Rigorous Error Bounds for Ewald Summation of Electrostatics at Planar Interfaces

We present a rigorous Ewald summation formula to evaluate the electrostatic interactions in two-dimensionally periodic planar interfaces of three-dimensional systems. By rewriting the Fourier part of the summation formula of the original Ewald2D expression with an explicit order N2 complexity to a closed form Fourier integral, we find that both the previously developed electrostatic layer correction term and the boundary correction term naturally arise from the expression of a rigorous trapezoidal summation of the Fourier integral part. We derive the exact corrections to the trapezoidal summation in a form of contour integrals offering precise error bounds with given parameter sets of mesh size and system length. Numerical calculations of Madelung constants in model ionic crystals of slab geometry have been performed to support our analytical results.