Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 1.3, p. 69   | 1 | 2 |

Section Parseval's theorem with crystallographic symmetry

G. Bricognea

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Parseval's theorem with crystallographic symmetry

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The general statement of Parseval's theorem given in Section[link] may be rewritten in terms of symmetry-unique structure factors and electron densities by means of orbit decomposition.

In reciprocal space, [{\textstyle\sum\limits_{{\bf h} \in {\bb Z}^{3}}} \overline{F_{1} ({\bf h})} F_{2} ({\bf h}) = {\textstyle\sum\limits_{l \in L}}\; {\textstyle\sum\limits_{\gamma_{l} \in G/G_{{\bf h}_{l}}}} \overline{F_{1} ({\bf R}_{\gamma_{l}}^{T} {\bf h}_{l})} F_{2} ({\bf R}_{\gamma_{l}}^{T} {\bf h}_{l})\hbox{;}] for each l, the summands corresponding to the various [\gamma_{l}] are equal, so that the left-hand side is equal to [\eqalign{&F_{1} ({\bf 0}) F_{2} ({\bf 0})\cr &\quad + {\textstyle\sum\limits_{c \in L_{c}}} 2|(G/G_{{\bf h}_{c}})^{+} \|F_{1} ({\bf h}_{c})\| F_{2} ({\bf h}_{c})| \cos [\varphi_{1} ({\bf h}_{c}) - \varphi_{2} ({\bf h}_{c})]\cr &\quad + {\textstyle\sum\limits_{a \in L_{a}^{+}}} 2|G/G_{{\bf h}_{a}} \|F_{1} ({\bf h}_{a})\| F_{2} ({\bf h}_{a})| \cos [\varphi_{1} ({\bf h}_{a}) - \varphi_{2} ({\bf h}_{a})].}]

In real space, the triple integral may be rewritten as [{\textstyle\int\limits_{{\bb R}^{3}/{\bb Z}^{3}}} \overline{\rho\llap{$-\!$}_{1} ({\bf x})} \rho\llap{$-\!$}_{2} ({\bf x}) \hbox{ d}^{3} {\bf x} = |G| {\textstyle\int\limits_{D}} \overline{\rho\llap{$-\!$}_{1} ({\bf x})} \rho\llap{$-\!$}_{2} ({\bf x}) \hbox{ d}^{3} {\bf x}] (where D is the asymmetric unit) if [\rho\llap{$-\!$}_{1}] and [\rho\llap{$-\!$}_{2}] are smooth densities, since the set of special positions has measure zero. If, however, the integral is approximated as a sum over a G-invariant grid defined by decimation matrix N, special positions on this grid must be taken into account: [\eqalign{&{1 \over |{\bf N}|} \sum\limits_{{\bf k} \in {\bb Z}^{3}/{\bf N}{\bb Z}^{3}} \overline{\rho\llap{$-\!$}_{1} ({\bf x})} \rho\llap{$-\!$}_{2} ({\bf x})\cr &\qquad = {1 \over |{\bf N}|} \sum\limits_{{\bf x} \in D}\; [G:G_{\bf x}] \overline{\rho\llap{$-\!$}_{1} ({\bf x})} \rho\llap{$-\!$}_{2} ({\bf x})\cr &\qquad = {|G| \over |{\bf N}|} \sum\limits_{{\bf x} \in D} {1 \over |G_{\bf x}|} \overline{\rho\llap{$-\!$}_{1} ({\bf x})} \rho\llap{$-\!$}_{2} ({\bf x}),}] where the discrete asymmetric unit D contains exactly one point in each orbit of G in [{\bb Z}^{3}/{\bf N}{\bb Z}^{3}].

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