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calculate madelung constant by ewald method
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<i class="fa fa-calendar"></i><time datetime="2014-04-22T14:31:00+02:00"> Tue 22 April 2014</time>
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<p>I would like to understand the details of ewald summation. So, I took the
calculation of madelung constant (<a href="http://en.wikipedia.org/wiki/Madelung_constant">Wikipedia</a>)
as an example. I took two method to calculate madelung constant.
One is calculate it by expanding cubes method, the other is ewald summation method.
A good description of ewald summation can be found <a href="http://micro.stanford.edu/mediawiki/images/4/46/Ewald_notes.pdf">here</a>.</p>
<p>After the derivation, the total Coulomb interaction energy can be finally written as,
</p>
<div class="math">\begin{array}
E E &=& E^S + E^L - E^{self} \\
\end{array}</div>
<div class="math">\begin{array}
&=& \frac{1}{4\pi\epsilon_0}\frac{1}{2}
\sum_\mathbf{n} \sum_{i=1}^N \sum_{j=1}^N
\frac{q_iq_j}{|\mathbf{r}_i - \mathbf{r}_j + \mathbf{n}L|}
erfc\left( \frac{|\mathbf{r}_i - \mathbf{r}_j + \mathbf{n}L|}{\sqrt{2}\sigma}\right)
\end{array}</div>
<div class="math">\begin{array}
+ + \frac{1}{2V\epsilon_0} \sum_{\mathbf{k}\neq \mathbf{0}}
\frac{e^{-\sigma^2k^2/2}}{k^2}|S(\mathbf{k})|
-\frac{1}{4\pi\epsilon_0}\frac{1}{\sqrt{2\pi\sigma}}
\sum_{i=1}^{N}q_i^2
\end{array}</div>
<p>In the <a href="http://micro.stanford.edu/mediawiki/images/4/46/Ewald_notes.pdf">paper</a>
I just mentioned, we can find all the details for this formula.</p>
<p>While, we need potential instead of total energy to calculate Madelung constant.
We should take derivative over q to calculate the potential. Notice that some
term exists which include 1/2. These are used to deal with double counting.
When we want to calculate potential, these 1/2 should be erased.</p>
<p><span class="math">\(test^2\)</span></p>
<p><a href="./codes/madelung_realspace.m">madelung_realspace.m</a>
<a href="./codes/madelung_ewald.m">madelung_ewald.m</a></p>
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