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

Equivariant analytical mapping of first principles Hamiltonians to accurate and transferable materials models

“…Hedge and Bowen 32 employed Kernel ridge regression with a bispectrum representation 33 for an analytical representation of a minimal basis DFT Hamiltonian for bulk copper and diamond. Equivariant parameterisations for molecular systems along similar lines to what we describe here have been reported, learning either from the Hamiltonian 34 or from wavefunctions and electronic densities 35 . These works apply linear or nonlinear equivariant models, respectively, to the MD17 molecular dataset, both of which improve on the non-equivariant SchNOrb approach of ref.…”

UV–Visible Absorption Spectra of Solvated Molecules by Quantum Chemical Machine Learning

“…While this is auspicious for future works, it is not guaranteed that this will be the case for different types of molecules in different solvation environments. To extend the transferability of this approach to other classes of molecules, given the nonlocal nature of electronic excitations in large molecules, it may turn out to be necessary to supplement our approach by including richer physically based descriptors, e.g., electronic orbitals, and to compute excitation energies as the eigenvalues of ML effective Hamiltonians. − …”

Orbital Mixer: Using Atomic Orbital Features for Basis-Dependent Prediction of Molecular Wavefunctions

“…Our approach currently requires training with data augmentation to implicitly learn sensitivity to rigid rotations of the system, which can become computationally costly for large systems and increasingly large atomic orbital basis sets. These data augmentations can be avoided by incorporating equivariance into the model design, which has previously been achieved for orbital prediction with the SE(3)-equivariant neural networks of PhisNet and the symmetry-adapted representations employed by Ceriotti and co-workers . Operating on optimized minimal basis sets also provides benefits in scaling to larger systems, as these representations reduce the effective dimensionality of the learning target while also achieving speedups when solving the eigenvalue problem for the predicted Hamiltonian matrix .…”

“…These data augmentations can be avoided by incorporating equivariance into the model design, which has previously been achieved for orbital prediction with the SE(3)equivariant neural networks of PhisNet 42 and the symmetryadapted representations employed by Ceriotti and co-workers. 55 Operating on optimized minimal basis sets also provides benefits in scaling to larger systems, as these representations reduce the effective dimensionality of the learning target while also achieving speedups when solving the eigenvalue problem for the predicted Hamiltonian matrix. 54 While incorporating more inductive biases into the Orbital Mixer model design could improve data efficiency and performance, the simple MLP-based construction of our architecture already positions Orbital Mixer as an effective rapid inference model with competitive prediction accuracy.…”

“…More recently, Unke et al 42 propose PhiSNet, which draws upon insights of SE(3)-equivariant models to maintain that Hamiltonian predictions remain explicitly covariant with respect to rigid rotations or translations while also reporting significantly improved prediction accuracies. Notably, Nigam et al 55 devised N-center permutation and rotationally equivariant representations of atomic systems, which are also applied to Hamiltonian prediction using relatively simple linear ridge regression and kernel-based Gaussian-process regression models.…”