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
, 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
of the electrostatic potential.
Since the Fourier representation of a delta-function requires infinitely
many terms, the Fourier space calculation would again lead to
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,
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
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.
The potential sum is evaluated in
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
Proceeding to three-dimensional model systems, we consider
a cubic base cell with side length containing charges.
with integer etc.
Interparticle vectors: including all periodic images of the base
cell, we have
is a translation vector in the periodic lattice.
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
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