Equivariant representations for molecular Hamiltonians and N-center atomic-scale properties
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Symmetry considerations are at the core of the major frameworks used to provide an effective mathematical representation of atomic configurations, that are 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 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 lay the foundations for symmetry-adapted machine-learning models of new classes of properties of molecules and materials.
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