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.
[1] Ewald summation rewrites the interaction potential as the sum of two terms,
represents the short-range term whose sum quickly converges in real space and
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).
is the volume of the central unit cell (if it is geometrically a parallelepiped, which is often but not necessarily the case).
The most common reason for lack of convergence is a poorly defined unit cell, which must be charge neutral to avoid infinite sums.
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.
In the particle mesh method, just as in standard Ewald summation, the generic interaction potential is separated into two terms
represent the Fourier transforms of the potential and the charge density (this is the Ewald part).
Since both summations converge quickly in their respective spaces (real and Fourier), they may be truncated with little loss of accuracy and great improvement in required computational time.
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).
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.
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.
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
of the dipole in a central unit cell with that surface charge density can be written[3]
Generally, different Ewald summation methods give different time complexities.