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Point clouds are versatile representations of 3D objects and have found widespread application in science and engineering. Many successful deep-learning models have been proposed that use them as input. Some application domains require incorporating exactly physical constraints, including chemical and materials modeling which we focus on in this paper. These constraints include smoothness, and symmetry with respect to translations, rotations, and permutations of identical particles. Most existing architectures in other domains do not fulfill simultaneously all of these requirements and thus are not applicable to atomic-scale simulations. Many of them, however, can be straightforwardly made to incorporate all the physical constraints except for rotational symmetry. We propose a general symmetrization protocol that adds rotational equivariance to any given model while preserving all the other constraints. As a demonstration of the potential of this idea, we introduce the Point Edge Transformer (PET) architecture, which is not intrinsically equivariant but achieves state-of-the-art performance on several benchmark datasets of molecules and solids. A-posteriori application of our general protocol makes PET exactly equivariant, with minimal changes to its accuracy. By alleviating the need to explicitly incorporate rotational symmetry within the model, our method bridges the gap between the approaches used in different communities, and simplifies the design of deep-learning schemes for chemical and materials modeling.

Machine-learning models based on a point-cloud representation of a physical object are ubiquitous in scientific applications and particularly well-suited to the atomic-scale description of molecules and materials. Among the many different approaches that have been pursued, the description of local atomic environments in terms of their neighbor densities has been used widely and very succesfully. We propose a novel density-based method which involves computing ``Wigner kernels''. These are fully equivariant and body-ordered kernels that can be computed iteratively with a cost that is independent of the radial-chemical basis and grows only linearly with the maximum body-order considered. This is in marked contrast to feature-space models, which comprise an exponentially-growing number of terms with increasing order of correlations. We present several examples of the accuracy of models based on Wigner kernels in chemical applications, for both scalar and tensorial targets, reaching state-of-the-art accuracy on the popular QM9 benchmark dataset, and we discuss the broader relevance of these ideas to equivariant geometric machine-learning.

Achieving a complete and symmetric description of a group of point particles, such as atoms in a molecule, is a common problem in physics and theoretical chemistry. The introduction of machine learning to science has made this issue even more critical, as it underpins the ability of a model to reproduce arbitrary physical relationships, and to do so while being consistent with basic symmetries and conservation laws. However, the descriptors that are commonly used to represent point clouds -- most notably those adopted to describe matter at the atomic scale -- are unable to distinguish between special arrangements of particles. This makes it impossible to machine learn their properties. Frameworks that are provably complete exist, but are only so in the limit in which they simultaneously describe the mutual relationship between all atoms, which is impractical. We introduce, and demonstrate on a particularly insidious class of atomic arrangements, a strategy to build descriptors that rely solely on information on the relative arrangement of triplets of particles, but can be used to construct symmetry-adapted models that have universal approximation power.

Spherical harmonics provide a smooth, orthogonal, and symmetry-adapted basis to expand functions on a sphere, and they are used routinely in physical and theoretical chemistry as well as in different fields of science and technology, from geology and atmospheric sciences to signal processing and computer graphics. More recently, they have become a key component of rotationally equivariant models in geometric machine learning, including applications to atomic-scale modeling of molecules and materials. We present an elegant and efficient algorithm for the evaluation of the real-valued spherical harmonics. Our construction features many of the desirable properties of existing schemes and allows to compute Cartesian derivatives in a numerically stable and computationally efficient manner. To facilitate usage, we implement this algorithm in sphericart, a fast C++ library which also provides C bindings, a Python API, and a PyTorch implementation that includes a GPU kernel.

Machine learning (ML) models for molecules and materials commonly rely on a decomposition of the global target quantity into local, atom-centered contributions. This approach is convenient from a computational perspective, enabling large-scale ML-driven simulations with a linear-scaling cost, and also allow for the identification and post-hoc interpretation of contributions from individual chemical environments and motifs to complicated macroscopic properties. However, even though there exist practical justifications for these decompositions, only the global quantity is rigorously defined, and thus it is unclear to what extent the atomistic terms predicted by the model can be trusted. Here, we introduce a quantitative metric, which we call the local prediction rigidity (LPR), that allows one to assess how robust the locally decomposed predictions of ML models are. We investigate the dependence of LPR on the aspects of model training, particularly the composition of training dataset, for a range of different problems from simple toy models to real chemical systems. We present strategies to systematically enhance the LPR, which can be used to improve the robustness, interpretability, and transferability of atomistic ML models.

Due to the subtle balance of intermolecular interactions that govern structure-property relations, predicting the stability of crystal structures formed from molecular building blocks is a highly non-trivial scientific problem. A particularly active and fruitful approach involves classifying the different combinations of interacting chemical moieties, as understanding the relative energetics of different interactions enables the design of molecular crystals and fine-tuning their stabilities. While this is usually performed based on the empirical observation of the most commonly encountered motifs in known crystal structures, we propose to apply a combination of supervised and unsupervised machine-learning techniques to automate the construction of an extensive library of molecular building blocks. We introduce a structural descriptor tailored to the prediction of the binding (lattice) energy and apply it to a curated dataset of organic crystals and exploit its atom-centered nature to obtain a data-driven assessment of the contribution of different chemical groups to the lattice energy of the crystal. We then interpret this library using a low-dimensional representation of the structure-energy landscape and discuss selected examples of the insights into crystal engineering that can be extracted from this analysis, providing a complete database to guide the design of molecular materials.

Alloys composed of several elements in roughly equimolar composition, often referred to as high-entropy alloys, have long been of interest for their thermodynamics and peculiar mechanical properties, and more recently for their potential application in catalysis. They are a considerable challenge to traditional atomistic modeling, and also to data-driven potentials that for the most part have memory footprint, computational effort and data requirements which scale poorly with the number of elements included. We apply a recently proposed scheme to compress chemical information in a lower-dimensional space, which reduces dramatically the cost of the model with negligible loss of accuracy, to build a potential that can describe 25 d-block transition metals. The model shows semi-quantitative accuracy for prototypical alloys, and is remarkably stable when extrapolating to structures outside its training set. We use this framework to study element segregation in a computational experiment that simulates an equimolar alloy of all 25 elements, mimicking the seminal experiments by Cantor et al., and use our observations on the short-range order relations between the elements to define a data-driven set of Hume-Rothery rules that can serve as guidance for alloy design. We conclude with a study of three prototypical alloys, CoCrFeMnNi, CoCrFeMoNi and IrPdPtRhRu, determining their stability and the short-range order behavior of their constituents.

The "quasi-constant" SOAP and ACSF fingerprint manifolds recently discovered by Parsaeifard and Goedecker are a direct consequence of the presence of degenerate pairs of configurations, a known shortcoming of all low-body-order atom-density correlation representations of molecular structures. Contrary to the configurations that are rigorously singular -- that we demonstrate can only occur in finite, discrete sets -- the continuous "quasi-constant" manifolds exhibit low, but non-zero, sensitivity to atomic displacements. Thus, it is possible to build interpolative machine-learning models of high-order interactions along the manifold, even though the numerical instabilities associated with proximity to the exact singularities affect the accuracy and transferability of such models, to an extent that depends on numerical details of the implementation., Comment: Comment on arXiv:2102.06915

Over the past decade inter-atomic potentials based on machine-learning (ML) techniques have become an indispensable tool in the atomic-scale modeling of materials. Trained on energies and forces obtained from electronic-structure calculations, they inherit their predictive accuracy, and extend greatly the length and time scales that are accessible to explicit atomistic simulations. Inexpensive predictions of the energetics of individual configurations have facilitated greatly the calculation of the thermodynamics of materials, including finite-temperature effects and disorder. More recently, machine-learning models have been closing the gap with first-principles calculations in another area: the prediction of arbitrarily complicated functional properties, from vibrational and optical spectroscopies to electronic excitations. The implementation of integrated machine-learning models, that combine energetic and functional predictions with statistical and dynamical sampling of atomic-scale properties is bringing the promise of predictive, uncompromising simulations of existing and novel materials closer to its full realisation.

Selecting the most relevant features and samples out of a large set of candidates is a task that occurs very often in the context of automated data analysis, where it can be used to improve the computational performance, and also often the transferability, of a model. Here we focus on two popular sub-selection schemes which have been applied to this end: CUR decomposition, that is based on a low-rank approximation of the feature matrix and Farthest Point Sampling, that relies on the iterative identification of the most diverse samples and discriminating features. We modify these unsupervised approaches, incorporating a supervised component following the same spirit as the Principal Covariates Regression (PCovR) method. We show that incorporating target information provides selections that perform better in supervised tasks, which we demonstrate with ridge regression, kernel ridge regression, and sparse kernel regression. We also show that incorporating aspects of simple supervised learning models can improve the accuracy of more complex models, such as feed-forward neural networks. We present adjustments to minimize the impact that any subselection may incur when performing unsupervised tasks. We demonstrate the significant improvements associated with the use of PCov-CUR and PCov-FPS selections for applications to chemistry and materials science, typically reducing by a factor of two the number of features and samples which are required to achieve a given level of regression accuracy.

Many-body descriptors are widely used to represent atomic environments in the construction of machine learned interatomic potentials and more broadly for fitting, classification and embedding tasks on atomic structures. It was generally believed that 3-body descriptors uniquely specify the environment of an atom, up to a rotation and permutation of like atoms. We produce several counterexamples to this belief, with the consequence that any classifier, regression or embedding model for atom-centred properties that uses 3 (or 4)-body features will incorrectly give identical results for different configurations. Writing global properties (such as total energies) as a sum of many atom-centred contributions mitigates, but does not eliminate, the impact of this fundamental deficiency -- explaining the success of current "machine-learning" force fields. We anticipate the issues that will arise as the desired accuracy increases, and suggest potential solutions.

Statistical learning algorithms are finding more and more applications in science and technology. Atomic-scale modeling is no exception, with machine learning becoming commonplace as a tool to predict energy, forces and properties of molecules and condensed-phase systems. This short review summarizes recent progress in the field, focusing in particular on the problem of representing an atomic configuration in a mathematically robust and computationally efficient way. We also discuss some of the regression algorithms that have been used to construct surrogate models of atomic-scale properties. We then show examples of how the optimization of the machine-learning models can both incorporate and reveal insights onto the physical phenomena that underlie structure-property relations.

Infrared pump-probe spectroscopy provides detailed information on the dynamics of hydrogen-bonded liquids. Due to dissipation of the absorbed pump pulse energy, also thermal equilibration dynamics contributes to the observed signal. Disentangling this contribution from the molecular response remains an open challenge. Performing non-equilibrium molecular dynamics simulat ions of liquid deuterated methanol, we show that faster molecular vibrational relaxation and slower heat diffusion are decoupled and occur on different length scale. Transient structures of the hydrogen bonding network influence thermal relaxation by affecting thermal diffusivity over the length-scale of several nanometers.

The calculation of chemical shifts in solids has enabled methods to determine crystal structures in powders. The dependence of chemical shifts on local atomic environments sets them among the most powerful tools for structure elucidation of powdered solids or amorphous materials. Unfortunately, this dependency comes with the cost of high accuracy first-principle calculations to qualitatively predict chemical shifts in solids. Machine learning methods have recently emerged as a way to overcome the need for explicit high accuracy first-principle calculations. However, the vast chemical and combinatorial space spanned by molecular solids, together with the strong dependency of chemical shifts of atoms on their environment, poses a huge challenge for any machine learning method. Here we propose a machine learning method based on local environments to accurately predict chemical shifts of different molecular solids and of different polymorphs within DFT accuracy (RMSE of 0.49 ppm ( 1 H), 4.3ppm ( 13 C), 13.3 ppm ( 15 N), and 17.7 ppm ( 17 O) with $R^2$ of 0.97 for 1 H, 0.99 for 13 C, 0.99 for 15 N, and 0.99 for 17 O). We also demonstrate that the trained model is able to correctly determine, based on the match between experimentally-measured and ML-predicted shifts, structures of cocaine and the drug 4-[4-(2-adamantylcarbamoyl)-5-tert-butylpyrazol-1-yl]benzoic acid in an chemical shift based NMR crystallography approach., Includes supporting information (only the PDF section)

The curvature dependence of interfacial free energy, which is crucial in quantitatively predicting nucleation kinetics and the stability of bubbles and droplets, can be described in terms of the Tolman length {\delta}. For solid-liquid interfaces, however,{\delta} has never been computed directly due to various theoretical and practical challenges. Here we present a general method that enables the direct evaluation of the Tolman length from atomistic simulations of a solid-liquid planar interface in out-of-equilibrium conditions. This method works by first measuring the surface tension from the amplitude of thermal capillary fluctuations of a localized version of Gibbs dividing surface, and bythen computing the free energy difference between the surface of tension and the equimolar dividing surface. For benchmark purposes, we computed {\delta}for a model potential, and compared the results to less rigorous indirect approaches.

One of the many peculiar properties of water is the pronounced deviation of the proton momentum distribution from Maxwell-Boltzmann behaviour. This deviation from the classical limit is a manifestation of the quantum mechanical nature of protons. Its extent, which can be probed directly by Deep Inelastic Neutron Scattering (DINS) experiments, gives important insight on the potential of mean force felt by H atoms. The determination of the full distribution of particle momenta, however, is a real tour de force for both experiments and theory, which has led to unresolved discrepancies between the two. In this Letter we present comprehensive, fully-converged momentum distributions for water at several thermodynamic state points, focusing on the components that cannot be described in terms of a scalar contribution to the quantum kinetic energy, and providing a benchmark that can serve as a reference for future simulations and experiments. In doing so, we also introduce a number of technical developments that simplify and accelerate greatly the calculation of momentum distributions by means of atomistic simulations., Comment: 19 pages, 4 figures

We investigate the thermodynamics and kinetics of a hydrogen interstitial in magnetic ��-iron, taking account of the quantum fluctuations of the proton as well as the anharmonicities of lattice vibrations and hydrogen hopping. We show that the diffusivity of hydrogen in the lattice of BCC iron deviates strongly from an Arrhenius behavior at and below room temperature. We compare a quantum transition state theory to explicit ring polymer molecular dynamics in the calculation of diffusivity and we find that the role of phonons is to inhibit, not to enhance, diffusivity at intermediate temperatures in constrast to the usual polaron picture of hopping. We then address the trapping of hydrogen by a vacancy as a prototype lattice defect. By a sequence of steps in a thought experiment, each involving a thermodynamic integration, we are able to separate out the binding free energy of a proton to a defect into harmonic and anharmonic, and classical and quantum contributions. We find that about 30% of a typical binding free energy of hydrogen to a lattice defect in iron is accounted for by finite temperature effects and about half of these arise from quantum proton fluctuations. This has huge implications for the comparison between thermal desorption and permeation experiments and standard electronic structure theory. The implications are even greater for the interpretation of muon spin resonance experiments.

Including quantum mechanical effects on the dynamics of nuclei in the condensed phase is challenging, because the complexity of exact methods grows exponentially with the number of quantum degrees of freedom. Efforts to circumvent these limitations can be traced down to two approaches: methods that treat a small subset of the degrees of freedom with rigorous quantum mechanics, considering the rest of the system as a static or classical environment, and methods that treat the whole system quantum mechanically, but using approximate dynamics. Here we perform a systematic comparison between these two philosophies for the description of quantum effects in vibrational spectroscopy, taking the Embedded Local Monomer (LMon) model and a mixed quantum-classical (MQC) model as representatives of the first family of methods, and centroid molecular dynamics (CMD) and thermostatted ring polymer molecular dynamics (TRPMD) as examples of the latter. We use as benchmarks D$_2$O doped with HOD and pure H$_2$O at three distinct thermodynamic state points (ice Ih at 150K, and the liquid at 300K and 600K), modeled with the simple q-TIP4P/F potential energy and dipole moment surfaces. With few exceptions the different techniques yield IR absorption frequencies that are consistent with one another within a few tens of cm$^{-1}$. Comparison with classical molecular dynamics demonstrates the importance of nuclear quantum effects up to the highest temperature, and a detailed discussion of the discrepancies between the various methods let us draw some (circumstantial) conclusions about the impact of the very different approximations that underlie them. Such cross validation between radically different approaches could indicate a way forward to further improve the state of the art in simulations of condensed-phase quantum dynamics., Comment: Bundled together with the Supporting Materials

Long-range ion induced water-water correlations were recently observed in femtosecond elastic second harmonic scattering experiments of electrolyte solutions. To further the qualitative understanding of these correlations, we derive an analytical expression that quantifies ion induced dipole-dipole correlations in a non-interacting gas of dipoles. This model is a logical extension of Debye-H\"uckel theory that can be used to qualitatively understand how the combined electric field of the ions induces correlations in the orientational distributions of the water molecules in an aqueous solution. The model agrees with results from molecular dynamics simulations and provides an important starting point for further theoretical work.

Recently, we have shown how a colored-noise Langevin equation can be used in the context of molecular dynamics as a tool to obtain dynamical trajectories whose properties are tailored to display desired sampling features. In the present paper, after having reviewed some analytical results for the stochastic differential equations forming the basis of our approach, we describe in detail the implementation of the generalized Langevin equation thermostat and the fitting procedure used to obtain optimal parameters. We discuss in detail the simulation of nuclear quantum effects, and demonstrate that, by carefully choosing parameters, one can successfully model strongly anharmonic solids such as neon. For the reader's convenience, a library of thermostat parameters and some demonstrative code can be downloaded from an on-line repository.

Based on density functional perturbation theory, we have studied the vibrational properties of three crystalline phases of tellurium dioxide: paratellurite alpha-TeO2, tellurite beta-TeO2, and the new phase gamma-TeO2, recently identified experimentally. Calculated Raman and IR spectra are in good agreement with available experimental data. The vibrational spectra of alpha- and beta-TeO2 can be interpreted in terms of vibrations of TeO2 molecular units.

The properties of molecules and materials containing light nuclei are affected by their quantum mechanical nature. Accurate modeling of these quantum nuclear effects requires computationally demanding path integral techniques. Considerable success has been achieved in reducing the cost of such simulations by using generalized Langevin dynamics to induce frequency-dependent fluctuations. Path integral generalized Langevin equation methods, however, have this far been limited to the study of static, thermodynamic properties due to the large perturbation to the system's dynamics induced by the aggressive thermostatting. Here, we introduce a post-processing scheme, based on analytical estimates of the dynamical perturbation induced by the generalized Langevin dynamics, which makes it possible to recover meaningful time correlation properties from a thermostatted trajectory. We show that this approach yields spectroscopic observables for model and realistic systems that have an accuracy comparable to much more demanding approximate quantum dynamics techniques based on full path integral simulations.

Statistical learning based on a local representation of atomic structures provides a universal model of chemical stability., Determining the stability of molecules and condensed phases is the cornerstone of atomistic modeling, underpinning our understanding of chemical and materials properties and transformations. We show that a machine-learning model, based on a local description of chemical environments and Bayesian statistical learning, provides a unified framework to predict atomic-scale properties. It captures the quantum mechanical effects governing the complex surface reconstructions of silicon, predicts the stability of different classes of molecules with chemical accuracy, and distinguishes active and inactive protein ligands with more than 99% reliability. The universality and the systematic nature of our framework provide new insight into the potential energy surface of materials and molecules.