Infinite Boundary Terms of Ewald Sums and Pairwise Interactions for Electrostatics in Bulk and at Interfaces

We present a unified derivation of the Ewald sum for electrostatics in a three-dimensional infinite system that is periodic in one, two, or three dimensions. The derivation leads to the Ewald3D sum being expressed as a sum of a real space contribution and a reciprocal space contribution, as in previous work. However, the k → 0 term in the reciprocal space contribution is analyzed further and found to give an additional contribution that is not part of previous reciprocal space contributions. The transparent derivation provides a unified view of the existing conducting infinite boundary term, the vacuum spherical infinite boundary term and the vacuum planar infinite boundary term for the Ewald3D sum. The derivation further explains that the infinite boundary term is conditional for the Ewald3D sum because it depends on the asymptotic behavior that the system approaches the infinite in 3D but it becomes a definite term for the Ewald2D or Ewald1D sum irrespective of the asymptotic behavior in the reduced dimensions. Moreover, the unified derivation yields two formulas for the Ewald sum in one-dimensional periodicity, and we rigorously prove that the two formulas are equivalent. These formulas might be useful for simulations of organic crystals with wirelike shapes or liquids confined in uniform cylinders. More importantly, the Ewald3D, Ewald2D, and Ewald1D sums are further written as sums of well-defined pairwise potentials overcoming the difficulty in splitting the total Coulomb potential energy into contributions from each individual group of charges. The pairwise interactions with their clear physical meaning of the explicit presence of the periodic images thus can be used to consistently perform analysis based on the trajectories from computer simulations of bulk or interfaces.