Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions (e.g., Coulombic interactions) in periodic systems. 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. 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 in order to calculate accurately 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.
Contents
Derivation
Ewald summation rewrites the interaction potential as the sum of two terms,
where
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
and the total charge density field
Here,
Since this is a convolution, the Fourier transformation of
where the Fourier transform of the lattice function is another sum over delta functions
where the reciprocal space vectors are defined
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
This is the essential result. Once
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. Recently, PME has also been used to calculate the
In the particle mesh method, just as in standard Ewald summation, the generic interaction potential is separated into two terms
with two summations, a direct sum
(that is the particle part of particle mesh Ewald) and a summation in Fourier space of the long-ranged part
where
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
The electrostatic energy of a polar crystal (i.e., a crystal with a net dipole
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
The negative sign derives from the definition of
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. Direct calculation gives