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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 allows for the identification and posthoc interpretation of contributions from individual chemical environments and motifs to complicated macroscopic properties. However, even though practical justifications exist for the local decomposition, only the global quantity is rigorously defined. Thus, when the atom-centered contributions are used, their sensitivity to the training strategy or the model architecture should be carefully considered. To this end, 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 the LPR on the aspects of model training, particularly the composition of training data set, 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.

The electron density of a molecule or material has recently received major attention as a target quantity of machine-learning models. A natural choice to construct a model that yields transferable and linear-scaling predictions is to represent the scalar field using a multicentered atomic basis analogous to that routinely used in density fitting approximations. However, the nonorthogonality of the basis poses challenges for the learning exercise, as it requires accounting for all the atomic density components at once. We devise a gradient-based approach to directly minimize the loss function of the regression problem in an optimized and highly sparse feature space. In so doing, we overcome the limitations associated with adopting an atom-centered model to learn the electron density over arbitrarily complex data sets, obtaining very accurate predictions using a comparatively small training set. The enhanced framework is tested on 32-molecule periodic cells of liquid water, presenting enough complexity to require an optimal balance between accuracy and computational efficiency. We show that starting from the predicted density a single Kohn-Sham diagonalization step can be performed to access total energy components that carry an error of just 0.1 meV/atom with respect to the reference density functional calculations. Finally, we test our method on the highly heterogeneous QM9 benchmark data set, showing that a small fraction of the training data is enough to derive ground-state total energies within chemical accuracy.

The application of machine learning to theoretical chemistry has made it possible to combine the accuracy of quantum chemical energetics with the thorough sampling of finite-temperature fluctuations. To reach this goal, a diverse set of methods has been proposed, ranging from simple linear models to kernel regression and highly nonlinear neural networks. Here we apply two widely different approaches to the same, challenging problem: the sampling of the conformational landscape of polypeptides at finite temperature. We develop a local kernel regression (LKR) coupled with a supervised sparsity method and compare it with a more established approach based on Behler-Parrinello type neural networks. In the context of the LKR, we discuss how the supervised selection of the reference pool of environments is crucial to achieve accurate potential energy surfaces at a competitive computational cost and leverage the locality of the model to infer which chemical environments are poorly described by the DFTB baseline. We then discuss the relative merits of the two frameworks and perform Hamiltonian-reservoir replica-exchange Monte Carlo sampling and metadynamics simulations, respectively, to demonstrate that both frameworks can achieve converged and transferable sampling of the conformational landscape of complex and flexible biomolecules with comparable accuracy and computational cost.

We introduce a local machine-learning method for predicting the electron densities of periodic systems. The framework is based on a numerical, atom-centered auxiliary basis, which enables an accurate expansion of the all-electron density in a form suitable for learning isolated and periodic systems alike. We show that, using this formulation, the electron densities of metals, semiconductors, and molecular crystals can all be accurately predicted using symmetry-adapted Gaussian process regression models, properly adjusted for the nonorthogonal nature of the basis. These predicted densities enable the efficient calculation of electronic properties, which present errors on the order of tens of meV/atom when compared to ab initio density-functional calculations. We demonstrate the key power of this approach by using a model trained on ice unit cells containing only 4 water molecules to predict the electron densities of cells containing up to 512 molecules and see no increase in the magnitude of the errors of derived electronic properties when increasing the system size. Indeed, we find that these extrapolated derived energies are more accurate than those predicted using a direct machine-learning model. Finally, on heterogeneous data sets SALTED can predict electron densities with errors below 4%.

We present a generally applicable computational framework for the efficient and accurate characterization of molecular structural patterns and acid properties in an explicit solvent using H 2 O 2 and CH 3 SO 3 H in phenol as an example. To address the challenges posed by the complexity of the problem, we resort to a set of data-driven methods and enhanced sampling algorithms. The synergistic application of these techniques makes the first-principle estimation of the chemical properties feasible without renouncing to the use of explicit solvation, involving extensive statistical sampling. Ensembles of neural network (NN) potentials are trained on a set of configurations carefully selected out of preliminary simulations performed at a low-cost density functional tight-binding (DFTB) level. The energy and forces of these configurations are then recomputed at the hybrid density functional theory (DFT) level and used to train the neural networks. The stability of the NN model is enhanced by using DFTB energetics as a baseline, but the efficiency of the direct NN ( i.e ., baseline-free) is exploited via a multiple-time-step integrator. The neural network potentials are combined with enhanced sampling techniques, such as replica exchange and metadynamics, and used to characterize the relevant protonated species and dominant noncovalent interactions in the mixture, also considering nuclear quantum effects.

Machine learning encompasses tools and algorithms that are now becoming popular in almost all scientific and technological fields. This is true for molecular dynamics as well, where machine learning offers promises of extracting valuable information from the enormous amounts of data generated by simulation of complex systems. We provide here a review of our current understanding of goals, benefits, and limitations of machine learning techniques for computational studies on atomistic systems, focusing on the construction of empirical force fields from ab initio databases and the determination of reaction coordinates for free energy computation and enhanced sampling.

Imaginary time path-integral (PI) simulations that account for nuclear quantum effects (NQE) beyond the harmonic approximation are increasingly employed together with modern electronic-structure calculations. Existing PI methods are applicable to molecules, liquids, and solids; however, the computational cost of such simulations increases dramatically with decreasing temperature. To address this challenge, here, we propose to combine high-order PI factorization with perturbation theory (PT). Already for conventional second-order PI simulations, the PT ansatz increases the accuracy 2-fold compared to fourth-order schemes with the same settings. In turn, applying PT to high-order path integrals (HOPI) further improves the efficiency of simulations for molecular and condensed matter systems especially at low temperatures. We present results for bulk liquid water, the aspirin molecule, and the CH 5 + molecule. Perturbed HOPI simulations remain both efficient and accurate down to 20 K and provide a convenient method to estimate the convergence of quantum-mechanical observables.

Quantitative evaluation of the thermodynamic properties of materials-most notably their stability, as measured by the free energy-must take into account the role of thermal and zero-point energy fluctuations. While these effects can easily be estimated within a harmonic approximation, corrections arising from the anharmonic nature of the interatomic potential are often crucial and require computationally costly path integral simulations to obtain results that are essentially exact for a given potential. Consequently, different approximate frameworks for computing affordable estimates of the anharmonic free energies have been developed over the years. Understanding which of the approximations involved are justified for a given system, and therefore choosing the most suitable method, is complicated by the lack of comparative benchmarks. To facilitate this choice we assess the accuracy and efficiency of some of the most commonly used approximate methods: the independent mode framework, the vibrational self-consistent field, and self-consistent phonons. We compare the anharmonic correction to the Helmholtz free energy against reference path integral calculations. These benchmarks are performed for a diverse set of systems, ranging from simple weakly anharmonic solids to flexible molecular crystals with freely rotating units. The results suggest that, for simple solids such as allotropes of carbon, these methods yield results that are in excellent agreement with the reference calculations, at a considerably lower computational cost. For more complex molecular systems such as polymorphs of ice and paracetamol the methods do not consistently provide a reliable approximation of the anharmonic correction. Despite substantial cancellation of errors when comparing the stability of different phases, we do not observe a systematic improvement over the harmonic approximation even for relative free energies. We conclude that, at least for the classes of materials considered here, efforts toward obtaining computationally feasible anharmonic free energies should therefore be directed toward reducing the expense of path integral methods.

Metal-organic frameworks show both fundamental interest and great promise for applications in adsorption-based technologies, such as the separation and storage of gases. The flexibility and complexity of the molecular scaffold pose a considerable challenge to atomistic modeling, especially when also considering the presence of guest molecules. We investigate the role played by quantum and anharmonic fluctuations in the archetypical case of MOF-5, comparing the material at various levels of methane loading. Accurate path integral simulations of such effects are made affordable by the introduction of an accelerated simulation scheme and the use of an optimized force field based on first-principles reference calculations. We find that the level of statistical treatment that is required for predictive modeling depends significantly on the property of interest. The thermal properties of the lattice are generally well described by a quantum harmonic treatment, with the adsorbate behaving in a classical but strongly anharmonic manner. The heat capacity of the loaded framework-which plays an important role in the characterization of the framework and in determining its stability to thermal fluctuations during adsorption/desorption cycles-requires, however, a full quantum and anharmonic treatment, either by path integral methods or by a simple but approximate scheme. We also present molecular-level insight into the nanoscopic interactions contributing to the material's properties and suggest design principles to optimize them.

We present a scheme to obtain an inexpensive and reliable estimate of the uncertainty associated with the predictions of a machine-learning model of atomic and molecular properties. The scheme is based on resampling, with multiple models being generated based on subsampling of the same training data. The accuracy of the uncertainty prediction can be benchmarked by maximum likelihood estimation, which can also be used to correct for correlations between resampled models and to improve the performance of the uncertainty estimation by a cross-validation procedure. In the case of sparse Gaussian Process Regression models, this resampled estimator can be evaluated at negligible cost. We demonstrate the reliability of these estimates for the prediction of molecular and materials energetics and for the estimation of nuclear chemical shieldings in molecular crystals. Extension to estimate the uncertainty in energy differences, forces, or other correlated predictions is straightforward. This method can be easily applied to other machine-learning schemes and will be beneficial to make data-driven predictions more reliable and to facilitate training-set optimization and active-learning strategies.

The Density Overlap Region Indicator (DORI) is a density-based scalar field that reveals covalent bonding patterns and noncovalent interactions in the same value range. This work goes beyond the traditional static quantum chemistry use of scalar fields and illustrates the suitability of DORI for analyzing geometrical and electronic signatures in highly fluxional molecular systems. Examples include a dithiocyclophane, which possesses multiple local minima with differing extents of π-stacking interactions and a temperature dependent rotation of a molecular rotor, where the descriptor is employed to capture fingerprints of CH-π and π-π interactions. Finally, DORI serves to examine the fluctuating π-conjugation pathway of a photochromic torsional switch (PTS). Attention is also placed on postprocessing the large amount of generated data and juxtaposing DORI with a data-driven low-dimensional representation of the structural landscape.

Most of the current understanding of structure-property relations at the molecular and the supramolecular scales can be formulated in terms of the stability of and the interactions between a limited number of recurring structural motifs (e.g., H-bonds, coordination polyhedra, and protein secondary structure). Here we demonstrate an algorithm to automatically recognize such patterns, based on the identification of local maxima in the probability distributions observed in atomistic computer simulations, which is robust to the dimensionality and the sparsity of the reference atomistic data. We first discuss its main features, demonstrating some on artificial data sets, and then show how it can be applied to identify coordination environments in Lennard-Jones clusters and to recognize secondary-structure patterns in the simulation of an oligopeptide. To assess the applicability of this algorithm for motifs that involve several interdependent degrees of freedom, we also employ it to identify groups of conformers of the cluster and the polypeptide, considered in their entirety. The motifs identified by analyzing atomistic simulations can be used to interpret and rationalize the stability and behavior of the system at hand, and also as a tool to accelerate sampling, in association with biased molecular dynamics schemes.

We introduce a nonequilibrium molecular dynamics simulation approach, based on the generalized Langevin equation, to study vibrational energy relaxation in pump-probe spectroscopy. A colored noise thermostat is used to selectively excite a set of vibrational modes, leaving the other modes nearly unperturbed, to mimic the effect of a monochromatic laser pump. Energy relaxation is probed by analyzing the evolution of the system after excitation in the microcanonical ensemble, thus providing direct information about the energy redistribution paths at the molecular level and their time scale. The method is applied to hydrogen-bonded molecular liquids, specifically deuterated methanol and water, providing a robust picture of energy relaxation at the molecular scale.

The hydrogen-bond network of water is characterized by the presence of coordination defects relative to the ideal tetrahedral network of ice, whose fluctuations determine the static and time-dependent properties of the liquid. Because of topological constraints, such defects do not come alone but are highly correlated coming in a plethora of different pairs. Here we discuss in detail such correlations in the case of ab initio water models and show that they have interesting similarities to regular and defective solid phases of water. Although defect correlations involve deviations from idealized tetrahedrality, they can still be regarded as weaker hydrogen bonds that retain a high degree of directionality. We also investigate how the structure and population of coordination defects is affected by approximations to the interatomic potential, finding that, in most cases, the qualitative features of the hydrogen-bond network are remarkably robust.

This work examines the conformational ensemble involved in β-hairpin folding by means of advanced molecular dynamics simulations and dimensionality reduction. A fully atomistic description of the protein and the surrounding solvent molecules is used, and this complex energy landscape is sampled by means of parallel tempering metadynamics simulations. The ensemble of configurations explored is analyzed using the recently proposed sketch-map algorithm. Further simulations allow us to probe how mutations affect the structures adopted by this protein. We find that many of the configurations adopted by a mutant are the same as those adopted by the wild-type protein. Furthermore, certain mutations destabilize secondary-structure-containing configurations by preventing the formation of hydrogen bonds or by promoting the formation of new intramolecular contacts. Our analysis demonstrates that machine-learning techniques can be used to study the energy landscapes of complex molecules and that the visualizations that are generated in this way provide a natural basis for examining how the stabilities of particular configurations of the molecule are affected by factors such as temperature or structural mutations.

Increasingly, it is recognized that new automated forms of analysis are required to understand the high-dimensional output obtained from atomistic simulations. Recently, we introduced a new dimensionality reduction algorithm, sketch-map, that was designed specifically to work with data from molecular dynamics trajectories. In what follows, we provide more details on how this algorithm works and on how to set its parameters. We also test it on two well-studied Lennard-Jones clusters and show that the coordinates we extract using this algorithm are extremely robust. In particular, we demonstrate that the coordinates constructed for one particular Lennard-Jones cluster can be used to describe the configurations adopted by a second, different cluster and even to tell apart different phases of bulk Lennard-Jonesium.