Computing the atomic geometry of lattice defects (e.g., point defects, dislocations, crack tips, surfaces, boundaries) required 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 might not accurately reproduce the correct bulk response to an isolated defect; this was especially true for dislocations. Flexible boundary conditions were 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 required the calculation of the bulk response with the lattice Green function. While the lattice Green function could be computed from the force-constant matrix, the force-constant matrix was only known to a maximum range. This work showed how to calculate accurately the lattice Green function and estimate the error using a truncated force-constant matrix combined with knowledge of the long-range behaviour of the lattice Green function. The effective range of deviation of the lattice Green function from the long-range elastic behaviour provided an important length scale in multiscale quasi-continuum and flexible boundary-condition calculations, and measured the error introduced with periodic-boundary conditions.

Lattice Green Function for Extended Defect Calculations - Computation and Error Estimation with Long-Range Forces. D.R.Trinkle: Physical Review B, 2008, 78[1], 014110