We present a multiscale atomistic-to-continuum method for ionic crystals with defects. Defects often play a central role in ionic and electronic solids, not only to limit reliability, but more importantly to enable the functionalities that make these materials of critical importance. Examples include solid electrolytes that conduct current through the motion of charged point defects, and complex oxide ferroelectrics that display multifunctionality through the motion of domain wall defects. Therefore, it is important to understand the structure of defects and their response to electrical and mechanical fields. A central hurdle, however, is that interactions in ionic solids include both short-range atomic interactions as well as long-range electrostatic interactions. Existing atomistic-to-continuum multiscale methods, such as the Quasicontinuum method, are applicable only when the atomic interactions are short-range. In addition, empirical reductions of quantum mechanics to density functional models are unable to capture key phenomena of interest in these materials.To address this open problem, we develop a multiscale atomistic method to coarse-grain the longrange electrical interactions in ionic crystals with defects. In these settings, the charge density is rapidly varying, but in an almost-periodic manner. The key idea is to use the polarization density field as a multiscale mediator that enables efficient coarse-graining by exploiting the almost-periodic nature of the variation. In regions far from the defect, where the crystal is close-to-perfect, the polarization field serves as a proxy that enables us to avoid accounting for the details of the charge variation. We combine this approach for long-range electrostatics with the standard Quasicontinuum method for short-range interactions to achieve an efficient multiscale atomistic-to-continuum method. As a side note, we examine an important issue that is critical to our method: namely, the dependence of the computed polarization field on the choice of unit cell. Potentially, this is fatal to our coarse-graining scheme; however, we show that consistently accounting for boundary charges leaves the continuum electrostatic fields invariant to choice of unit cell.

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This work develops a multiscale modeling framework for defects in crystals with general geometries and boundary conditions in which ionic interactions are important, with potential application to, e.g., ionic solids and electric field interactions with materials. The overall strategy is posed in the framework of the Quasicontinuum multiscale method; specifically, the use of a finite-element inspired kinematic description enables a significant reduction in the large number of degrees of freedom to describe the atomic positions. The key advance of this work is a method for the efficient and accurate treatment of nonlocal electrostatic charge-charge interactions without restrictions on the geometry or boundary conditions. Electrostatic interactions are long-range with slow decay, and hence require consideration of all pairs of charges making a brute-force approach computationally prohibitive. The method proposed here accounts for the exact charge-charge interactions in the near-field and uses a coarse-grained approximation in the far-field. The coarse-grained approximation and the associated errors are rigorously derived based on the limit of a finite body with a small periodic lengthscale, thereby enabling the errors in the approximation to be controlled to a desired tolerance. The method is applied to a simple model of Gallium Nitride and it is shown that electrostatic interactions can be approximated with a desired level of accuracy using the proposed methodology.