Triplet relationships using structural information

Giacovazzo, C.

doi:10.1107/97809553602060000555

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

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, p. 220 | 1 | 2 |

Section 2.2.5.4. Triplet relationships using structural information

C. Giacovazzo^a ^*

^aDipartimento Geomineralogico, Campus Universitario, I-70125 Bari, Italy
Correspondence e-mail: c.giacovazzo@area.ba.cnr.it

2.2.5.4. Triplet relationships using structural information

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A strength of direct methods is that no knowledge of structure is required for their application. However, when some a priori information is available, it should certainly be a weakness of the methods not to make use of this knowledge. The conditional distribution of Φ given $[R_{\bf h}R_{\bf k}R_{{\bf h}-{\bf k}}]$ and the first three of the five kinds of a priori information described in Section 2.2.4.1 is (Main, 1976; Heinermann, 1977a) $[P (\Phi) \simeq {\exp \{2QR_{1}R_{2}R_{3} \cos (\Phi - q)\}\over 2\pi I_{0} (2QR_{1}R_{2}R_{3})}, \eqno(2.2.5.14)]$ where $[Q \exp (iq) = {{\textstyle\sum_{i=1}^{p}} g_{i}({\bf h}_{1},{\bf h}_{2},{\bf h}_{3})\over \langle |F_{{\bf h}_{1}}|^{2} \rangle^{1/2} \langle |F_{{\bf h}_{2}}|^{2} \rangle^{1/2} \langle |F_{{\bf h}_{3}}|^{2} \rangle^{1/2}}.]$ $[{\bf h}_{1}, {\bf h}_{2}, {\bf h}_{3}]$ stand for h, $[-{\bf k}]$ , $[-{\bf h} + {\bf k}]$ , and $[R_{1}, R_{2}, R_{3}]$ for $[R_{\bf h}, R_{\bf k}, R_{{\bf h} - {\bf k}}]$ . The quantities $[\langle |F_{{\bf h}_{i}}|^{2} \rangle]$ have been calculated in Section 2.2.4.1 according to different categories: $[g_{i}({\bf h}_{1}, {\bf h}_{2}, {\bf h}_{3})]$ is a suitable average of the product of three scattering factors for the ith atomic group, p is the number of atomic groups in the cell including those related by symmetry elements. We have the following categories.

(a) No structural information

(2.2.5.14) then reduces to (2.2.5.6).
(b) Randomly positioned and randomly oriented atomic groups

Then $[g_{i}({\bf h}_{1}, {\bf h}_{2}, {\bf h}_{3}) = \textstyle\sum\limits_{j, \, k, \, l} f_{j}\;f_{k}\;f_{l} \langle \exp [2\pi i({\bf h}_{1} \cdot {\bf r}_{kj} + {\bf h}_{2} \cdot {\bf r}_{lj})]\rangle_{R},]$ where $[\langle \ldots \rangle_{R}]$ means rotational average. The average of the exponential term extends over all orientations of the triangle formed by the atoms j, k and l, and is given (Hauptman, 1965) by $[\eqalign{B(z, t) &= \langle \exp [2\pi i ({\bf h} \cdot {\bf r} + {\bf h}' \cdot {\bf r}')] \rangle\cr &= \left({\pi\over 2z}\right)^{1/2} \sum\limits_{n=0}^{\infty} {t^{2n}\over (n!)^{2}} J_{(4n+1)/2} (z),}]$ where $[z = 2 \pi [q^{2} r^{2} + 2 qrq' r' \cos \varphi_{q} \cos \varphi_{r} + q'^{2} r'^{2}]^{1/2}]$ and $[t = [2 \pi^{2} qrq' r' \sin \varphi_{q} \sin \varphi_{r}]/z\hbox{;}]$ q, q′, r and r′ are the magnitudes of h, h′, r and r′, respectively; $[\varphi_{q}]$ and $[\varphi_{r}]$ are the angles $[({\bf h},{\bf h}')]$ and $[({\bf r},{\bf r}')]$ , respectively.
(c) Randomly positioned but correctly oriented atomic groups

Then $[\eqalign{g_{i} ({\bf h}_{1}, {\bf h}_{2}, {\bf h}_{3}) &= \textstyle\sum\limits_{s = 1}^{m} \textstyle\sum\limits_{j, \, k, \, l} f_{j}\; f_{k}\; f_{l}\cr &\quad \times \exp [2\pi i ({\bf h}_{1} \cdot {\bf R}_{s} {\bf r}_{kj} + {\bf h}_{2} \cdot {\bf R}_{s} {\bf r}_{lk})],}]$ where the summations over j, k, l are taken over all the atoms in the ith group.

A modified expression for $[g_{i}]$ has to be used in polar space groups for special triplets (Giacovazzo, 1988).

Translation functions [see Chapter 2.3 ; for an overview, see also Beurskens et al. (1987)] are also used to determine the position of a correctly oriented molecular fragment.

Such functions can work in direct space [expressed as Patterson convolutions (Buerger, 1959; Nordman, 1985) or electron-density convolutions (Rossmann et al., 1964; Argos & Rossmann, 1980)] or in reciprocal space [expressed as correlation functions (Crowther & Blow, 1967; Karle, 1972; Langs, 1985) or residual functions (Rae, 1977)]. Both the probabilistic methods and the translation functions are quite efficient tools: the decision as to which one to use is often a personal choice.
(d) Atomic groups correctly positioned

Let p be the number of atoms with known position, q the number of atoms with unknown position, $[F_{p}]$ and $[F_{q}]$ the corresponding structure factors.

Tangent recycling methods (Karle, 1970b) may be used for recovering the complete crystal structure. The phase $[\varphi_{p, \, {\bf h}}]$ is accepted in the starting set as a useful approximation of $[\varphi_{\bf h}]$ if $[|F_{p, \, {\bf h}}|\gt \eta |F_{\bf h}|]$ , where η is the fraction of the total scattering power contained in the fragment and where $[|F_{\bf h}|]$ is associated with $[|E_{\bf h}|\gt 1.5]$ .

Tangent recycling methods are applied (Beurskens et al., 1979) with greater effectiveness to difference s.f.'s $[\Delta F =]$ $[(|F| - |F_{p}|) \exp (i \varphi_{p})]$ . The weighted tangent formula uses $[\Delta F_{\bf h}]$ values in order to convert them to more probable $[F_{q, \, {\bf h}}]$ values.

From a probabilistic point of view (Giacovazzo, 1983a; Camalli et al., 1985) the distribution of $[\varphi_{\bf h}]$ , given $[E'_{p, \, {\bf h}}]$ and some products $[(E'_{\bf k} - E'_{p, \, {\bf k}}) (E'_{{\bf h}-{\bf k}} - E'_{p, \, {\bf h}-{\bf k}})]$ , is the von Mises function $[P(\varphi_{\bf h}| \ldots) = [2\pi I_{0} (\alpha)]^{-1} \exp [\alpha \cos (\varphi_{\bf h} - \theta_{\bf h})], \eqno(2.2.5.15)]$ where $[\theta_{\bf h}]$ , the most probable value of $[\varphi_{\bf h}]$ , is given by $[\eqalignno{\tan \theta_{\bf h}& \simeq \alpha'_{2}/\alpha'_{1}, &(2.2.5.16)\cr \alpha^2&=\alpha_1^{'2}+\alpha_2^{'2}&\cr}]$ and $[ \eqalign{\alpha'_{1} &= 2 R'_{\bf h} \left\{\hbox{\scr R} \left[E'_{p, \, {\bf h}} + q^{-1/2} \textstyle\sum_{\bf k} (E'_{\bf k} - E'_{p, \, {\bf k}})\right.\right.\cr &\qquad \vphantom{\sum_{k}}\times (E'_{{\bf h}-{\bf k}} - E'_{p, \, {\bf h}- {\bf k}})\Big]\Big\}\cr \alpha'_{2} &= 2 R'_{\bf h} \left\{\hbox{\scr I} \left[E'_{p, \, {\bf h}} + q^{-1/2} \textstyle\sum_{\bf k} (E'_{\bf k} - E'_{p, \, {\bf k}}) \right.\right.\cr &\qquad \vphantom{\sum_{k}}\times (E'_{{\bf h}-{\bf k}} - E'_{p, \, {\bf h}-{\bf k}})\Big]\Big\}.}]$ $[ \hbox{\scr R}]$ and $[ \hbox{\scr I}]$ stand for `real and imaginary part of', respectively. Furthermore, $[E' = F/\sum_{q}^{1/2}]$ is a pseudo-normalized s.f. If no pair $[(\varphi_{\bf k}, \varphi_{{\bf h}-{\bf k}})]$ is known, then $[\eqalign{\alpha'_{1} &= 2 R'_{\bf h} R'_{p, \, {\bf h}} \cos \varphi_{p, \, {\bf h}}\cr \alpha'_{2} &= 2 R'_{\bf h} R'_{p, \, {\bf h}} \sin \varphi_{p, \, {\bf h}}}]$ and (2.2.5.15) reduces to Sim's (1959) equation $[P(\varphi_{\bf h}) \simeq [2\pi I_{0} (G)]^{-1} \exp [G \cos (\varphi_{\bf h} - \varphi_{p, \, {\bf h}})], \eqno(2.2.5.17)]$ where $[G = 2 R'_{\bf h} R'_{p, \, {\bf h}}]$ . In this case $[\varphi_{p, \, {\bf h}}]$ is the most probable value of $[\varphi_{\bf h}]$ .
(e) Pseudotranslational symmetry is present

Substructure and superstructure reflections are then described by different forms of the structure-factor equation (Böhme, 1982; Gramlich, 1984; Fan et al., 1983), so that probabilistic formulae estimating triplet cosines derived on the assumption that atoms are uniformly dispersed in the unit cell cannot hold. In particular, the reliability of each triplet also depends on, besides $[R_{\bf h}, R_{\bf k}, R_{{\bf h} - {\bf k}}]$ , the actual h, k, $[{\bf h}-{\bf k}]$ indices and on the nature of the pseudotranslation. It has been shown (Cascarano et al., 1985b; Cascarano, Giacovazzo & Luić, 1987) that (2.2.5.7), (2.2.5.8), (2.2.5.9) still hold provided $[G_{{{\bf h}, \, {\bf k}}_{j}}]$ is replaced by $[G'_{{{\bf h}, \, {\bf k}}_{j}} = {2 R_{\bf h} R_{{\bf k}_{j}} R_{{\bf h}-{\bf k}_{j}}\over \sqrt{N_{{\bf h}, \, {\bf k}}}},]$ where factors E and $[n_{i}]$ are defined according to Section 2.2.4.1, $[N_{{\bf h},{\bf k}} = {(\zeta_{\bf h} [\sigma_{2}]_{p} + [\sigma_{2}]_{q}) (\zeta_{\bf k} [\sigma_{2}]_{p} + [\sigma_{2}]_{q}) (\zeta_{{\bf h} - {\bf k}} [\sigma_{2}]_{p} + [\sigma_{2}]_{q})\over \{(\beta / m) [\sigma_{3}]_{p} (n_{1}^{2} n_{2}^{2} n_{3}^{2} \ldots) + [\sigma_{3}]_{q}\}^{2}},]$ and β is the number of times for which $[\displaylines{{\bf hR}_{s} \cdot {\bf u}_{1} \equiv 0 \ (\hbox{mod} \ 1) \qquad {\bf hR}_{s} \cdot {\bf u}_{2} \equiv 0 \ (\hbox{mod} \ 1) \qquad {\bf hR}_{s} \cdot {\bf u}_{3} \equiv 0 \ (\hbox{mod} \ 1) \ldots\cr {\bf kR}_{s} \cdot {\bf u}_{1} \equiv 0 \ (\hbox{mod} \ 1) \qquad {\bf kR}_{s} \cdot {\bf u}_{2} \equiv 0 \ (\hbox{mod} \ 1) \qquad {\bf kR}_{s} \cdot {\bf u}_{3} \equiv 0 \ (\hbox{mod} \ 1) \ldots\cr ({\bf h - k}){\bf R}_{s} \cdot {\bf u}_{1} \equiv 0 \ (\hbox{mod} \ 1)\qquad ({\bf h - k}){\bf R}_{s} \cdot {\bf u}_{2} \equiv 0 \ (\hbox{mod} \ 1)\cr ({\bf h} - {\bf k}) {\bf R}_{s} \cdot {\bf u}_{3} \equiv 0 \ (\hbox{mod} \ 1) \ldots}]$ are simultaneously satisfied when s varies from 1 to m. The above formulae have been generalized (Cascarano et al., 1988b) to the case in which deviations both of replacive and of displacive type from ideal pseudo-translational symmetry occur.

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, p. 220