Authors: Lars Pastewka, Tristan A. Sharp, Mark O. Robbins

The elastic response of an infinite bulk crystalline solid with a free surface can be renormalized into a linear operator, the surface Green’s function, that acts on the surface’s degrees of freedom only. We here present a general transfer matrix formulation that allows the computation of the elastic surface Green’s function from the knowledge of the interatomic interaction. Our approach naturally allows a treatment of multi-atom unit cells that are necessary to couple a renormalized substrate to a full atomistic system that interacts via long-ranged forces. The substrate interaction is treated exactly up to harmonic order and the full system comprising atomistic and elastic boundary regions is described by a single Hamiltonian. This concurrent multi-scale coupling enables simple, seamless semi-infinite elastic boundary conditions for atomistic simulations where near-surface deformations occur, such as nanoindentation, contact, friction, or fracture.