Ewald summation

Ewald summation
Ewald summation is a method for computing the interaction energies of periodic systems
(e.g. crystals), particularly electrostatic energies. 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. The advantage of this approach is the rapid
convergence of the Fourier-space summation compared to its real-space equivalent when the
real-space interactions are long-range. Because electrostatic energies consist of both short-
and long-range interactions, it is maximally efficient to decompose the interaction potential
into a short-range component summed in real space and a long-range component summed in
Fourier space.
Derivation
Ewald summation rewrites the interaction potential as the sum of two terms
where represents the short-range term that sums quickly in real space and
represents the long-range term that sums quickly in Fourier 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
integral over two charge density fields representing the fields of the unit cell and the crystal
lattice
where the unit-cell charge density field is a sum over the positions of the charges
qk in the central unit cell
and the total charge density field is the same sum over the unit-cell charges qk and
their periodic images

Here, is the Dirac delta function, , and are the lattice vectors and n1, n2 and n3
range over all integers. The total field can be represented as a convolution of
with a lattice function
Since this is a convolution, the Fourier transformation of is a product
where the Fourier transform of the lattice function is another sum over delta functions
where the reciprocal space vectors are defined (and cyclic permutations)
where is the volume of the central unit cell (if it is geometrically a
parallelepiped, which is often but not necessarily the case). Note that both and are
real, even functions.
For brevity, define an effective single-particle potential
Since this is also a convolution, the Fourier transformation of the same equation is a product
where the Fourier transform is defined
The energy can now be written as a single field integral

Using Parseval's theorem, the energy can also be summed in Fourier space
where in the final summation.
This is the essential result. Once is calculated, the summation/integration over 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.
[edit] Particle mesh Ewald (PME) method
Ewald summation was developed as a method of theoretical physics, long before the advent of
computers. However, the Ewald method has enjoyed widespread use since the 1970's in
computer simulations of particle systems, especially those interacting via an inverse square
force law such as gravity or electrostatics. Applications include simulations of plasmas,
galaxies and biomolecules.
As in normal Ewald summation, a generic interaction potential is separated into two terms
- a short-ranged part that sums quickly in real space
and a long-ranged part that sums quickly in Fourier space. The basic idea of particle
mesh Ewald summation is to replace the direct summation of interaction energies between
point particles
with two summations, a direct sum Esr of the short-ranged potential in real space
(that's the particle part of particle mesh Ewald) and a summation in Fourier space of the
long-ranged part
where and represent the Fourier transforms of the potential and the charge density
(that's 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. To evaluate the Fourier transform 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 (that's 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 one of a three-dimensional version of the Asteroids game,
in which each dimension "wraps around" on itself. This property of the cell 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
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 R grows like R2; (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 U of the dipole in a central unit cell with that surface charge density can be
written
where and Vuc are the net dipole moment and volume of the unit cell, dS 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.
[edit] 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.
La sommation d'Ewald (ou parfois somme d'Ewald) est une méthode de calcul des énergies
d'interaction de systèmes périodiques (et particulier des cristaux), et tout particulièrement les
énergies électrostatiques. La sommation d'Ewald est un cas particulier de la formule
sommatoire de Poisson, avec le remplacement de la sommation des énergies d'interaction
dans l'espace réel par une sommation équivalente dans un espace de Fourier. L'avantage de
cette approche est la convergence rapide de la sommation dans l'espace de Fourier comparée à
son équivalent dans l'espace réel lorsque les interactions se font à longue distance. Les
énergies électrostatiques comprenant à la fois des termes d'interactions de courtes et de
longues portées, il est très intéressant de décomposer le potentiel d'interaction en termes de
courte portée - dont la sommation se fait dans l'espace réel - et de longue portée - dont la
sommation se fait dans l'espace de Fourier.
La méthode fut développée par Paul Peter Ewald en 1921 (voir références) afin de déterminer
l'énergie électrostatique (et, par là, la constante de Madelung) des cristaux ioniques.
La sommation d'Ewald réécrit le potentiel d'interaction comme la somme de deux termes :
où représente le terme de courte portée dont la sommation est rapide dans l'espace
réel et le terme de longue portée dont la sommation est rapide est rapide dans l'espace
de Fourier. La partie à longue portée doit être finie pour tous les arguments (en particulier
pour r = 0) mais peuvent avoir une forme mathématique adéquate quelconque, le plus
généralement une distribution gaussienne. La méthode postule que la partie à courte portée
peut être sommée facilement, ainsi, le problème est de traiter la sommation du terme à longue
portée. En raison de l'utilisation d'une intégrale de Fourier, la méthode postule implicitement
que le système étudié est infiniment périodique (postulat pertinent pour un cristal parfait - i.e.