Lattice Green function for extended defect calculations: Computation and error estimation with long-range forces

Computing the atomic geometry of lattice defects---e.g., point defects, dislocations, crack tips, surfaces, or boundaries---requires an accurate coupling of the local deformations to the long-range elastic field. Periodic or fixed boundary conditions used by classical potentials or density-functional theory may not accurately reproduce the correct bulk response to an isolated defect; this is especially true for dislocations. Flexible boundary conditions have been developed to produce the correct long-range strain field from a defect---effectively ``embedding'' a finite-sized defect with infinite bulk response, isolating it from either periodic images or free surfaces. Flexible boundary conditions require the calculation of the bulk response with the lattice Green function (LGF). While the LGF can be computed from the force-constant matrix, the force-constant matrix is only known to a maximum range. This paper illustrates how to accurately calculate the lattice Green function and estimate the error using a truncated force-constant matrix combined with knowledge of the long-range behavior of the lattice Green function. The effective range of deviation of the lattice Green function from the long-range elastic behavior provides an important length scale in multiscale quasicontinuum and flexible boundary-condition calculations, and measures the error introduced with periodic-boundary conditions.