Most multi-scale embedding schemes (A/C, QM/MM, . . .) schemes can be understood within a single framework: a coarse model for a far-field is used to provide a boundary condition for a core region of interest. It turns out that, for crystalline defects, boundary conditions of arbitrary accuracy can be constructed *almost* analytically, without ever having to resort to the complex mechanisms and many pitfalls of concurrent multi-scale schemes. We achieve this by constructing an expansion of an atomistic model in terms of higher-order continuum models and then extracting an effective far-field model at the desired accuracy. We then prescribe a self-consistent equation coupling the defect core to the far-field.

The starting point is a multi-pole expansion of the discrete, nonlinear elastic field. The first three terms in the expansion are typically (1) linearised elasticity, (2) nonlinear elasticity, (3) strain-gradient elasticity. This yields high accuracy simulation results with small computational domains, e.g., in the following figure for a screw dislocation (bcc, W, 111, EAM); see [73], [48] for more details.

It is particular interesting though (work in progress) to explore higher-order generalisations as well as applications to electronic structure models.