One of the most exciting tools that have entered the material science toolbox in recent years is machine learning. This collection of statistical methods has already proved to be capable of considerably speeding up both fundamental and applied research. At present, we are witnessing an explosion of works that develop and apply machine learning to solid-state systems. We provide a comprehensive overview and analysis of the most recent research in this topic. As a starting point, we introduce machine learning principles, algorithms, descriptors, and databases in materials science. We continue with the description of different machine learning approaches for the discovery of stable materials and the prediction of their crystal structure. Then we discuss research in numerous quantitative structure-property relationships and various approaches for the replacement of first-principle methods by machine learning. We review how active learning and surrogate-based optimization can be applied to improve the rational design process and related examples of applications. Two major questions are always the interpretability of and the physical understanding gained from machine learning models. We consider therefore the different facets of interpretability and their importance in materials science. Finally, we propose solutions and future research paths for various challenges in computational materials science.

We perform a large scale benchmark of machine learning methods for the prediction of the thermodynamic stability of solids. We start by constructing a data set that comprises density functional theory calculations of around 250000 cubic perovskite systems. This includes all possible perovskite and antiperovskite crystals that can be generated with elements from hydrogen to bismuth, excluding rare gases and lanthanides. Incidentally, these calculations already reveal a large number of systems (around 500) that are thermodynamically stable but that are not present in crystal structure databases. Moreover, some of these phases have unconventional compositions and define completely new families of perovskites. This data set is then used to train and test a series of machine learning algorithms to predict the energy distance to the convex hull of stability. In particular, we study the performance of ridge regression, random forests, extremely randomized trees (including adaptive boosting), and neural networks. We find that extremely randomized trees give the smallest mean absolute error of the distance to the convex hull (121 meV/atom) in the test set of 230000 perovskites, after being trained in 20000 samples. Surprisingly, the machine already works if we give it as sole input features the group and row in the periodic table of the three elements composing the perovskite. Moreover, we find that the prediction accuracy is not uniform across the periodic table, being worse for first-row elements and elements forming magnetic compounds. Our results suggest that machine learning can be used to speed up considerably (by at least a factor of 5) high-throughput DFT calculations, by restricting the space of relevant chemical compositions without degradation of the accuracy.

We re-examine the analysis of the equilibrium limits of the fewest switches surface hopping algorithm for mixed quantum-classical dynamics. In contrast with previously reported results, we show that surface hopping does not, in general, exactly yield Boltzmann equilibrium, but that in practice the observed deviations are quite small. We also demonstrate that surface hopping does approach the exact equilibrium distribution in both the limits of small adiabatic splitting and/or strong nonadiabatic coupling. We verify these analytical results with numerical simulations for a simple two-level quantum system connected to a bath of classical particles.

Natural bond orbital (NBO) analysis is a powerful analysis technique capable of generating intuitive chemical representations of otherwise complex quantum mechanical electronic structure results, yielding a localized "Lewis-like" description of bonding and reactivity. We generalize this algorithm to periodic systems, thus expanding the scope of NBO analysis to bulk materials and/or periodic surface models. We employ a projection scheme to further expand the algorithm's applicability to ubiquitous plane-wave density functional theory (PW DFT) calculations. We also present a variety of example applications: examining bulk bonding and surface reconstruction and elucidating fundamental aspects of heterogeneous catalysis-all derived from rigorous underlying PW DFT calculations.

We conducted a large-scale density-functional theory study on the influence of the exchange-correlation functional in the calculation of electronic band gaps of solids. First, we use the large materials data set that we have recently proposed to benchmark 21 different functionals, with a particular focus on approximations of the meta-generalized-gradient family. Combining these data with the results for 12 functionals in our previous work, we can analyze in detail the characteristics of each approximation and identify its strong and/or weak points. Beside confirming that mBJ, HLE16 and HSE06 are the most accurate functionals for band gap calculations, we reveal several other interesting functionals, chief among which are the local Slater potential approximation, the GGA AK13LDA, and the meta-GGAs HLE17 and TASK. We also compare the computational efficiency of these different approximations. Relying on these data, we investigate the potential for improvement of a promising subset of functionals by varying their internal parameters. The identified optimal parameters yield a family of functionals fitted for the calculation of band gaps. Finally, we demonstrate how to train machine learning models for accurate band gap prediction, using as input structural and composition data, as well as approximate band gaps obtained from density-functional theory.