   # 9.1 Ewald summation

Let be the ion-ion interaction between charged particles. In the Ewald summation approach [EWALD 21] the basic cell with containing each of positive and negative charges in some spatial arrangement is interpreted as a single crystallographic element surrounded by an infinite number of identical copies of itself. The entire system is then neutral and contains an infinite number of charges situated at points and , respectively. The total potential at the position of some ion residing in the basic cell is given by the finite difference of two infinite, diverging series: We are facing the problem of an undetermined form . Instead of evaluating the potential as a sum over the point charges we may first rewrite these charges as delta-like charge densities, and expand these in a Fourier series whose terms determine the Fourier components of the electrostatic potential. Since the Fourier representation of a delta-function requires infinitely many terms, the Fourier space calculation would again lead to convergence problems.

The solution is to split up the potential in two well-behaved parts, one being represented in -space and the other in -space by rapidly converging series. We demonstrate this on a one-dimensional ion lattice with a charge distribution as depicted in Figure 9.1. We augment the delta-like point charges by Gaussian charge clouds'' of opposite sign, (9.1)

to form an auxiliary lattice . A further lattice ( ) is then introduced to compensate the additional Gaussian charges, such that lattice 1 + lattice 2 = original lattice''.

The contributions of the two lattices to the potential are computed separately:
• Lattice : Seen from a greater distance, a Gaussian charge cloud resembles a delta-like point charge, effectively compensating the original charge it accompanies. The effect of lattice is therefore best computed in -space, where the series will converge quite rapidly. The convergence will be faster if the Gaussians are narrow, i.e. if the parameter in 9.1 is large.
• Lattice : The potential sum is evaluated in -space. When the Gaussians are broad, i.e. when is small, we will need a smaller number of Fourier components.
By suitably adjusting , optimal convergence of both series may be achieved.

Proceeding to three-dimensional model systems, we consider a cubic base cell with side length containing charges.
Fourier vectors: with integer etc.
Interparticle vectors: including all periodic images of the base cell, we have where is a translation vector in the periodic lattice.

Ewald sum: (9.2)

with (9.3)

Note: Two details need attention:

• The Gaussian charge clouds will formally interact with themselves, giving rise to a spurious contribution to the potential energy; this contribution must be subtracted in the final formula.
• The consistent way of taking the infinite-size limit is the following:
- consider a finite (roughly spherical) array of image cells; surround them by a continuum with some arbitrary dielectric constant , which is usually taken to be ;
- take the limit of an infinitely large repeated array; this limit still contains a contribution from .

Considering these two corrections, we have for the total potential energy (9.4)

Another interesting class of particles are those with embedded point dipoles. Several methods have been devised to deal with the long range contributions in these model systems. One is a modification of the Ewald sum; it is known as the Ewald-Kornfeld summation'' technique. Other strategies are the reaction field method and Ladd's multipole expansion method; see [VESELY 78] and [ALLEN 90].   F. J. Vesely / University of Vienna