# 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.

### 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.