Gaussian approximation potentials: A brief tutorial introduction

We present a swift walk-through of our recent work that uses machine learning to fit interatomic potentials based on quantum mechanical data. We describe our Gaussian Approximation Potentials (GAP) framework, discussing a variety of descriptors, how to train the model on total energies and derivatives and the simultaneous use of multiple models. We also show a small example using QUIP, the software sandbox implementation of GAP that is available for non-commercial use.A.P.B. is supported by a Leverhulme Early Career Fellowship and the Isaac Newton Trust. We would like to thank our referees for their comments during the revision process.This is the author accepted manuscript. The final version is available via Wiley at http://onlinelibrary.wiley.com/doi/10.1002/qua.24927/abstract

Many-Body Dispersion Correction Effects on Bulk and Surface Properties of Rutile and Anatase TiO2_2

Titanium dioxide (titania, TiO$_2$) is a widely studied material with diverse applications. Here, we explore how pairwise and many-body descriptions of van der Waals dispersion interactions perform in atomistic modeling of the two most important TiO$_2$ polymorphs, rutile and anatase. In particular, we obtain an excellent description of both bulk structures from density-functional theory (DFT) computations with the many-body dispersion (MBD) method of Tkatchenko and co-workers coupled to an iterative Hirshfeld partitioning scheme ("Hirshfeld-I"). Beyond the bulk, we investigate the most important crystal surfaces, namely, rutile (110), (101), and (100) and anatase (101), (100), and (001). Dispersion has a highly anisotropic effect on the different ($\textit{hkl}$) surfaces; this directly changes the predicted nanocrystal morphology as determined from Wulff constructions. The periodic DFT+MBD method combined with Hirshfeld-I partitioning appears to be promising for future large-scale atomistic studies of this technologically important material.V.L.D. gratefully acknowledges a postdoctoral fellowship from the Alexander von Humboldt Foundation. This work used the ARCHER UK National Supercomputing Service, access to which was granted via support for the UKCP consortium (Engineering and Physical Sciences Research Council Grant EP/K014560/1)

Nested Transition Path Sampling.

We introduce a novel transition path (TPS) sampling scheme employing nested sampling. Analogous to how nested sampling explores the entire configurational phase space for atomistic systems, nested TPS samples the entire available trajectory space in one simulation. Thermodynamic and path observables can be constructed a posteriori for all temperatures simultaneously. We exploit this to compute the rate of rare processes at arbitrarily low temperature through the coupling to easily accessible rates at high temperature. We illustrate the method on several model systems

Bayesian inference of the spatial distributions of material properties

The inverse problem of estimating the spatial distributions of elastic material properties from noisy strain measurements is ill-posed. However, it is still typically treated as an optimisation problem to maximise a likelihood function that measures the agreement between the measured and theoretically predicted strains. Here we propose an alternative approach employing Bayesian inference with Nested Sampling used to explore parameter space and compute Bayesian evidence. This approach not only aids in identifying the basis function set (referred to here as a model) that best describes the spatial material property distribution but also allows us to estimate the uncertainty in the predictions. Increasingly complex models with more parameters generate very high likelihood solutions and thus are favoured by a maximum likelihood approach. However, these models give poor predictions of the material property distributions with a large associated uncertainty as they overfit the noisy data. On the other hand, the Bayes’ factor peaks for a relatively simple model and indicates that this model is most appropriate even though its likelihood is comparatively low. Intriguingly, even for the appropriate model that has a unique maximum likelihood solution, the measurement noise is amplified to give large errors in the predictions of the maximum likelihood solution. By contrast, the mean of the posterior probability distribution reduces the effect of noise in the data and predicts the material properties with significantly higher fidelity. Simpler model selection criteria such as the Bayesian information criterion are shown to fail due to the non-Gaussian nature of the posterior distribution of the parameters. This makes accurate evaluation of the posterior distribution and the associated Bayesian evidence integral (by Nested Sampling or other means) imperative for this class of problems. The output of the Nested Sampling algorithm is also used to construct likelihood landscapes. These landscapes show the existence of multiple likelihood maxima when there is paucity of data and/or for overly complex models. They thus graphically illustrate the pitfalls in using optimisation methods to search for maximum likelihood solutions in such inverse problems.Royal Societ

Partitioning of sulfur between solid and liquid iron under Earth's core conditions: Constraints from atomistic simulations with machine learning potentials

Partition coefficients of light elements between the solid and liquid iron phases are crucial for uncovering the state and dynamics of the Earth's core. As one of the major light element candidates, sulfur has attracted extensive interests for measuring its partitioning and phase behaviors over the last several decades, but the relevant experimental data under Earth's core conditions are still scarce. In this study, using a toolkit consisting of electronic structure theory, high-accuracy machine learning potentials and rigorous free energy calculations, we establish an efficient and extendible framework for predicting complex phase behaviors of iron alloys under extreme conditions. As a first application of this framework, we predict the partition coefficients of sulfur over wide range of temperatures and pressures (from 4000 K, 150 GPa to 6000 K, 330 GPa), which are demonstrated to be in good agreement with previous experiments and ab initio simulations. After a continuous increase below ∼250 GPa, the partition coefficient is found to be around 0.75 ± 0.07 at higher pressures and are essentially temperature-independent. Given these predictions, the partitioning of sulfur is confirmed to be insufficient to account for the observed density jump across the Earth's inner core boundary and its roles on the geodynamics of the Earth's core should be minor

Hydrogen induced fast-fracture

One of the recurring anomalies in the hydrogen induced fracture of high strength steels is the apparent disconnect between their toughness and uniaxial tensile strength in identical hydrogen environments. Here we propose, supported by detailed atomistic and continuum calculations, that unlike macroscopic toughness, hydrogen-mediated tensile failure is a result of a fast-fracture mechanism. Specifically, we show that failure originates from the fast propagation of cleavage cracks that initiate from cavities that form around inclusions such as carbide particles. The failure process occurs in two stages. In stage-A, hydrides rapidly form around the roots of stressed notches on the cavity surfaces with hydrogen fed from the hydrogen gas within the cavity. These hydrides promote cleavage fracture with the cracks propagating at >100 ms^(-1) until the hydrogen gas in the cavity is exhausted. Predictions of this hydrogen-assisted crack growth mechanism are supported by atomistic calculations of binding energies, mobility barriers and molecular dynamics calculations of the fracture process. Typically, cracks grow by less than 1 μm via this hydrogen-assisted mechanism and thus insufficient to cause macroscopic fracture of the specimen. However, this stage is then followed by a stage-B process where these fast propagating cracks can continue to grow, now in the absence of hydrogen supply, given an appropriate level of remote tensile stress. This is surprising because the fracture energy is now that of Fe in the absence of H and cleavage fracture requires opening tractions on the order of 15 GPa to be generated. Thus, fracture is usually precluded due to plasticity around the crack-tip. Here we show via macroscopic continuum crack growth calculations in a rate dependent elastic-plastic solid with fracture modelled using a cohesive zone that cleavage is possible if the crack propagates fast enough. This is because strain-rates at the tips of fast propagating cracks are sufficiently high for the drag on the motion of dislocations resulting from phonon scattering to limit plasticity. This combined atomistic/continuum model is used to explain a host of well-established experimental observations including (but not limited to): (i) insensitivity of the strength to the concentration of trapped hydrogen; (ii) the extensive microcracking in addition to the final cleavage fracture event and (iii) the higher susceptibility of high strength steels to hydrogen embrittlement. Importantly, we also show that the stage-A hydrogen-assisted fracture process only occurs in certain crystallographic orientations with crack-tip plasticity processes, such as twinning, blunting cracks in other orientations. This inhibits the fast-fracture mechanism in a macroscopic toughness on a polycrystalline material and thus explains the apparent contradiction between the hydrogen-assisted macroscopic toughness and tensile strength of steels.EPSRC EP/L014742/

Understanding the thermal properties of amorphous solids using machine-learning-based interatomic potentials

© 2018 Informa UK Limited, trading as Taylor & Francis Group. Understanding the thermal properties of disordered systems is of fundamental importance for condensed matter physics - and for practical applications as well. While quantities such as the thermal conductivity are usually well characterised experimentally, their microscopic origin is often largely unknown - hence the pressing need for molecular simulations. However, the time and length scales involved with thermal transport phenomena are typically well beyond the reach of ab initio calculations. On the other hand, many amorphous materials are characterised by a complex structure, which prevents the construction of classical interatomic potentials. One way to get past this deadlock is to harness machine-learning (ML) algorithms to build interatomic potentials: these can be nearly as computationally efficient as classical force fields while retaining much of the accuracy of first-principles calculations. Here, we discuss neural network potentials (NNPs) and Gaussian approximation potentials (GAPs), two popular ML frameworks. We review the work that has been devoted to investigate, via NNPs, the thermal properties of phase-change materials, systems widely used in non-volatile memories. In addition, we present recent results on the vibrational properties of amorphous carbon, studied via GAPs. In light of these results, we argue that ML-based potentials are among the best options available to further our understanding of the vibrational and thermal properties of complex amorphous solids

First-principles energetics of water clusters and ice: A many-body analysis

Standard forms of density-functional theory (DFT) have good predictive power for many materials, but are not yet fully satisfactory for cluster, solid, and liquid forms of water. Recent work has stressed the importance of DFT errors in describing dispersion, but we note that errors in other parts of the energy may also contribute. We obtain information about the nature of DFT errors by using a many-body separation of the total energy into its 1-body, 2-body, and beyond-2-body components to analyze the deficiencies of the popular PBE and BLYP approximations for the energetics of water clusters and ice structures. The errors of these approximations are computed by using accurate benchmark energies from the coupled-cluster technique of molecular quantum chemistry and from quantum Monte Carlo calculations. The systems studied are isomers of the water hexamer cluster, the crystal structures Ih, II, XV, and VIII of ice, and two clusters extracted from ice VIII. For the binding energies of these systems, we use the machine-learning technique of Gaussian Approximation Potentials to correct successively for 1-body and 2-body errors of the DFT approximations. We find that even after correction for these errors, substantial beyond-2-body errors remain. The characteristics of the 2-body and beyond-2-body errors of PBE are completely different from those of BLYP, but the errors of both approximations disfavor the close approach of non-hydrogen-bonded monomers. We note the possible relevance of our findings to the understanding of liquid water