Ewald summation

Ewald summation, named after Paul Peter Ewald, is a method for computing long range interactions (e. g. electrostatic interactions) in periodic systems. It was first developed as the method for calculating the electrostatic energies of ionic crystals, and is now commonly used for calculating long range interactions in computational chemistry. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. In this method, the long range interaction is divided into two parts: a short range contribution, and a long range contribution which does not have a singularity. The short range contribution is calculated in real space, whereas the long range contribution is calculated using a Fourier transform. The advantage of this method is the rapid convergence of the energy compared with that of a direct summation. This means that the method has high accuracy and reasonable speed when computing long range interactions, and it is thus the de facto standard method for calculating long range interactions in periodic systems. The method requires charge neutrality of the molecular system to accurately calculate the total Coulombic interaction. A study of the truncation errors introduced in the energy and force calculations of disordered point charge systems is provided by Kolafa and Perram.

The electrostatic energy of a polar crystal (i. e. a crystal with a net dipole in the unit cell) is conditionally convergent, i. e. depends on the order of the summation. For example, if the dipole dipole interactions of a central unit cell with unit cells located on an ever increasing cube, the energy converges to a different value than if the interaction energies had been summed spherically. Roughly speaking, this conditional convergence arises because (1) the number of interacting dipoles on a shell of radius grows like ; (2) the strength of a single dipole dipole interaction falls like ; and (3) the mathematical summation diverges. This somewhat surprising result can be reconciled with the finite energy of real crystals because such crystals are not infinite, i. e. have a particular boundary. More specifically, the boundary of a polar crystal has an effective surface charge density on its surface where is the surface normal vector and represents the net dipole moment per volume. The interaction energy of the dipole in a central unit cell with that surface charge density can be written

where and are the net dipole moment and volume of the unit cell, is an infinitesimal area on the crystal surface and is the vector from the central unit cell to the infinitesimal area. This formula results from integrating the energy where represents the infinitesimal electric field generated by an infinitesimal surface charge (Coulomb's law)

The negative sign derives from the definition of , which points towards the charge, not away from it.

The Ewald summation was developed by Paul Peter Ewald in 1921 (see References below) to determine the electrostatic energy (and, hence, the Madelung constant) of ionic crystals.

Generally, different Ewald summation methods give different time complexities. Direct calculation gives , where is the number of atoms in the system. The PME method gives .