Ewald summation

particle mesh EwaldEwald sumsEwaldEwald methodEwald sumParticle mesh Ewald methodparticle-mesh
Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions (e.g., electrostatic interactions) in periodic systems.wikipedia
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Periodic boundary conditions

periodiccircularcyclic boundary conditions
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).
PBCs can be used in conjunction with Ewald summation methods (e.g., the particle mesh Ewald method) to calculate electrostatic forces in the system.

Poisson summation formula

Poisson series
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.
(A broad function in real space becomes a narrow function in Fourier space and vice versa.) This is the essential idea behind Ewald summation.

Paul Peter Ewald

P. P. EwaldPaul P. EwaldEwald
Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions (e.g., electrostatic interactions) in periodic systems. 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.

Molecular dynamics

dynamicsMDmolecular dynamic
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.
This computational cost can be reduced by employing electrostatics methods such as particle mesh Ewald summation, particle–particle-particle–mesh (P3M), or good spherical cutoff methods ( O(n) ).

Madelung constant

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.
There are many practical methods for calculating Madelung's constant using either direct summation (for example, the Evjen method ) or integral transforms, which are used in the Ewald method.

Wolf summation

This method is generally more computationally efficient than the Ewald summation.

Ionic crystal

ionicionic crystalline compounds
It was first developed as the method for calculating electrostatic energies of ionic crystals, and is now commonly used for calculating long-range interactions in computational chemistry.

Computational chemistry

computational chemistcomputationalcomputational methods
It was first developed as the method for calculating electrostatic energies of ionic crystals, and is now commonly used for calculating long-range interactions in computational chemistry.

Frequency domain

frequency-domainFourier spaceFourier domain
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.

Singularity (mathematics)

singularitiessingularitysingular
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.

Convergent series

convergenceconvergesconverge
The advantage of this method is the rapid convergence of the energy compared with that of a direct summation.

Normal distribution

normally distributedGaussian distributionnormal
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.

Dirac delta function

Dirac deltadelta functionimpulse
Here, is the Dirac delta function, and are the lattice vectors and n_1, n_2 and n_3 range over all integers.

Convolution

convolvedconvolvingconvolution kernel
The total field can be represented as a convolution of with a lattice function

Fourier transform

continuous Fourier transformFourierFourier transforms
The short-range contribution is calculated in real space, whereas the long-range contribution is calculated using a Fourier transform.

Parallelepiped

parallelotopeparallelopipedparallelepipeds
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).

Plancherel theorem

Plancherel's theoremPlancherel formula
Using Plancherel theorem, the energy can also be summed in Fourier space

Theoretical physics

theoretical physicisttheoreticaltheoretical physicists
Ewald summation was developed as a method in theoretical physics, long before the advent of computers.

Computer

computerscomputer systemdigital computer
Ewald summation was developed as a method in theoretical physics, long before the advent of computers.

Computer simulation

computer modelsimulationcomputer modeling
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.

Inverse-square law

inverse square lawinverse squareinverse-square
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.

Force

forcesattractiveelastic force
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.

Gravity

gravitationgravitationalgravitational force
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.

Electrostatics

electrostaticelectrostatic repulsionelectrostatic interactions
Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions (e.g., electrostatic interactions) in periodic systems. 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.

Lennard-Jones potential

Lennard-JonesLennard-Jones interactionLennard-Jones interatomic potential
Recently, PME has also been used to calculate the r^{-6} part of the Lennard-Jones potential in order to eliminate artifacts due to truncation.