Equivariant representations for molecular Hamiltonians and N-center atomic-scale properties

TL;DR

This paper introduces N-centers equivariant descriptors for molecular Hamiltonians, enabling machine learning models to efficiently predict quantum properties involving multi-atom interactions with symmetry considerations.

Contribution

It generalizes atom-centered density features to N-centers, allowing for symmetry-aware learning of multi-atom quantum properties like Hamiltonian matrix elements.

Findings

N-centers features are fully equivariant under translation, rotation, and permutation.

The approach efficiently learns matrix elements of the single-particle Hamiltonian.

Applicable to a new class of molecular and material properties.

Abstract

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.