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 Electron-density calculations

G. Bricognea

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Electron-density calculations

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A formula for the Fourier synthesis of electron-density maps from symmetry-unique structure factors is readily obtained by orbit decomposition: [\eqalign{\rho\llap{$-\!$} ({\bf x}) &= {\textstyle\sum\limits_{{\bf h} \in {\bb Z}^{3}}} F({\bf h}) \exp (-2\pi i{\bf h} \cdot {\bf x})\cr &= {\textstyle\sum\limits_{l \in L}} \left[{\textstyle\sum\limits_{\gamma_{l} \in G/G_{{\bf h}_{l}}}} F({\bf R}_{\gamma_{l}}^{T} {\bf h}_{l}) \exp [-2\pi i({\bf R}_{\gamma_{l}}^{T} {\bf h}_{l}) \cdot {\bf x}]\right]\cr &= {\textstyle\sum\limits_{l \in L}} \;F({\bf h}_{l}) \left[{\textstyle\sum\limits_{\gamma_{l} \in G/G_{{\bf h}_{l}}}} \exp \{-2\pi i{\bf h}_{l} \cdot [S_{\gamma_{l}} ({\bf x})]\}\right],}] where L is a subset of [{\bb Z}^{3}] such that [\{{\bf h}_{l}\}_{l \in L}] contains exactly one point of each orbit for the action [\theta^{*}: (g, {\bf h}) \;\longmapsto\; ({\bf R}_{g}^{-1})^{T} {\bf h}] of G on [{\bb Z}^{3}]. The physical electron density per cubic ångström is then [\rho ({\bf X}) = {1 \over V} \rho\llap{$-\!$} ({\bf Ax})] with V in Å3.

In the absence of anomalous scatterers in the crystal and of a centre of inversion −I in Γ, the spectrum [\{F({\bf h})\}_{{\bf h} \in {\bb Z}^{3}}] has an extra symmetry, namely the Hermitian symmetry expressing Friedel's law (Section[link]). The action of a centre of inversion may be added to that of Γ to obtain further simplification in the above formula: under this extra action, an orbit [G{\bf h}_{l}] with [{\bf h}_{l} \neq {\bf 0}] is either mapped into itself or into the disjoint orbit [G(-{\bf h}_{l})]; the terms corresponding to [+{\bf h}_{l}] and [-{\bf h}_{l}] may then be grouped within the common orbit in the first case, and between the two orbits in the second case.

  • Case 1: [G (-{\bf h}_{l}) = G{\bf h}_{l}, {\bf h}_{l}] is centric. The cosets in [G/G_{{\bf h}_{l}}] may be partitioned into two disjoint classes by picking one coset in each of the two-coset orbits of the action of −I. Let [(G/G_{{\bf h}_{l}})^{+}] denote one such class: then the reduced orbit [\{{\bf R}_{\gamma_{l}}^{T} {\bf h}_{l} | \gamma_{l} \in (G/G_{{\bf h}_{l}})^{+}\}] contains exactly once the Friedel-unique half of the full orbit [G{\bf h}_{l}], and thus [|(G/G_{{\bf h}_{l}})^{+}| = {\textstyle{1 \over 2}} |G/G_{{\bf h}_{l}}|.] Grouping the summands for [+{\bf h}_{l}] and [- {\bf h}_{l}] yields a real-valued summand [2F({\bf h}_{l}) {\textstyle\sum\limits_{\gamma_{l} \in (G/G_{{\bf h}_{l}})^{+}}} \cos [2\pi {\bf h}_{l} \cdot [S_{\gamma_{l}} ({\bf x})] - \varphi_{{\bf h}{l}}].]

  • Case 2: [G(- {\bf h}_{l}) \neq G{\bf h}_{l},\ {\bf h}_{l}] is acentric. The two orbits are then disjoint, and the summands corresponding to [+ {\bf h}_{l}] and [- {\bf h}_{l}] may be grouped together into a single real-valued summand [2F({\bf h}_{l}) {\textstyle\sum\limits_{\gamma_{l} \in G/G_{{\bf h}_{l}}}} \cos [2\pi {\bf h}_{l} \cdot [S_{\gamma_{l}} ({\bf x})] - \varphi_{{\bf h}_{l}}].]

    In order to reindex the collection of all summands of [\rho\llap{$-\!$}], put [L = L_{c} \cup L_{a},] where [L_{c}] labels the Friedel-unique centric reflections in L and [L_{a}] the acentric ones, and let [L_{a}^{+}] stand for a subset of [L_{a}] containing a unique element of each pair [\{+ {\bf h}_{l}, - {\bf h}_{l}\}] for [l \in L_{a}]. Then [\eqalign{\rho\llap{$-\!$} ({\bf x}) &= F ({\bf 0})\cr &\quad + {\textstyle\sum\limits_{c \in L_{c}}} \left[2F ({\bf h}_{c}) {\textstyle\sum\limits_{\gamma_{c} \in (G/G_{{\bf h}_{c}})^{+}}} \cos [2\pi {\bf h}_{c} \cdot [S_{\gamma_{c}} ({\bf x})] - \varphi_{{\bf h}_{c}}]\right]\cr &\quad + {\textstyle\sum\limits_{a \in L_{a}^{+}}} \left[2F ({\bf h}_{a}) {\textstyle\sum\limits_{\gamma_{a} \in G/G_{{\bf h}_{a}}}} \cos [2 \pi {\bf h}_{a} \cdot [S_{\gamma_{a}} ({\bf x})] - \varphi_{{\bf h}_{a}}]\right].}]

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