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Physics-based, atom-centered machine learning (ML) representations have been instrumental to the effective integration of ML within the atomistic simulation community. Many of these representations build off the idea of atoms as having spherical, or isotropic, interactions. In many communities, there is often a need to represent groups of atoms, either to increase the computational efficiency of simulation via coarse-graining or to understand molecular influences on system behavior. In such cases, atom-centered representations will have limited utility, as groups of atoms may not be well-approximated as spheres. In this work, we extend the popular Smooth Overlap of Atomic Positions (SOAP) ML representation for systems consisting of non-spherical anisotropic particles or clusters of atoms. We show the power of this anisotropic extension of SOAP, which we deem \AniSOAP, in accurately characterizing liquid crystal systems and predicting the energetics of Gay-Berne ellipsoids and coarse-grained benzene crystals. With our study of these prototypical anisotropic systems, we derive fundamental insights into how molecular shape influences mesoscale behavior and explain how to reincorporate important atom-atom interactions typically not captured by coarse-grained models. Moving forward, we propose \AniSOAP as a flexible, unified framework for coarse-graining in complex, multiscale simulation., Comment: The following article has been submitted to the Journal of Chemical Physics. After it is published, the updated version can be found through their website

Data-driven techniques are increasingly used to replace electronic-structure calculations of matter. In this context, a relevant question is whether machine learning (ML) should be applied directly to predict the desired properties or be combined explicitly with physically-grounded operations. We present an example of an integrated modeling approach, in which a symmetry-adapted ML model of an effective Hamiltonian is trained to reproduce electronic excitations from a quantum-mechanical calculation. The resulting model can make predictions for molecules that are much larger and more complex than those that it is trained on, and allows for dramatic computational savings by indirectly targeting the outputs of well-converged calculations while using a parameterization corresponding to a minimal atom-centered basis. These results emphasize the merits of intertwining data-driven techniques with physical approximations, improving the transferability and interpretability of ML models without affecting their accuracy and computational efficiency, and providing a blueprint for developing ML-augmented electronic-structure methods.

In this paper, we address the challenge of obtaining a comprehensive and symmetric representation of point particle groups, such as atoms in a molecule, which is crucial in physics and theoretical chemistry. The problem has become even more important with the widespread adoption of machine-learning techniques in science, as it underpins the capacity of models to accurately reproduce physical relationships while being consistent with fundamental symmetries and conservation laws. However, some of the descriptors that are commonly used to represent point clouds -- most notably those based on discretized correlations of the neighbor density, that underpin most of the existing ML models of matter at the atomic scale -- are unable to distinguish between special arrangements of particles in three dimensions. This makes it impossible to machine learn their properties. Atom-density correlations are provably complete in the limit in which they simultaneously describe the mutual relationship between all atoms, which is impractical. We present a novel approach to construct descriptors of \emph{finite} correlations based on the relative arrangement of particle triplets, which can be employed to create symmetry-adapted models with universal approximation capabilities, which have the resolution of the neighbor discretization as the sole convergence parameter. Our strategy is demonstrated on a class of atomic arrangements that are specifically built to defy a broad class of conventional symmetric descriptors, showcasing its potential for addressing their limitations.

Data-driven schemes that associate molecular and crystal structures with their microscopic properties share the need for a concise, effective description of the arrangement of their atomic constituents. Many types of models rely on descriptions of atom-centered environments, that are associated with an atomic property or with an atomic contribution to an extensive macroscopic quantity. Frameworks in this class can be understood in terms of atom-centered density correlations (ACDC), that are used as a basis for a body-ordered, symmetry-adapted expansion of the targets. Several other schemes, that gather information on the relationship between neighboring atoms using "message-passing" ideas, cannot be directly mapped to correlations centered around a single atom. We generalize the ACDC framework to include multi-centered information, generating representations that provide a complete linear basis to regress symmetric functions of atomic coordinates, and provides a coherent foundation to systematize our understanding of both atom-centered and message-passing, invariant and equivariant machine-learning schemes.

Symmetry considerations are at the core of the major frameworks used to provide an effective mathematical representation of atomic configurations that is then used in machine-learning models to predict the properties associated with each structure. In most cases, the models rely on a description of atom-centered environments, and are suitable to learn atomic properties, or global observables that can be decomposed into atomic contributions. Many quantities that are relevant for quantum mechanical calculations, however -- most notably the single-particle Hamiltonian matrix when written in an atomic-orbital basis -- are not associated with a single center, but with two (or more) atoms in the structure. We discuss a family of structural descriptors that generalize the very successful atom-centered density correlation features to the N-centers case, and show in particular how this construction can be applied to efficiently learn the matrix elements of the (effective) single-particle Hamiltonian written in an atom-centered orbital basis. These N-centers features are fully equivariant -- not only in terms of translations and rotations, but also in terms of permutations of the indices associated with the atoms -- and are suitable to construct symmetry-adapted machine-learning models of new classes of properties of molecules and materials.

Electronic nearsightedness is one of the fundamental principles governing the behavior of condensed matter and supporting its description in terms of local entities such as chemical bonds. Locality also underlies the tremendous success of machine-learning schemes that predict quantum mechanical observables -- such as the cohesive energy, the electron density, or a variety of response properties -- as a sum of atom-centred contributions, based on a short-range representation of atomic environments. One of the main shortcomings of these approaches is their inability to capture physical effects, ranging from electrostatic interactions to quantum delocalization, which have a long-range nature. Here we show how to build a multi-scale scheme that combines in the same framework local and non-local information, overcoming such limitations. We show that the simplest version of such features can be put in formal correspondence with a multipole expansion of permanent electrostatics. The data-driven nature of the model construction, however, makes this simple form suitable to tackle also different types of delocalized and collective effects. We present several examples that range from molecular physics, to surface science and biophysics, demonstrating the ability of this multi-scale approach to model interactions driven by electrostatics, polarization and dispersion, as well as the cooperative behavior of dielectric response functions.

Mapping an atomistic configuration to an $N$-point correlation of a field associated with the atomic positions (e.g. an atomic density) has emerged as an elegant and effective solution to represent structures as the input of machine-learning algorithms. While it has become clear that low-order density correlations do not provide a complete representation of an atomic environment, the exponential increase in the number of possible $N$-body invariants makes it difficult to design a concise and effective representation. We discuss how to exploit recursion relations between equivariant features of different orders (generalizations of $N$-body invariants that provide a complete representation of the symmetries of improper rotations) to compute high-order terms efficiently. In combination with the automatic selection of the most expressive combination of features at each order, this approach provides a conceptual and practical framework to generate systematically-improvable, symmetry adapted representations for atomistic machine learning.