The Poisson summation formula

Bricogne, G.

doi:10.1107/97809553602060000551

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 42-43 | 1 | 2 |

Section 1.3.2.6.7. The Poisson summation formula

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.6.7. The Poisson summation formula

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Let $[\varphi \in {\scr S}]$ , so that $[{\scr F}[\varphi] \in {\scr S}]$ . Let R be the lattice distribution associated to lattice Λ, with period matrix A, and let $[R^{*}]$ be associated to the reciprocal lattice $[\Lambda^{*}]$ . Then we may write: $[\eqalignno{\langle R, \varphi \rangle &= \langle R, \bar{\scr F}[{\scr F}[\varphi]]\rangle\cr &= \langle \bar{\scr F}[R], {\scr F}[\varphi]\rangle\cr &= |\det {\bf A}|^{-1} \langle R^{*}, {\scr F}[\varphi]\rangle}]$ i.e. $[{\textstyle\sum\limits_{{\bf x} \in \Lambda}} \varphi ({\bf x}) = |\det {\bf A}|^{-1} {\textstyle\sum\limits_{{\boldxi} \in \Lambda^{*}}} {\scr F}[\varphi] ({\boldxi}).]$

This identity, which also holds for $[\bar{\scr F}]$ , is called the Poisson summation formula. Its usefulness follows from the fact that the speed of decrease at infinity of φ and $[{\scr F}[\varphi]]$ are inversely related (Section 1.3.2.4.4.3), so that if one of the series (say, the left-hand side) is slowly convergent, the other (say, the right-hand side) will be rapidly convergent. This procedure has been used by Ewald (1921) [see also Bertaut (1952), Born & Huang (1954)] to evaluate lattice sums (Madelung constants) involved in the calculation of the internal electrostatic energy of crystals (see Chapter 3.4 in this volume on convergence acceleration techniques for crystallographic lattice sums).

When φ is a multivariate Gaussian $[\varphi ({\bf x}) = G_{\bf B} ({\bf x}) = \exp (-\textstyle{{1 \over 2}} {\bf x}^{T} {\bf Bx}),]$ then $[{\scr F}[\varphi] (\boldxi) = |\det (2 \pi {\bf B}^{-1})|^{1/2} G_{{\bf B}^{-1}} (\boldxi),]$ and Poisson's summation formula for a lattice with period matrix A reads: $[\eqalignno{{\textstyle\sum\limits_{{\bf m} \in {\bb Z}^{n}}} G_{\bf B} ({\bf Am}) &= |\det {\bf A}|^{-1}| \det (2 \pi {\bf B}^{-1})|^{1/2}\cr &\quad \times \textstyle\sum\limits_{{\boldmu} \in {\bb Z}^{n}} G_{4 \pi^{2}{\bf B}^{-1}} [({\bf A}^{-1})^{T} {\boldmu}]}]$ or equivalently $[{\textstyle\sum\limits_{{\bf m} \in {\bb Z}^{n}}} G_{C} ({\bf m}) = |\det (2 \pi {\bf C}^{-1})|^{1/2} {\textstyle\sum\limits_{{\boldmu} \in {\bb Z}^{n}}} G_{4 \pi^{2}}{{_{{\bf C}^{-1}}}} ({\boldmu})]$ with $[{\bf C} = {\bf A}^{T} {\bf BA}.]$

References

Bertaut, E. F. (1952). L'énergie électrostatique de réseaux ioniques. J. Phys. Radium, 13, 499–505.Google Scholar

Born, M. & Huang, K. (1954). Dynamical theory of crystal lattices. Oxford University Press.Google Scholar

Ewald, P. P. (1921). Die Berechnung optischer und electrostatischer Gitterpotentiale. Ann. Phys. Leipzig, 64, 253–287.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 42-43