  We propose a scheme to construct predictive models for Hamiltonian matrices
in atomic orbital representation from ab initio data as a function of 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
Hamiltonian matrix blocks that transform equivariantly with respect to the full
rotation group. The approach produces analytical linear models for the
Hamiltonian and overlap matrices. Through an application to aluminium, we
demonstrate that it is possible to train models from a handful of structures
computed with density functional theory, and apply them to produce accurate
predictions for the electronic structure. The model generalises well and is
able to predict defects accurately from only bulk training data.
