A new Ewald-type formula for thin film electrostatics

Abstract A new analytic formula, 2DE, is derived for the electrostatic potential of a thin slab formed from a unit cell composed of arbitrarily positioned point charges repeated in the xy plane, which is of the Ewald form with real and reciprocal space lattice summations except that the artificial charge broadening function acts only in the xy plane. Expressions are derived for Gaussian and exponential (in 2D Fourier space) charge spreading functions. A comparison of the convergence of 2DE with direct summation (DS) and the Parry formulae (Ewald method applied to a slab geometry) is made. The 2DE reciprocal space lattice term has a simpler analytic form than the equivalent term in the Parry formulae, but the real space term would be more problematic in certain applications where computational efficiency in regard to multiple applications is an important issue (such as molecular simulation). Convergence to the exact answer is initially comparable to Parry (especially for the forces) but Parry converges more rapidly to reach very high precision (relative error ~ 10 − 8 ).