely used tool for modeling configurational disorder in alloys. CE relies on machine learning algorithms to train Hamiltonians and uses first-principles calculated data as training sets. In this paper we present a new compressive sensing-based algorithm for the efficient construction of CE Hamiltonians of multicomponent alloys. Our algorithm constructs highly sparse and physically reasonable models from a carefully selected small training set of alloy structures. Compared to conventional fitting algorithms, the algorithm achieves more than 50% reduction in the training set size. The resultant sparse models can sample the configuration space at least 3 × faster. We demonstrate this algorithm on 4 different alloy systems, namely Ag-Au, Ag-Au-Cu, Ag-Au-Cu-Pd and (Ge,Sn)(S,Se,Te).The sparse CE models for these alloys can rapidly reproduce known ground state orderings and order-disorder transitions. Our method can truly enable high-throughput multicomponent alloy thermodynamics by reducing the cost associated with model construction and configuration sampling.

Machine Learning Force Field Aided Cluster Expansion Approach to Configurationally Disordered Materials: Critical Assessment of Training Set Selection and Size Convergence

Cluster expansion (CE) is a powerful theoretical tool to study the configuration-dependent properties of substitutionally disordered systems. Typically, a CE model is built by fitting a few tens or hundreds of target quantities calculated by first-principles approaches. To validate the reliability of the model, a convergence test of the cross-validation (CV) score to the training set size is commonly conducted to verify the sufficiency of the training data. However, such a test only confirms the convergence of the predictive capability of the CE model within the training set, and it is unknown whether the convergence of the CV score would lead to robust thermodynamic simulation results such as order-disorder phase transition temperature T_c. In this work, using carbon defective MoC_1-x as a model system and aided by the machine-learning force field technique, a training data pool with about 13000 configurations has been efficiently obtained and used to generate different training sets of the same size randomly. By conducting parallel Monte Carlo simulations with the CE models trained with different randomly selected training sets, the uncertainty in calculated T_c can be evaluated at different training set sizes. It is found that the training set size that is sufficient for the CV score to converge still leads to a significant uncertainty in the predicted T_c and that the latter can be considerably reduced by enlarging the training set to that of a few thousand configurations. This work highlights the importance of using a large training set to build the optimal CE model that can achieve robust statistical modeling results and the facility provided by the machine-learning force field approach to efficiently produce adequate training data.

Room-temperature oxygen vacancy migration induced reversible phase transformation during the anelastic deformation in CuO

From the mechanical perspectives, the influence of point defects is generally considered at high temperature, especially when the creep deformation dominates. Here, we show the stress-induced reversible oxygen vacancy migration in CuO nanowires at room temperature, causing the unanticipated anelastic deformation. The anelastic strain is associated with the nucleation of oxygen-deficient CuO_x phase, which gradually transforms back to CuO after stress releasing, leading to the gradual recovery of the nanowire shape. Detailed analysis reveals an oxygen deficient metastable CuO_x phase that has been overlooked in the literatures. Both theoretical and experimental investigations faithfully predict the oxygen vacancy diffusion pathways in CuO. Our finding facilitates a better understanding of the complicated mechanical behaviors in materials, which could also be relevant across multiple scientific disciplines, such as high-temperature superconductivity and solid-state chemistry in Cu-O compounds, etc.

The effect of point defects on mechanical behaviour of materials is generally considered at high temperatures. This work reports a reversible stress-induced migration of point defects during anelastic deformation in CuO nanowires at room temperature resulting from heterogeneous strain distribution.

The thermodynamic stability and mechanical properties of TiCxN1-x(0x⩽ 1) compounds by cluster expansion method and first-principles calculations

The thermodynamic stability and mechanical properties of titanium carbonitrides TiC_xN_1-x(0 ⩽x⩽ 1) are investigated by a combination of the universal cluster expansion method and the first-principles calculations. By considering the ordering of the N/C distributions on the anion sublattice sites of TiC_xN_1-x, a binary diagram of the heat of formation is constructed, and seven kinds of ground-state structures are predicted in the whole range of 0 ⩽x⩽ 1. These predicted ground-state TiC_xN_1-xstructures are further proved to be dynamically and mechanically stable by examining their phonon dispersion spectra and elastic constants. Further studies indicate that the mechanical and thermodynamic properties of the ternary TiC_xN_1-xstructures are generally better than those of the binary TiC or TiN, while the differences within the ternary systems are insignificant. The possible origin of the enhancement of the mechanical and thermodynamic properties of the predicted ground-state TiC_xN_1-xare discussed together with the electronic structures.

Chemical short-range order in derivative Cr-Ta-Ti-V-W high entropy alloys from the first-principles thermodynamic study

The development of high-entropy alloys (HEAs) focuses on exploring compositional regions in multi-component systems with all alloy elements in equal or near-equal atomic concentrations. Initially it was based on the main idea that high mixing configurational entropy contributions to the alloy free energy could promote the formation of a single solid solution phase. By using the ab-initio based Cluster Expansion (CE) Hamiltonian model constructed for the quinary bcc Cr-Ta-Ti-V-W system in combination with Monte Carlo (MC) simulations, we show that the phase stability and chemical short-range order (SRO) of the equiatomic quinary and five sub-quaternary systems, as well as their derivative alloys, can dramatically change the order-disorder transition temperatures (ODTT) as a function of alloy compositions. In particular, it has been found, that the equiatomic quaternary Ta-Ti-V-W and Cr-Ta-Ti-W alloys had the lowest order-disorder transition temperature (500 K) among all the analysed equiatomic compositions. In all investigated alloy systems, the strongest chemical ordering has been observed between Cr and V, which led to the conclusion that decreasing the concentration of either Cr or V might be beneficial in terms of decreasing the ODTT. It also predicts that increasing concentration of Ti significantly decreases the ODTT. Our analysis of chemical SRO as a function of alloy composition allows to understand the microstructure evolution of HEAs as a function of temperature in excellent agreement with available experimental observations. Importantly, our free energy of mixing and SRO calculations predict that the origin of precipitates formed by Cr- and V-rich in the sub-quaternary Cr-Ta-V-W system is driven by the thermodynamics. The modelling results are in an excellent agreement with experimental observation of Cr and V segregation in the W0.38Ta0.36Cr0.15V0.11 alloy which in turns shows an exceptional radiation resistance.

Finite-Temperature Structures of Supported Subnanometer Catalysts Inferred via Statistical Learning and Genetic Algorithm-Based Optimization

Single-atom catalysts (SACs) minimize noble metal utilization and can alter the activity and selectivity of supported metal nanoparticles. However, the morphology of active centers, including single atoms and subnanometer clusters of a few atoms, remains elusive due to experimental challenges. The computational cost to describe numerous cluster shapes and sizes makes direct first-principles calculations impractical. We present a computational framework to enable structure determination for single-atom and subnanometer cluster catalysts. As a case study, we obtained the low-energy structures of Pd_n (n = 1-21) clusters supported on CeO₂(111), which are critical components of automobile three-way catalysts. Trained on density functional theory data, a three-dimensional cluster expansion is established using statistical learning to describe the Hamiltonian and predict energies of supported Pd_n clusters of any structure. Low-energy stable and metastable structures are identified using a Metropolis Monte Carlo-based genetic algorithm in the canonical ensemble at 300 K. We observe that supported single atoms sinter to form bilayer clusters, and large cluster isomers share similarities in both shape and energy. The findings elucidate the significance of the support and microstructure on cluster stability. We discovered a simple surrogate structure-energy model, where the energy per atom scales with the square root of the average first coordination number, which can be used to estimate energies and compare the stability of clusters. Our framework, applicable to any metal/support system, fills an important methodological gap to predict the stability of supported metal catalysts in the subnanometer regime.

Atom Classification Model for Total Energy Evaluation of Two-Dimensional Multicomponent Materials

The tunable properties of materials originate from variety of structures; however, it is still a challenge to give an accurate and fast evaluation of stabilities for screening numerous candidates. Herein, we propose an atom classification model to describe the multicomponent materials based on the structural recognition, in which the atoms are classified to estimate the total energies. Taking two-dimensional planar C_1-xB_x and C_1-2x(BN)_x as examples, we have found that the test error of total energies is about 3 meV per atom. Notably, the distributions of classified atoms demonstrate the evolution of configurations as a function of temperature, providing a clearer picture of phase transition. In addition, our method is universal, which can be flexibly extended to the bulk structures with more components.

SnO–Sn3O4 heterostructural gas sensor with high response and selectivity to parts-per-billion-level NO2 at low operating temperature

Considering the harmfulness of nitrogen dioxide (NO₂), it is important to develop NO₂ sensors with high responses and low limits of detection. In this study, we synthesize a novel SnO–Sn₃O₄ heterostructure through a one-step solvothermal method, which is used for the first time as an NO₂ sensor. The material exhibits three-dimensional flower-like microparticles assembled by two-dimensional nanosheets, in situ-formed SnO–Sn₃O₄ heterostructures, and large specific surface area. Gas sensing measurements show that the responses of the SnO–Sn₃O₄ heterostructure to 500 ppb NO₂ are as high as 657.4 and 63.4 while its limits of detection are as low as 2.5 and 10 parts per billion at 75 °C and ambient temperature, respectively. In addition, the SnO–Sn₃O₄ heterostructure has an excellent selectivity to NO₂, even if exposed to mixture gases containing interferential part with high concentration. The superior sensing properties can be attributed to the in situ formation of SnO–Sn₃O₄ p–n heterojunctions and large specific surface area. Therefore, the SnO–Sn₃O₄ heterostructure having excellent NO₂ sensing performances is very promising for applications as an NO₂ sensor or alarm operated at a low operating temperature.

A novel SnO–Sn₃O₄ heterostructural gas sensor with high response and selectivity to ppb-level NO₂ at 75 °C and room temperature.

CLEASE: a versatile and user-friendly implementation of cluster expansion method

Materials exhibiting a substitutional disorder such as multicomponent alloys and mixed metal oxides/oxyfluorides are of great importance in many scientific and technological sectors. Disordered materials constitute an overwhelmingly large configurational space, which makes it practically impossible to be explored manually using first-principles calculations such as density functional theory due to the high computational costs. Consequently, the use of methods such as cluster expansion (CE) is vital in enhancing our understanding of the disordered materials. CE dramatically reduces the computational cost by mapping the first-principles calculation results on to a Hamiltonian which is much faster to evaluate. In this work, we present our implementation of the CE method, which is integrated as a part of the atomic simulation environment (ASE) open-source package. The versatile and user-friendly code automates the complex set up and construction procedure of CE while giving the users the flexibility to tweak the settings and to import their own structures and previous calculation results. Recent advancements such as regularization techniques from machine learning are implemented in the developed code. The code allows the users to construct CE on any bulk lattice structure, which makes it useful for a wide range of applications involving complex materials. We demonstrate the capabilities of our implementation by analyzing the two example materials with varying complexities: a binary metal alloy and a disordered lithium chromium oxyfluoride.

Cluster expansion based configurational averaging approach to bandgaps of semiconductor alloys

Configurationally disordered semiconducting materials including semiconductor alloys [e.g., (GaN)_1-x(ZnO)_x] and stoichiometric materials with fractional occupation (e.g., LaTiO₂N) have attracted a lot of interest recently in search for efficient visible light photo-catalysts. First-principles modeling of such materials poses great challenges due to the difficulty in treating the configurational disorder efficiently. In this work, a configurational averaging approach based on the cluster expansion technique has been exploited to describe bandgaps of ordered, partially disordered (with short-range order), and fully disordered phases of semiconductor alloys on the same footing. We take three semiconductor alloys [Cd_1-xZn_xS, ZnO_1-xS_x, and (GaN)_1-x(ZnO)_x] as model systems and clearly demonstrate that semiconductor alloys can have a system-dependent short-range order that has significant effects on their electronic properties.

An extended cluster expansion for ground states of heterofullerenes

It is challenging to determine the ground states of heterofullerenes due to the numerous isomers. Taking the C_60-nB_n heterofullerenes (1 ≤ n ≤ 4) as an example, our first-principles calculations with the isomer enumeration present the most stable structure of C₅₇B₃, which is energetically favored by 0.73 eV than the reported counterpart. It was difficult to conduct the enumeration for the isomers with n beyond 4 because of the expensive first-principle calculations. Here, we propose a nomenclature to enhance structural recognition and adopt an extended cluster expansion to describe the structural stabilities, in which the energies of the heterofullerenes with various concentrations are predicted by linear combination of the multi-body interactions. Unlike the conventional cluster expansion, the interaction parameters are derived from the enumeration of C_60-nB_n (n = 1~4), where there are only 4 coefficients to be fitted as a function of composition for the consideration of local bonding. The cross-validation scores are 1~2 meV per atom for both C₅₅B₅ and C₅₄B₆, ensuring the ground states obtained from our model are in line with the first-principles results. With the help of the structural recognition, the extended cluster expansion could be further applied to other binary systems as an effective complement to the first-principle calculations.

First Principles Neural Network Potentials for Reactive Simulations of Large Molecular and Condensed Systems

Modern simulation techniques have reached a level of maturity which allows a wide range of problems in chemistry and materials science to be addressed. Unfortunately, the application of first principles methods with predictive power is still limited to rather small systems, and despite the rapid evolution of computer hardware no fundamental change in this situation can be expected. Consequently, the development of more efficient but equally reliable atomistic potentials to reach an atomic level understanding of complex systems has received considerable attention in recent years. A promising new development has been the introduction of machine learning (ML) methods to describe the atomic interactions. Once trained with electronic structure data, ML potentials can accelerate computer simulations by several orders of magnitude, while preserving quantum mechanical accuracy. This Review considers the methodology of an important class of ML potentials that employs artificial neural networks.

The many-body expansion combined with neural networks

Fragmentation methods such as the many-body expansion (MBE) are a common strategy to model large systems by partitioning energies into a hierarchy of decreasingly significant contributions. The number of calculations required for chemical accuracy is still prohibitively expensive for the ab initio MBE to compete with force field approximations for applications beyond single-point energies. Alongside the MBE, empirical models of ab initio potential energy surfaces have improved, especially non-linear models based on neural networks (NNs) which can reproduce ab initio potential energy surfaces rapidly and accurately. Although they are fast, NNs suffer from their own curse of dimensionality; they must be trained on a representative sample of chemical space. In this paper we examine the synergy of the MBE and NN's and explore their complementarity. The MBE offers a systematic way to treat systems of arbitrary size while reducing the scaling problem of large systems. NN's reduce, by a factor in excess of 10⁶, the computational overhead of the MBE and reproduce the accuracy of ab initio calculations without specialized force fields. We show that for a small molecule extended system like methanol, accuracy can be achieved with drastically different chemical embeddings. To assess this we test a new chemical embedding which can be inverted to predict molecules with desired properties. We also provide our open-source code for the neural network many-body expansion, Tensormol.

How silver segregation stabilizes 1D surface gold oxide: a cluster expansion study combined with ab initio MD simulations

Gold surprises us again by the unusual stability of one-dimensional gold oxide structures supported on bimetallic surfaces of gold and silver.

Solute segregation and deviation from bulk thermodynamics at nanoscale crystalline defects

It has long been known that solute segregation at crystalline defects can have profound effects on material properties. Nevertheless, quantifying the extent of solute segregation at nanoscale defects has proven challenging due to experimental limitations. A combined experimental and first-principles approach has been used to study solute segregation at extended intermetallic phases ranging from 4 to 35 atomic layers in thickness. Chemical mapping by both atom probe tomography and high-resolution scanning transmission electron microscopy demonstrates a markedly different composition for the 4-atomic-layer-thick phase, where segregation has occurred, compared to the approximately 35-atomic-layer-thick bulk phase of the same crystal structure. First-principles predictions of bulk free energies in conjunction with direct atomistic simulations of the intermetallic structure and chemistry demonstrate the breakdown of bulk thermodynamics at nanometer dimensions and highlight the importance of symmetry breaking due to the proximity of interfaces in determining equilibrium properties.

The free energy of mechanically unstable phases

Phase diagrams provide 'roadmaps' to the possible states of matter. Their determination traditionally rests on the assumption that all phases, even unstable ones, have well-defined free energies under all conditions. However, this assumption is commonly violated in condensed phases due to mechanical instabilities. This long-standing problem impedes thermodynamic database development, as pragmatic attempts at solving this problem involve delicate extrapolations that are highly nonunique and that lack an underlying theoretical justification. Here we propose an efficient computational solution to this problem that has a simple interpretation, both as a topological partitioning of atomic configuration space and as a minimally constrained physical system. Our natural scheme smoothly extends the free energy of stable phases, without relying on extrapolation, thus enabling a formal assessment of widely used extrapolation schemes.

How to determine accurate chemical ordering in several nanometer large bimetallic crystallites from electronic structure calculations

The proposed method allows to efficiently determine the atomic arrangement in bimetallic nanoparticles based on electronic structure calculations and unravels the relationship between structural preferences of atoms and binding in nanoalloys.

Chemical and physical properties of binary metallic nanoparticles (nanoalloys) are to a great extent defined by their chemical ordering, i.e. the pattern in which atoms of the two elements are located in a given crystal lattice. The reliable determination of the lowest-energy chemical ordering is a challenge that impedes in-depth studies of several-nm large bimetallic particles. We propose a method to efficiently optimize the chemical ordering based solely on results of electronic structure (density functional) calculations. We show that the accuracy of this method is practically the same as the accuracy of the underlying quantum mechanical approach. This method, due to its simplicity, immediately reveals why one or another chemical ordering is preferred and unravels the nature of the binding within the nanoparticles. For instance, our results provide very intuitive understanding of why gold and silver segregate on low-coordinated sites in Pd₇₀Au₇₀ and Pd₇₀Ag₇₀ particles, while Pd₇₀Cu₇₀ exhibits matryoshka-like structure and Pd₇₀Zn₇₀ features Zn and Pd atoms arranged in layers. To illustrate the power of the new method we optimized the chemical ordering in much larger Pd₇₃₂Au₇₃₁, Pd₇₃₂Ag₇₃₁, Pd₇₃₂Cu₇₃₁, and Pd₇₃₂Zn₇₃₁ nanocrystals, whose size ∼4.4 nm is common for catalytic applications.

Identifying structural flow defects in disordered solids using machine-learning methods

We use machine-learning methods on local structure to identify flow defects-or particles susceptible to rearrangement-in jammed and glassy systems. We apply this method successfully to two very different systems: a two-dimensional experimental realization of a granular pillar under compression and a Lennard-Jones glass in both two and three dimensions above and below its glass transition temperature. We also identify characteristics of flow defects that differentiate them from the rest of the sample. Our results show it is possible to discern subtle structural features responsible for heterogeneous dynamics observed across a broad range of disordered materials.

Kinetic modelling of heterogeneous catalytic systems

The importance of heterogeneous catalysis in modern life is evidenced by the fact that numerous products and technologies routinely used nowadays involve catalysts in their synthesis or function. The discovery of catalytic materials is, however, a non-trivial procedure, requiring tedious trial-and-error experimentation. First-principles-based kinetic modelling methods have recently emerged as a promising way to understand catalytic function and aid in materials discovery. In particular, kinetic Monte Carlo (KMC) simulation is increasingly becoming more popular, as it can integrate several sources of complexity encountered in catalytic systems, and has already been used to successfully unravel the underlying physics of several systems of interest. After a short discussion of the different scales involved in catalysis, we summarize the theory behind KMC simulation, and present the latest KMC computational implementations in the field. Early achievements that transformed the way we think about catalysts are subsequently reviewed in connection to latest studies of realistic systems, in an attempt to highlight how the field has evolved over the last few decades. Present challenges and future directions and opportunities in computational catalysis are finally discussed.

Nanoalloy electrocatalysis: simulating cyclic voltammetry from configurational thermodynamics with adsorbates

Simulated 2-dimensional cyclic voltammetry for nanoalloys with a hybrid ensemble scheme in Monte Carlo simulation based on the cluster expansion method.

Configurational thermodynamics of alloyed nanoparticles with adsorbates

Changes in the chemical configuration of alloyed nanoparticle (NP) catalysts induced by adsorbates under working conditions, such as reversal in core-shell preference, are crucial to understand and design NP functionality. We extend the cluster expansion method to predict the configurational thermodynamics of alloyed NPs with adsorbates based on density functional theory data. Exemplified with PdRh NPs having O-coverage up to a monolayer, we fully detail the core-shell behavior across the entire range of NP composition and O-coverage with quantitative agreement to in situ experimental data. Optimally fitted cluster interactions in the heterogeneous system are the key to enable quantitative Monte Carlo simulations and design.

Cluster expansion of multicomponent ionic systems with controlled accuracy: importance of long-range interactions in heterovalent ionic systems

We have been examining factors determining the accuracy of cluster expansion (CE), which is used in combination with many density functional theory (DFT) calculations. With the exception of multicomponent metallic or isovalent ionic systems, the contributions of long-range effective cluster interactions (ECIs) to configurational energetics are not negligible, which is ascribed to long-range electrostatic interactions. The truncation of ECIs in such systems leads to systematic errors. A typical problem with such errors can be seen in Monte Carlo simulations, since simulation supercells composed of a larger number of atoms than those of the input DFT structures are used. The prediction errors for long-period structures beyond the cell size of the input DFT structures in addition to those for short-period structures within the cell size of the input DFT structures need to be carefully examined to control the accuracy of CE. In this study, we quantitatively discuss the contribution of the truncation of long-range ECIs to the accuracy of CE. Two types of system, namely a point-charge spinel lattice and a real MgAl2O4 spinel crystal, are examined. The prediction error of the long-period structures can be improved both by increasing the number of pairs and by also considering the effective screened electrostatic energy.

Density functional theory in the solid state

Density functional theory (DFT) has been used in many fields of the physical sciences, but none so successfully as in the solid state. From its origins in condensed matter physics, it has expanded into materials science, high-pressure physics and mineralogy, solid-state chemistry and more, powering entire computational subdisciplines. Modern DFT simulation codes can calculate a vast range of structural, chemical, optical, spectroscopic, elastic, vibrational and thermodynamic phenomena. The ability to predict structure-property relationships has revolutionized experimental fields, such as vibrational and solid-state NMR spectroscopy, where it is the primary method to analyse and interpret experimental spectra. In semiconductor physics, great progress has been made in the electronic structure of bulk and defect states despite the severe challenges presented by the description of excited states. Studies are no longer restricted to known crystallographic structures. DFT is increasingly used as an exploratory tool for materials discovery and computational experiments, culminating in ex nihilo crystal structure prediction, which addresses the long-standing difficult problem of how to predict crystal structure polymorphs from nothing but a specified chemical composition. We present an overview of the capabilities of solid-state DFT simulations in all of these topics, illustrated with recent examples using the CASTEP computer program.

A first-principles study of CO oxidation by surface oxygen on Pt-incorporated perovskite catalyst (CaPtxTi1−xO3)

In the present work, we investigated the structural and catalytic properties of a prototype system Pt-doped CaTiO₃ by means of first principles calculations.

Unveiling stable group IV alloy nanowires via a comprehensive search and their electronic band characteristics

By means of density functional theory calculations, the cluster expansion method, and Monte Carlo simulations, we identify the stable spatial configurations (ground states) for [100] CSi, GeSi, and SnSi alloy nanowires (NWs) across compositions. In particular, we find that stable configurations of GeSiNWs and SnSiNWs exhibit core-shell segregation tendencies, while those of CSiNWs favor ordering. Moreover, we show compositional ranges where the band gaps are expected to vary linearly with composition, allowing predictable band gap fine-tuning. We also predict composition ranges where the spatial separation of near-band gap states are imminent, making it possible for electron-hole charge separation. By addressing both the issues of stability and the compositional trend of electronic band structure, our work should prove useful for designing alloy NWs of smaller dimensions.

Accelerating materials property predictions using machine learning

The materials discovery process can be significantly expedited and simplified if we can learn effectively from available knowledge and data. In the present contribution, we show that efficient and accurate prediction of a diverse set of properties of material systems is possible by employing machine (or statistical) learning methods trained on quantum mechanical computations in combination with the notions of chemical similarity. Using a family of one-dimensional chain systems, we present a general formalism that allows us to discover decision rules that establish a mapping between easily accessible attributes of a system and its properties. It is shown that fingerprints based on either chemo-structural (compositional and configurational information) or the electronic charge density distribution can be used to make ultra-fast, yet accurate, property predictions. Harnessing such learning paradigms extends recent efforts to systematically explore and mine vast chemical spaces, and can significantly accelerate the discovery of new application-specific materials.

The high-throughput highway to computational materials design

High-throughput computational materials design is an emerging area of materials science. By combining advanced thermodynamic and electronic-structure methods with intelligent data mining and database construction, and exploiting the power of current supercomputer architectures, scientists generate, manage and analyse enormous data repositories for the discovery of novel materials. In this Review we provide a current snapshot of this rapidly evolving field, and highlight the challenges and opportunities that lie ahead.

A canonical stability-elasticity relationship verified for one million face-centred-cubic structures

Any thermodynamically stable or metastable phase corresponds to a local minimum of a potentially very complicated energy landscape. But however complex the crystal might be, this energy landscape is of parabolic shape near its minima. Roughly speaking, the depth of this energy well with respect to some reference level determines the thermodynamic stability of the system, and the steepness of the parabola near its minimum determines the system's elastic properties. Although changing alloying elements and their concentrations in a given material to enhance certain properties dates back to the Bronze Age, the systematic search for desirable properties in metastable atomic configurations at a fixed stoichiometry is a very recent tool in materials design. Here we demonstrate, using first-principles studies of four binary alloy systems, that the elastic properties of face-centred-cubic intermetallic compounds obey certain rules. We reach two conclusions based on calculations on a huge subset of the face-centred-cubic configuration space. First, the stiffness and the heat of formation are negatively correlated with a nearly constant Spearman correlation for all concentrations. Second, the averaged stiffness of metastable configurations at a fixed concentration decays linearly with their distance to the ground-state line (the phase diagram of an alloy at zero Kelvin). We hope that our methods will help to simplify the quest for new materials with optimal properties from the vast configuration space available.

A comprehensive search for stable Pt-Pd nanoalloy configurations and their use as tunable catalysts

Using density-functional theory, we predict stable alloy configurations (ground states) for a 1 nm Pt-Pd cuboctahedral nanoparticle across the entire composition range and demonstrate their use as tunable alloy catalysts via hydrogen-adsorption studies. Unlike previous works, we use simulated annealing with a cluster expansion Hamiltonian to perform a rapid and comprehensive search that encompasses both high and low-symmetry configurations. The ground states show Pt(core)-Pd(shell) type configurations across all compositions but with specific Pd patterns. For catalysis studies at room temperatures, the ground states are more realistic structural models than the commonly assumed random alloy configurations. Using the ground states, we reveal that the hydrogen adsorption energy increases (decreases) monotonically with at. % Pt for the {111} hollow ({100} bridge) adsorption site. Such trends are useful for designing tunable Pd-Pt nanocatalysts for the hydrogen evolution reaction.

Interplay between subsurface ordering, surface segregation, and adsorption on Pt-Ti(111) near-surface alloys

Using the first-principles cluster expansion (CE) method, we studied the subsurface ordering of Pt/Pt-Ti(111) surface alloys and the effect of this ordering on segregation and adsorption behavior. The clusters included in the CE are optimized by a genetic algorithm to better describe the interactions between Pt and Ti atoms in the subsurface layer. Similar to bulk Pt-Ti alloys, Pt-Ti(111) subsurface alloys show a strong ordering tendency. A series of stable ordered Pt-Ti subsurface structures are identified from the two-dimensional (2D) CE. As an indication of the connection between the 2D and the bulk ordering, the CE predicts a ground-state Pt(8)Ti structure in the (111) subsurface layer, which is the same ordering as the close-packed plane of the bulk Pt(8)Ti compound. We carried out Monte Carlo simulations (MC) using the CE Hamiltonian to study the finite temperature stability of the Pt-Ti subsurface structures. The MC results show that subsurface structures in the Pt-rich range have higher order-disorder transition temperatures than their Ti-rich subsurface counterparts. We calculate the binding energy of different adsorbates (O, S, H, and NO) on Pt-terminated and Ti-segregated surfaces of ordered PtTi and Pt(8)Ti subsurface alloys. The binding of these adsorbates is generally stronger on Ti-segregated surfaces than Pt-terminated surfaces. The adsorption-induced Ti surface segregation is determined by two factors: (i) the unfavorable energy penalty for the Ti atom to segregate to the clean surface and (ii) the favorable energy decrease from stronger adsorbate binding on the Ti-segregated surface. The two factors introduce similar magnitude in energy change for the S and NO adsorption on Ti-segregated surfaces of PtTi subsurface alloys. We predict an adsorption-induced Ti surface segregation that is dependent on the atomic configurations of the Ti-segregated surfaces resulting from the competition of the two factors.

Ab initio random structure searching

It is essential to know the arrangement of the atoms in a material in order to compute and understand its properties. Searching for stable structures of materials using first-principles electronic structure methods, such as density-functional-theory (DFT), is a rapidly growing field. Here we describe our simple, elegant and powerful approach to searching for structures with DFT, which we call ab initio random structure searching (AIRSS). Applications to discovering the structures of solids, point defects, surfaces, and clusters are reviewed. New results for iron clusters on graphene, silicon clusters, polymeric nitrogen, hydrogen-rich lithium hydrides, and boron are presented.

Phase relationships and structures of inorganic crystals by a combination of the cluster expansion method and first principles calculations

Properties of crystalline solutions are generally dependent not only on their chemical composition but also on the configurations of solute atoms and/or point defects. Quantitative knowledge of the configuration-dependent properties is therefore essential for materials design. The cluster expansion (CE) method has been widely used to describe the configurational properties. Increases in computational power and advances in numerical techniques enable us to perform a large set of systematic first principles calculations based on density functional theory (DFT) to be combined with CE calculations. In this paper, our procedure of CE with optimal selections of clusters and DFT structures is described. Two examples of such calculations are then shown. One is the cation arrangement in a series of spinel oxides. The other is arrangement of the oxygen vacancy in a series of tin sub-dioxides.

Predicting the segregation profile of the Pt25Rh75(100) surface from first-principles

The segregation profile of the Pt(25)Rh(75)(100) surface is studied by the combination of density functional theory calculations with the cluster-expansion method and Monte Carlo simulations. We construct the stability diagram for the surface layers, which allows the prediction of the most stable atomic configuration for a given average concentration in those layers. On this basis, we apply the cluster-expansion Hamiltonian in grand-canonical Monte Carlo simulations for the prediction of the temperature-dependent concentration profile. The experimentally found enrichment of Pt in the top layer and depletion in the second layer is nicely confirmed by the calculations.

Uncovering compounds by synergy of cluster expansion and high-throughput methods

Predicting from first-principles calculations whether mixed metallic elements phase-separate or form ordered structures is a major challenge of current materials research. It can be partially addressed in cases where experiments suggest the underlying lattice is conserved, using cluster expansion (CE) and a variety of exhaustive evaluation or genetic search algorithms. Evolutionary algorithms have been recently introduced to search for stable off-lattice structures at fixed mixture compositions. The general off-lattice problem is still unsolved. We present an integrated approach of CE and high-throughput ab initio calculations (HT) applicable to the full range of compositions in binary systems where the constituent elements or the intermediate ordered structures have different lattice types. The HT method replaces the search algorithms by direct calculation of a moderate number of naturally occurring prototypes representing all crystal systems and guides CE calculations of derivative structures. This synergy achieves the precision of the CE and the guiding strengths of the HT. Its application to poorly characterized binary Hf systems, believed to be phase-separating, defines three classes of alloys where CE and HT complement each other to uncover new ordered structures.

Cluster expansion approach for transmutative lattice systems

We propose a cluster expansion (CE) technique that can express any function of atomic arrangement on any given lattice with the same number of lattice points in a single formalism. In the proposed CE, two types of spin variable, σ and τ, on the base lattice and virtual lattice, respectively, are introduced. The former spin variable specifies the occupation of the constituent elements for each lattice point. The latter specifies the positions of each lattice point. Basis functions constructed from the two types of spin variable satisfy completeness and orthonormality for any atomic arrangement on given lattices. As examples, the proposed CE is applied to one- and three-dimensional lattices in a binary system, which clarifies the concept of base and virtual lattices, how the functions of atomic arrangements are expressed in terms of the two types of spin variable, and the efficiency and convergence of the proposed CE with a finite number of clusters and input structures.