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.

Derivation

represents the long-range term whose sum quickly converges in Fourier (reciprocal) space. The long-ranged part should be finite for all arguments (most notably r=0) but may have any convenient mathematical form, most typically a Gaussian distribution. The method assumes that the short-range part can be summed easily; hence, the problem becomes the summation of the long-range term. Due to the use of the Fourier sum, the method implicitly assumes that the system under study is infinitely periodic (a sensible assumption for the interiors of crystals). One repeating unit of this hypothetical periodic system is called a unit cell. One such cell is chosen as the "central cell" for reference and the remaining cells are called images.

The long-range interaction energy is the sum of interaction energies between the charges of a central unit cell and all the charges of the lattice. Hence, it can be represented as a double and the total

is the Dirac delta function can be represented as a convolution with a lattice function

Since this is a convolution, the Fourier transformation where the reciprocal space is the volume of the central unit cell (if it is geometrically a parallelepiped are real, even functions.

The energy can now be written as a single

Using Plancherel theorem

in the final summation.

is straightforward and should converge quickly. The most common reason for lack of convergence is a poorly defined unit cell, which must be charge neutral to avoid infinite sums.

Particle mesh Ewald (PME) method

Ewald summation was developed as a method in theoretical physics, long before the advent of computers. However, the Ewald method has enjoyed widespread use since the 1970s in computer simulations of particle systems, especially those whose particles interact via an inverse square force law such as gravity or electrostatics part of the Lennard-Jones potential in order to eliminate artifacts due to truncation. Applications include simulations of plasmass, galaxies and molecules.

In the particle mesh (this is the particle part of particle mesh Ewald

represent the Fourier transforms of the potential and the charge density (this is the Ewald of the charge density field efficiently, one uses the fast Fourier transform, which requires that the density field be evaluated on a discrete lattice in space (this is the mesh part).

Due to the periodicity assumption implicit in Ewald summation, applications of the PME method to physical systems require the imposition of periodic symmetry. Thus, the method is best suited to systems that can be simulated as infinite in spatial extent. In molecular dynamics simulations this is normally accomplished by deliberately constructing a charge-neutral unit cell that can be infinitely "tiled" to form images; however, to properly account for the effects of this approximation, these images are reincorporated back into the original simulation cell. The overall effect is called a periodic boundary condition. To visualize this most clearly, think of a unit cube; the upper face is effectively in contact with the lower face, the right with the left face, and the front with the back face. As a result, the unit cell size must be carefully chosen to be large enough to avoid improper motion correlations between two faces "in contact", but still small enough to be computationally feasible. The definition of the cutoff between short- and long-range interactions can also introduce artifacts.

The restriction of the density field to a mesh makes the PME method more efficient for systems with "smooth" variations in density, or continuous potential functions. Localized systems or those with large fluctuations in density may be treated more efficiently with the fast multipole method of Greengard and Rokhlin.

Dipole term

in the unit cell) is conditionally convergent diverges.

of the dipole in a central unit cell with that surface charge density can be written is an infinitesimal (Coulomb's law, which points towards the charge, not away from it.

History

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.

Scaling

Generally, different Ewald summation methods give different time complexities.

See also

• Paul Peter Ewald

• Thomas A. Darden

• Madelung constant

• Poisson summation formula

• Molecular modeling

• Wolf summation

References

• Frenkel, D., & Smit, B. (2001). Understanding molecular simulation: from algorithms to applications, Academic press.