Atomic cluster expansion of scalar, vectorial, and tensorial properties including magnetism and charge transfer

The atomic cluster expansion [R. Drautz, Phys. Rev. B 99, 014104 (2019)] is extended in two ways, the modeling of vectorial and tensorial atomic properties and the inclusion of atomic degrees of freedom in addition to the positions of the atoms. In particular, atomic species, magnetic moments, and charges are attached to the atomic positions, and an atomic cluster expansion that includes the different degrees of freedom on equal footing is derived. Expressions for the efficient evaluation of forces and torques are given.