Time history kernel functions for square lattice

We propose an exact non-reflecting boundary condition for simulations of a square lattice in finite atomic sub-domain. The boundary condition is based on the series expansion of the single source kernel functions. The time history kernel Laplace transform is first derived in a closed form, where the Fourier transform inversion is performed analytically. An efficient numerical procedure is also developed for the inverse Laplace transform. Differences in the finite boundary condition and the infinite boundary approximation elucidate the cause of the long-lasting difficulty of corner effects. Numerical tests verify accuracy of the proposed approach.