A formula and numerical study on Ewald 1D summation

Ewald summation is famous for its successful applications in molecular simulations for systems under 2 dimensional periodic boundary condition (2D PBC, e.g., planar interfaces) and systems under 3D PBC (e.g., bulk). However, the extension to systems under 1D PBC (like porous structures and tubes) is largely hindered by the special functions in the formula. In this work, a simple approximation of Ewald 1D sum is introduced with its error rigorously controlled. To investigate the impacts on the efficiency and accuracy by different parts, a pairwise potential is calculated for a series of screening parameters (alpha) and radial distances (rho) between two point charges. A mapping between the sum of trigonometric functions in Ewald 1D method and the sum of specific vectors further reveals the different converging speeds of different Fourier parts. When choosing alpha 1/4 0:2 A(-1), it is appropriate to ignore the insignificant parts in the sum to accelerate the method.