# Ewald's generalized theta function

Could anyone provide me some materials on the derivation of Ewald's generalized theta function (in English)? The original paper was written in German :-(

Die Berechnung optischer und elektrostatischer Gitterpotentiale

P. P. Ewald

Annalen der Physik

Volume 369, pages 253–287, 1921

http://onlinelibrary.wiley.com/doi/10.1002/andp.19213690304/abstract

(DOI: 10.1002/andp.19213690304)

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A step by step derivation of Ewald summation can be found in:

Williams, D. E. (1971). Accelerated convergence of crystal-lattice potential sums. Acta Crystallographica Section A, 27(5), 452–455. doi:10.1107/S0567739471000998

There's also an expanded version of the above paper:

Williams, D. E. (2006). Accelerated convergence treatment of $R^{-n}$ lattice sums. In U. Shmueli (Ed.), International Tables for Crystallography. Volume B. (pp. 385–397). Kluwer Academic Publishers.

De Leeuw, S. W., Perram, J. W., & Smith, E. R. (1980). Simulation of Electrostatic Systems in Periodic Boundary Conditions. I. Lattice Sums and Dielectric Constants. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 373(1752), 27–56. doi:10.1098/rspa.1980.0135

Nijboer, B. R. A., & De Wette, F. W. (1957). On the calculation of lattice sums. Physica, 23(1-5), 309–321. doi:10.1016/S0031-8914(57)92124-9

I think between the papers of Williams and the appendices in Nijboer and De Wette, it is possible to fill all the gaps (at least in the case where there's no dipole moment).

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Thanks! Actually I was looking for the following identity \sum_{\vec{n}} e^{-t||\vec{n}+\vec{r}||^2} = \left(\frac{\pi}{t}\right)^(\frac{3}{2}) \sum_{\vec{n}} e^{-\frac{pi^2}{t}\cdot ||\vec{n}||^2}+i\cdot 2\pi\cdot \vec{n}\cdot \vec{r}} , which is equation 3.10 in the 2nd paper you mentioned(i.e. Leeuw et. al.'s paper and they missed the summation operator on the left hand side in that equation). I was trying to find a derivation for this idenity. Do you know any other web materials related to the proof? –  Steven Jul 28 '13 at 0:27

Here you go: http://en.wikipedia.org/wiki/Ewald_summation . And here are some characters to fool the retarded length police.

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I wasn't looking for Ewald's trick for accelerating the convergence of the infinite sum (see above for my comments to mathoverflow member "jmbr"). Thanks, though! –  Steven Jul 28 '13 at 0:30
@Steven: the wikipedia article I pointed at is describing exactly the content of the paper you were curious about (see the reference list at the end), and the Ewald theta function arises in that context. –  Igor Rivin Jul 28 '13 at 1:08
Sorry, I still do not quite see the relation between the derivation in that wikipedia article and the generalized theta function identity. Are you saying replacing the delta function by an exponential function in the step where we do the Fourier transformation of the lattice function will give the proof the generalized delta-function identity? –  Steven Jul 28 '13 at 4:56