A comparison of Ewald summation techniques for planar surfaces

Ewald-like summation methods to compute the electrostatic Coulomb potential and the forces on charged particles in a two-dimensional periodic system with nonperiodic extension to the third dimension are investigated. The techniques by Hautman and Klein, Mol. Phys. 75 (1992) 379 (HK), Heyes, Barber, and Clarke, J. Chem. Soc. Faraday Trans. II 73 (1977) 1485 (HBC), and Nijboer and de Wette, Physica A 125 (1984) 275 (NdW) are compared in respect to their numerical accuracy and efficiency. The convergence behaviour of the methods is analysed for different magnitudes of the nonperiodic z-separation between the charged particles up to twice the length of the square simulation cell. The HK method is best suited for small z but becomes inefficient for large particle separations in the nonperiodic direction. HBC is not sensitive on the out-of-plane separation but is very slow. NdW is only suited for large out-of-plane distances because it fails completely if the z-separation of any two particles becomes zero. Ready-to-implement sets of equations for the three algorithms are presented in the appendices.