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A force field as accurate as quantum mechanics (QM) and as fast as molecular mechanics (MM), with which one can simulate a biomolecular system efficiently enough and meaningfully enough to get quantitative insights, is among the most ardent dreams of biophysicists -- a dream, nevertheless, not to be fulfilled any time soon. Machine learning force fields (MLFFs) represent a meaningful endeavor towards this direction, where differentiable neural functions are parametrized to fit ab initio energies, and furthermore forces through automatic differentiation. We argue that, as of now, the utility of the MLFF models is no longer bottlenecked by accuracy but primarily by their speed (as well as stability and generalizability), as many recent variants, on limited chemical spaces, have long surpassed the chemical accuracy of $1$ kcal/mol -- the empirical threshold beyond which realistic chemical predictions are possible -- though still magnitudes slower than MM. Hoping to kindle explorations and designs of faster, albeit perhaps slightly less accurate MLFFs, in this review, we focus our attention on the design space (the speed-accuracy tradeoff) between MM and ML force fields. After a brief review of the building blocks of force fields of either kind, we discuss the desired properties and challenges now faced by the force field development community, survey the efforts to make MM force fields more accurate and ML force fields faster, envision what the next generation of MLFF might look like.

The point cloud is a flexible representation for a wide variety of data types, and is a particularly natural fit for the 3D conformations of molecules. Extant molecule embedding/representation schemes typically focus on internal degrees of freedom, ignoring the global 3D orientation. For tasks that depend on knowledge of both molecular conformation and 3D orientation, such as the generation of molecular dimers, clusters, or condensed phases, we require a representation which is provably complete in the types and positions of atomic nuclei and roto-inversion equivariant with respect to the input point cloud. We develop, train, and evaluate a new type of autoencoder, molecular O(3) encoding net (Mo3ENet), for multi-type point clouds, for which we propose a new reconstruction loss, capitalizing on a Gaussian mixture representation of the input and output point clouds. Mo3ENet is end-to-end equivariant, meaning the learned representation can be manipulated on O(3), a practical bonus for downstream learning tasks. An appropriately trained Mo3ENet latent space comprises a universal embedding for scalar and vector molecule property prediction tasks, as well as other downstream tasks incorporating the 3D molecular pose., Comment: 16 pages, 8 figures, including main text, bibliography and supplemental material

Identifying local structural motifs and packing patterns of molecular solids is a challenging task for both simulation and experiment. We demonstrate two novel approaches to characterize local environments in different polymorphs of molecular crystals using learning models that employ either flexibly learned or handcrafted molecular representations. In the first case, we follow our earlier work on graph learning in molecular crystals, deploying an atomistic graph convolutional network, combined with molecule-wise aggregation, to enable per-molecule environmental classification. For the second model, we develop a new set of descriptors based on symmetry functions combined with a point-vector representation of the molecules, encoding information about the positions as well as relative orientations of the molecule. We demonstrate very high classification accuracy for both approaches on urea and nicotinamide crystal polymorphs, and practical applications to the analysis of dynamical trajectory data for nanocrystals and solid-solid interfaces. Both architectures are applicable to a wide range of molecules and diverse topologies, providing an essential step in the exploration of complex condensed matter phenomena.

We develop and test new machine learning strategies for accelerating molecular crystal structure ranking and crystal property prediction using tools from geometric deep learning on molecular graphs. Leveraging developments in graph-based learning and the availability of large molecular crystal datasets, we train models for density prediction and stability ranking which are accurate, fast to evaluate, and applicable to molecules of widely varying size and composition. Our density prediction model, MolXtalNet-D, achieves state of the art performance, with lower than 2% mean absolute error on a large and diverse test dataset. Our crystal ranking tool, MolXtalNet-S, correctly discriminates experimental samples from synthetically generated fakes and is further validated through analysis of the submissions to the Cambridge Structural Database Blind Tests 5 and 6. Our new tools are computationally cheap and flexible enough to be deployed within an existing crystal structure prediction pipeline both to reduce the search space and score/filter crystal candidates.

The theorems of density functional theory (DFT) and reduced density matrix functional theory (RDMFT) establish a bijective map between the external potential of a many-body system and its electron density or one-particle reduced density matrix. Building on this foundation, we show that machine learning can be used to generate surrogate electronic structure methods. In particular, we generate surrogates of local and hybrid DFT as well as Hartree-Fock and full configuration interaction theories for systems ranging from small molecules such as water to more complex compounds like propanol and benzene. The surrogate models use the one-electron reduced density matrix as the central quantity to be learned. From the predicted density matrices, we show that either standard quantum chemistry or a second machine-learning model can be used to compute molecular observables, energies, and atomic forces. We show that the surrogate models can generate essentially anything that a standard electronic structure method can, ranging from band gaps and Kohn-Sham orbitals to energy-conserving ab-initio molecular dynamics simulations and IR spectra, which account anharmonicity and thermal effects, without the need of computationally expensive algorithms such as self-consistent field theory. The algorithms developed here are packaged in an efficient and easy to use Python code, QMLearn, accessible by the broader community on popular platforms.

Imidazole and 1,2,3-triazole are promising hydrogen-bonded heterocycles that conduct protons via a structural mechanism and whose derivatives are present in systems ranging from biological proton channels to proton exchange membrane fuel cells. Here, we leverage multiple time-stepping to perform ab initio molecular dynamics of imidazole and 1,2,3-triazole at the nanosecond timescale. We show that despite the close structural similarities of these compounds, their proton diffusion constants vary by over an order of magnitude. Our simulations reveal the reasons for these differences in diffusion constants, which range from the degree of hydrogen-bonded chain linearity to the effect of the central nitrogen atom in 1,2,3-triazole on proton transport. In particular, we uncover evidence of two "blocking" mechanisms in 1,2,3-triazole, where covalent and hydrogen bonds formed by the central nitrogen atom limit the mobility of protons. Our simulations thus provide insights into the origins of the experimentally observed 10-fold difference in proton conductivity., Comment: 7-page manuscript, 7 figures; 7-page supporting information, 8 figures

Imidazole is a promising anhydrous proton conductor with a high conductivity comparable to that of water at a similar temperature relative to its melting point. Previous theoretical studies of the mechanism of proton transport in imidazole have relied either on empirical models or on ab initio trajectories that have been too short to draw significant conclusions. Here, we present the results of ab initio molecular dynamics simulations of an excess proton in liquid imidazole reaching 1 nanosecond in total simulation time. We find that the proton transport is dominated by structural diffusion, and the diffusion constant of the proton defect is ~8 times higher than the self-diffusion of the imidazole molecules. By using correlation function analysis, we decompose the mechanism for proton transport into a series of first-order processes and show that the proton transport mechanism occurs over three distinct time and length scales. Although the mechanism at intermediate times is dominated by hopping along pseudo one-dimensional chains, at longer times, the overall rate of diffusion is limited by the reformation of these chains, thus providing a more complete picture of the traditional, idealized Grotthuss structural diffusion mechanism., Comment: 7-page manuscript (5 figures) and 7-page supporting information (7 figures)

We propose a rigorous construction of a 1D path collective variable to sample structural phase transformations in condensed matter. The path collective variable is defined in a space spanned by global collective variables that serve as classifiers derived from local structural units. A reliable identification of local structural environments is achieved by employing a neural network based classification. The 1D path collective variable is subsequently used together with enhanced sampling techniques to explore the complex migration of a phase boundary during a solid-solid phase transformation in molybdenum.

The Kohn-Sham scheme of density functional theory is one of the most widely used methods to solve electronic structure problems for a vast variety of atomistic systems across different scientific fields. While the method is fast relative to other first principles methods and widely successful, the computational time needed is still not negligible, making it difficult to perform calculations for very large systems or over long time-scales. In this submission, we revisit a machine learning model capable of learning the electron density and the corresponding energy functional based on a set of training examples. It allows us to bypass solving the Kohn-Sham equations, providing a significant decrease in computation time. We specifically focus on the machine learning formulation of the Hohenberg-Kohn map and its decomposability. We give results and discuss challenges, limits and future directions.

Path integral-based simulation methodologies play a crucial role for the investigation of nuclear quantum effects by means of computer simulations. However, these techniques are significantly more demanding than corresponding classical simulations. To reduce this numerical effort, we recently proposed a method, based on a rigorous Hamiltonian formulation, which restricts the quantum modeling to a small but relevant spatial region within a larger reservoir where particles are treated classically. In this work, we extend this idea and show how it can be implemented along with state-of-the-art path integral simulation techniques, such as ring polymer and centroid molecular dynamics, which allow the approximate calculation of both quantum statistical and quantum dynamical properties. To this end, we derive a new integration algorithm which also makes use of multiple time-stepping. The scheme is validated via adaptive classical--path-integral simulations of liquid water. Potential applications of the proposed multiresolution method are diverse and include efficient quantum simulations of interfaces as well as complex biomolecular systems such as membranes and proteins.

Last year, at least 30,000 scientific papers used the Kohn-Sham scheme of density functional theory to solve electronic structure problems in a wide variety of scientific fields, ranging from materials science to biochemistry to astrophysics. Machine learning holds the promise of learning the kinetic energy functional via examples, by-passing the need to solve the Kohn-Sham equations. This should yield substantial savings in computer time, allowing either larger systems or longer time-scales to be tackled, but attempts to machine-learn this functional have been limited by the need to find its derivative. The present work overcomes this difficulty by directly learning the density-potential and energy-density maps for test systems and various molecules. Both improved accuracy and lower computational cost with this method are demonstrated by reproducing DFT energies for a range of molecular geometries generated during molecular dynamics simulations. Moreover, the methodology could be applied directly to quantum chemical calculations, allowing construction of density functionals of quantum-chemical accuracy.

Clathrate hydrates hold considerable promise as safe and economical materials for hydrogen storage. Here we present a quantum mechanical study of H$_2$ and D$_2$ diffusion through a hexagonal face shared by two large cages of clathrate hydrates over a wide range of temperatures. Path integral molecular dynamics simulations are used to compute the free-energy profiles for the diffusion of H$_2$ and D$_2$ as a function of temperature. Ring polymer molecular dynamics rate theory, incorporating both exact quantum statistics and approximate quantum dynamical effects, is utilized in the calculations of the H$_2$ and D$_2$ diffusion rates in a broad temperature interval. We find that the shape of the quantum free-energy profiles and their height relative to the classical free energy barriers at a given temperature, as well as the rate of diffusion, are profoundly affected by competing quantum effects: above 25 K, zero-point energy (ZPE) perpendicular to the reaction path for diffusion between cavities decreases the quantum rate compared to the classical rate, whereas at lower temperatures tunneling outcompetes the ZPE and as result the quantum rate is greater than the classical rate., Comment: 9 pages, 4 figures

In computer simulations, quantum delocalization of atomic nuclei can be modeled making use of the Path Integral (PI) formulation of quantum statistical mechanics. This approach, however, comes with a large computational cost. By restricting the PI modeling to a small region of space, this cost can be significantly reduced. In the present work we derive a Hamiltonian formulation for a bottom-up, theoretically solid simulation protocol that allows molecules to change their resolution from quantum-mechanical to classical and vice versa on the fly, while freely diffusing across the system. This approach renders possible simulations of quantum systems at constant chemical potential. The validity of the proposed scheme is demonstrated by means of simulations of low temperature parahydrogen. Potential future applications include simulations of biomolecules, membranes, and interfaces.

Molecular dynamics is one of the most commonly used approaches for studying the dynamics and statistical distributions of many physical, chemical, and biological systems using atomistic or coarse-grained models. It is often the case, however, that the interparticle forces drive motion on many time scales, and the efficiency of a calculation is limited by the choice of time step, which must be sufficiently small that the fastest force components are accurately integrated. Multiple time-stepping algorithms partially alleviate this inefficiency by assigning to each time scale an appropriately chosen step-size. However, such approaches are limited by resonance phenomena, wherein motion on the fastest time scales limits the step sizes associated with slower time scales. In atomistic models of biomolecular systems, for example, resonances limit the largest time step to around 5-6 fs. In this paper, we introduce a set of stochastic isokinetic equations of motion that are shown to be rigorously ergodic and that can be integrated using a multiple time-stepping algorithm that can be easily implemented in existing molecular dynamics codes. The technique is applied to a simple, illustrative problem and then to a more realistic system, namely, a flexible water model. Using this approach outer time steps as large as 100 fs are shown to be possible.

We present the results of first-principles molecular-dynamics simulations of molten silicates, based on the density functional formalism. In particular, the structural properties of a calcium aluminosilicate $ [$ CaO-Al$_2$O$_3$-SiO$_2$ $ ]$ melt are compared to those of a silica melt. The local structures of the two melts are in good agreement with the experimental understanding of these systems. In the calcium aluminosilicate melt, the number of non-bridging oxygens found is in excess of the number obtained from a simple stoichiometric prediction. In addition, the aluminum avoidance principle, which states that links between AlO$_4$ tetrahedra are absent or rare, is found to be violated. Defects such as 2-fold rings and 5-fold coordinated silicon atoms are found in comparable proportions in both liquids. However, in the calcium aluminosilicate melt, a larger proportion of oxygen atoms are 3-fold coordinated. In addition, 5-fold coordinated aluminum atoms are observed. Finally evidence of creation and anihilation of non-bridging oxygens is observed, with these oxygens being mostly connected to Si tetrahedra., Comment: 20 pages, 9 figures, 1 table. Physical Review B, in press