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6.5.1 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 6.8.
Figure 6.8:
Ewald summation
|
We augment the delta-like point charges by Gaussian charge ``clouds''
of opposite sign,
|
(6.11) |
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 6.11 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:
|
(6.12) |
with
|
(6.13) |
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
|
(6.14) |
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].
Next: 6.5.2 Particle-Mesh Methods (PM
Up: 6.5 Particles and Fields
Previous: 6.5 Particles and Fields
Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001