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9.1 Ewald summation



be the ion-ion interaction between charged particles. In the Ewald summation approach [EWALD 21] the basic cell with containing $N/2$ 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 $\mbox{$\bf r$}_{j+}$ and $\mbox{$\bf r$}_{j-} $, respectively. The total potential at the position of some ion $i$ residing in the basic cell is given by the finite difference of two infinite, diverging series:

\phi(\mbox{$\bf r$}_{i})=
q \sum_{j+=1}^{\infty} \frac{1}{\l...
...}{\left\vert\mbox{$\bf r$}_{i}-\mbox{$\bf r$}_{j-}\right\vert}

We are facing the problem of an undetermined form $\infty - \infty$. Instead of evaluating the potential as a sum over the point charges we may first rewrite these charges as delta-like charge densities,

\rho(\mbox{$\bf r$})=
q \sum_{j+=1}^{\infty} \delta \left(\m...
...infty} \delta \left(\mbox{$\bf r$}-\mbox{$\bf r$}_{j-} \right)

and expand these in a Fourier series whose terms determine the Fourier components $\phi(\mbox{$\bf k$})$ 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 $\mbox{$\bf r$}$-space and the other in $\mbox{$\bf k$}$-space by rapidly converging series. We demonstrate this on a one-dimensional ion lattice with a charge distribution as depicted in Figure 9.1.

Figure 9.1: Ewald summation

We augment the delta-like point charges by Gaussian charge ``clouds'' of opposite sign,
\rho'(\mbox{$\bf r$}) = -q_{j}\left(\frac{\eta^{2}}{\pi}\rig...
...{\textstyle - \eta^{2}(\mbox{$\bf r$}-\mbox{$\bf r$}_{j})^{2}}
\end{displaymath} (9.1)

to form an auxiliary lattice $1$. A further lattice ($2$) 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: By suitably adjusting $\eta$, optimal convergence of both series may be achieved.

Proceeding to three-dimensional model systems, we consider a cubic base cell with side length $L$ containing $N$ charges.
Fourier vectors:

\mbox{$\bf k$} \equiv \frac{2 \pi}{L} \left( k_{x}, k_{y}, k_{z}\right)

with integer $k_{x}$ etc.
Interparticle vectors: including all periodic images of the base cell, we have

\mbox{$\bf r$}_{i,j,\mbox{$\bf n$}} \equiv \mbox{$\bf r$}_{j...{$\bf n$} L - \mbox{$\bf r$}_{i}
\;\;\; (i,j = 1, \dots, N)

where $\mbox{$\bf n$}L $ is a translation vector in the periodic lattice.

Ewald sum:
\phi \left(\mbox{$\bf r$}_{i}\right) = \frac{4 \pi}{L^{3}}
...F(\eta \vert\mbox{$\bf r$}_{i,j,\mbox{$\bf n$}}\vert ) \right]
\end{displaymath} (9.2)

F(z) \equiv \frac{2}{\sqrt{\pi}} \int_{z}^{\infty} e^{\textstyle -t^{2}} dt
\end{displaymath} (9.3)

Note: Two details need attention:

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

E_{pot} = \frac{1}{2}   \sum_{i=1}^{N} q_{i} \phi \left(\mb...
...eft\vert \sum_{i=1}^{N} q_{i}\mbox{$\bf r$}_{i}\right\vert^{2}
\end{displaymath} (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].
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F. J. Vesely / University of Vienna