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

We propose a data-driven scheme to construct predictive models of Hamiltonian and overlap matrices in atomic orbital representation from ab initio data as a function of local atomic and bond environments. The scheme goes beyond conventional tight binding descriptions as it represents the ab initio model to full order, rather than in two-centre or three-centre approximations. We achieve this by introducing an extension to the Atomic Cluster Expansion (ACE) descriptor that represents intraatomic onsite and interatomic offsite blocks of Hamiltonian and overlap matrices that transform equivariantly with respect to the full rotation group in 3 dimensions. The approach produces equivariant analytical maps from first principles data to linear models for the Hamiltonian and overlap matrices. Through an application to FCC and BCC aluminium, we demonstrate that it is possible to train models from a handful of Hamiltonian and overlap matrices computed with density functional theory using the FHIaims code, and apply them to produce accurate predictions for the band structure and density of states in both phases, as well as along the Bain path that connects them.