Structure-factor calculation

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. 68-69 | 1 | 2 |

Section 1.3.4.2.2.6. Structure-factor calculation

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

1.3.4.2.2.6. Structure-factor calculation

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Structure factors may be calculated from a list of symmetry-unique atoms by Fourier transformation of the orbit decomposition formula for the motif $[\rho\llap{$-\!$}^{0}]$ given in Section 1.3.4.2.2.4: $[\eqalignno{F({\bf h}) &= \bar{{\scr F}}[\rho\llap{$-\!$}^{0}] ({\bf h})&\cr &= \bar{{\scr F}}\left[{\textstyle\sum\limits_{j \in J}} \left({\textstyle\sum\limits_{\gamma_{j} \in G/G_{{\bf x}_{j}}}} S_{\gamma_{j}}^{\#} (\tau_{{\bf x}_{j}} \rho\llap{$-\!$}_{j})\right)\right] ({\bf h})&\cr &= {\textstyle\sum\limits_{j \in J}} {\textstyle\sum\limits_{\gamma_{j} \in G/G_{{\bf x}_{j}}}} \bar{{\scr F}}[\tau_{{\bf t}_{\gamma_{j}}} {\bf R}_{\gamma_{j}}^{\#} \tau_{{\bf x}_{j}} \rho\llap{$-\!$}_{j}] ({\bf h})&\cr &= {\textstyle\sum\limits_{j \in J}} {\textstyle\sum\limits_{\gamma_{j} \in G/G_{{\bf x}_{j}}}} \exp (2\pi i {\bf h} \cdot {\bf t}_{\gamma_{j}})&\cr &\quad \times [({\bf R}_{\gamma_{j}}^{-1})^{T_{\#}} [\exp (2\pi i{\boldxi} \cdot {\bf x}_{j}) \bar{{\scr F}}[\rho\llap{$-\!$}_{j}]_{{\boldxi}}]] ({\bf h}) &\cr &= {\textstyle\sum\limits_{j \in J}}\; {\textstyle\sum\limits_{\gamma_{j} \in G/G_{{\bf x}_{j}}}} \exp (2\pi i{\bf h} \cdot {\bf t}_{\gamma_{j}})&\cr &\quad \times \exp [2\pi i({\bf R}_{\gamma_{j}}^{T} {\bf h}) \cdot {\bf x}_{j}] \bar{{\scr F}}[\rho\llap{$-\!$}_{j}] ({\bf R}_{\gamma_{j}}^{T} {\bf h})\hbox{;}&}]$ i.e. finally: $[F({\bf h}) = {\textstyle\sum\limits_{j \in J}}\; {\textstyle\sum\limits_{\gamma_{j} \in G/G_{{\bf x}_{j}}}} \exp \{2\pi i{\bf h} \cdot [S_{\gamma_{j}} ({\bf x}_{j})]\} \bar{{\scr F}}[\rho\llap{$-\!$}_{j}] ({\bf R}_{\gamma_{j}}^{T} {\bf h}).]$

In the case of Gaussian atoms, the atomic transforms are $[\bar{{\scr F}}[\rho\llap{$-\!$}_{j}] ({\bf h}) = Z_{j} \exp [- {\textstyle{1 \over 2}} {\bf h}^{T} (4\pi^{2} {\bf Q}_{j}) {\bf h}]]$ or equivalently $[\bar{{\scr F}}[\rho_{j}] ({\bf H}) = Z_{j} \exp [-{\textstyle{1 \over 2}} {\bf H}^{T} (4\pi^{2} {\bf U}_{j}) {\bf H}].]$

Two common forms of equivalent temperature factors (incorporating both atomic form and thermal motion) are

(i) isotropic B: $[\bar{{\scr F}}[\rho\llap{$-\!$}_{j}] ({\bf h}) = Z_{j} \exp (-{\textstyle{1 \over 4}} B_{j} {\bf H}^{T} {\bf H}),]$ so that $[{\bf U}_{j} = (B_{j}/8\pi^{2}) {\bf I}]$ , or $[{\bf Q}_{j} = (B_{j}/8\pi^{2}) {\bf A}^{-1} ({\bf A}^{-1})^{T}]$ ;
(ii) anisotropic β's: $[\bar{{\scr F}}[\rho\llap{$-\!$}_{j}] ({\bf h}) = Z_{j} \exp (-{\bf h}^{T} {\boldbeta}_{j} {\bf h}),]$ so that $[{\boldbeta}_{j} = 2\pi^{2} {\bf Q}_{j} = 2\pi^{2} {\bf A}^{-1} {\bf U}_{j} ({\bf A}^{-1})^{T}]$ , or $[{\bf U}_{j} = (1/2\pi^{2}) {\bf A}\beta_{j} {\bf A}^{T}]$ .

In the first case, $[\bar{{\scr F}}[\rho\llap{$-\!$}_{j}] ({\bf R}_{\gamma_{j}}^{T} {\bf h})]$ does not depend on $[\gamma_{j}]$ , and therefore: $[\eqalign{ F({\bf h}) &= {\textstyle\sum\limits_{j \in J}} \;Z_{j} \exp \{-{\textstyle{1 \over 4}} B_{j} {\bf h}^{T} [{\bf A}^{-1} ({\bf A}^{-1})^{T}] {\bf h}\}\cr &\quad \times {\textstyle\sum\limits_{\gamma_{j} \in G/G_{{\bf x}_{j}}}} \exp \{2\pi i{\bf h} \cdot [S_{\gamma_{j}} ({\bf x}_{j})]\}.}]$ In the second case, however, no such simplification can occur: $[\eqalign{ F({\bf h}) &= {\textstyle\sum\limits_{j \in J}} \;Z_{j} {\textstyle\sum\limits_{\gamma_{j} \in G/G_{{\bf x}_{j}}}} \exp [-{\bf h}^{T} ({\bf R}_{\gamma_{j}} {\boldbeta}_{j} {\bf R}_{\gamma_{j}}^{T}) {\bf h}]\cr &\quad \times \exp \{2\pi i{\bf h} \cdot [S_{\gamma_{j}} ({\bf x}_{j})]\}.}]$ These formulae, or special cases of them, were derived by Rollett & Davies (1955), Waser (1955b), and Trueblood (1956).

The computation of structure factors by applying the discrete Fourier transform to a set of electron-density values calculated on a grid will be examined in Section 1.3.4.4.5.

References

Rollett, J. S. & Davies, D. R. (1955). The calculation of structure factors for centrosymmetric monoclinic systems with anisotropic atomic vibration. Acta Cryst. 8, 125–128.Google Scholar

Trueblood, K. N. (1956). Symmetry transformations of general anisotropic temperature factors. Acta Cryst. 9, 359–361.Google Scholar

Waser, J. (1955b). The anisotropic temperature factor in triclinic coordinates. Acta Cryst. 8, 731.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 68-69