9.1 Ewald summation

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

We augment the delta-like point charges by Gaussian charge ``clouds'' of opposite sign,

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.

Proceeding to

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) |

(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

F. J. Vesely / University of Vienna