Monoclinic groups

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, p. 82 | 1 | 2 |

Section 1.3.4.3.6.2. Monoclinic groups

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

1.3.4.3.6.2. Monoclinic groups

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A general monoclinic transformation is of the form $[S_{g}: {\bf x} \;\longmapsto\; {\bf R}_{g} {\bf x} + {\bf t}_{g}]$ with $[{\bf R}_{g}]$ a diagonal matrix whose entries are [+1] or [-1] , and $[{\bf t}_{g}]$ a vector whose entries are 0 or $[{1 \over 2}]$ . We may thus decompose both real and reciprocal space into a direct sum of a subspace $[{\bb Z}^{n_{+}}]$ where $[{\bf R}_{g}]$ acts as the identity, and a subspace $[{\bb Z}^{n_{-}}]$ where $[{\bf R}_{g}]$ acts as minus the identity, with $[n_{+} + n_{-} = n = 3]$ . All usual entities may be correspondingly written as direct sums, for instance: $[\matrix{\eqalign{{\bf R}_{g} &= {\bf R}_{g}^{+} \oplus {\bf R}_{g}^{-},\cr {\bf t}_{g} &= {\bi t}_{g}^{+} \oplus {\bf t}_{g}^{-},\cr {\bf m} &= {\bf m}^{+} \oplus {\bf m}^{-}, \cr{\bf h} &= {\bf h}^{+} \oplus {\bf h}^{-},} &\eqalign{{\bf N} &= {\bf N}^{+} \oplus {\bf N}^{-}, \cr {\bf t}_{g}^{(1)} &= {\bf t}_{g}^{(1) +} \oplus {\bf t}_{g}^{(1) -},\cr {\bf m}_{1} &= {\bf m}_{1}^{+} \oplus {\bf m}_{1}^{-},\cr{\bf h}_{1} &= {\bf h}_{1}^{+} \oplus {\bf h}_{1}^{-}, } &\eqalign{{\bf M} &= {\bf M}^{+} \oplus {\bf M}^{-},\cr {\bf t}_{g}^{(2)} &= {\bf t}_{g}^{(2) +} \oplus {\bf t}_{g}^{(2) -},\cr {\bf m}_{2} &= {\bf m}_{2}^{+} \oplus {\bf m}_{2}^{-},\cr {\bf h}_{2} &= {\bf h}_{2}^{+} \oplus {\bf h}_{2}^{-}.}}]$

We will use factoring by 2, with decimation in frequency when computing structure factors, and decimation in time when computing electron densities; this corresponds to $[{\bf N} = {\bf N}_{1}{\bf N}_{2}]$ with $[{\bf N}_{1} = {\bf M}]$ , $[{\bf N}_{2} = 2{\bf I}]$ . The non-primitive translation vector $[{\bf Nt}_{g}]$ then belongs to $[{\bf M}{\bb Z}^{n}]$ , and thus $[{\bf t}_{g}^{(1)} = {\bf 0} \hbox{ mod } {\bf M}{\bb Z}^{n},\quad {\bf t}_{g}^{(2)} \in {\bb Z}^{n} / 2{\bb Z}^{n}.]$ The symmetry relations obeyed by $[\rho\llap{$-\!$}]$ and F are as follows: for electron densities $[\rho\llap{$-\!$} ({\bf m}^{+}, {\bf m}^{-}) = \rho\llap{$-\!$} ({\bf m}^{+} + {\bf N}^{+}{\bf t}_{g}^{+}, -{\bf m}^{-} - {\bf N}^{-}{\bf t}_{g}^{-})]$ or, after factoring by 2, $[\eqalign{ &\rho\llap{$-\!$} ({\bf m}_{1}^{+}, {\bf m}_{2}^{+}, {\bf m}_{1}^{-}, {\bf m}_{2}^{-}) \cr &\phantom{\rho\llap{$-\!$}} = \rho\llap{$-\!$} ({\bf m}_{1}^{+}, {\bf m}_{2}^{+} + {\bf t}_{g}^{(2) +}, {\bf M}^{-}{\boldzeta} ({\bf m}_{1}^{-}) - {\bf m}_{1}^{-} - {\bf m}_{2}^{-}, {\bf m}_{2}^{-} + {\bf t}_{g}^{(2) -})\hbox{;}}]$ while for structure factors $[F ({\bf h}^{+}, {\bf h}^{-}) = \exp [2 \pi i({\bf h}^{+} \cdot {\bf t}_{g}^{+} + {\bf h}^{-} \cdot {\bf t}_{g}^{-})] F ({\bf h}^{+}, -{\bf h}^{-})]$ with its Friedel counterpart $[F ({\bf h}^{+}, {\bf h}^{-}) = \exp [2 \pi i({\bf h}^{+} \cdot {\bf t}_{g}^{+} + {\bf h}^{-} \cdot {\bf t}_{g}^{-})] \overline{F (-{\bf h}^{+}, {\bf h}^{-})}]$ or, after factoring by 2, $[\eqalign{ F({\bf h}_{1}^{+}, {\bf h}_{2}^{+}, {\bf h}_{1}^{-}, {\bf h}_{2}^{-}) =\ &(-1)^{{\bf h}_{2}^{+} \cdot {\bf t}_{g}^{(2) +} + {\bf h}_{2}^{-} \cdot {\bf t}_{g}^{(2) -}}\cr &\times F({\bf h}_{1}^{+}, {\bf h}_{2}^{+}, {\bf M}^{-}{\boldzeta} ({\bf h}_{1}^{-}) - {\bf h}_{1}^{-} - {\bf h}_{2}^{-}, {\bf h}_{2}^{-})}]$ with Friedel counterpart $[\eqalign{&F({\bf h}_{1}^{+}, {\bf h}_{2}^{+}, {\bf h}_{1}^{-}, {\bf h}_{2}^{-})\cr &\quad= (-1)^{{\bf h}_{2}^{+} \cdot {\bf t}_{g}^{(2) +} + {\bf h}_{2}^{-} \cdot {\bf t}_{g}^{(2) -}} \overline{F [{\bf M}^{+}{\boldzeta} ({\bf h}_{1}^{+}) - {\bf h}_{1}^{+} - {\bf h}_{2}^{+}, {\bf h}_{2}^{+}, {\bf h}_{1}^{-}, {\bf h}_{2}^{-}].}}]$

When calculating electron densities, two methods may be used.

(i) Transform on $[{\bf h}^{-}]$ first.

The partial vectors defined by $[X_{{\bf h}^{+}, \, {\bf h}_{2}^{-}} = F({\bf h}^{+}, {\bf h}_{1}^{-}, {\bf h}_{2}^{-})]$ obey symmetry relations of the form $[X({\bf h}_{1}^{-} - {\bf h}_{2}^{-}) = \varepsilon X[{\bf M}^{-}{\boldzeta} ({\bf h}_{1}^{-}) - {\bf h}_{1}^{-}]]$ with $[\varepsilon = \pm 1]$ independent of $[{\bf h}_{1}^{-}]$ . This is the same relation as for the same parity class of data for a (complex or real) symmetric $[(\varepsilon = + 1)]$ or antisymmetric $[(\varepsilon = - 1)]$ transform. The same techniques can be used to decrease the number of $[F({\bf M}^{-})]$ by multiplexing pairs of such vectors and demultiplexing their transforms. Partial vectors with different values of ɛ may be mixed in this way (Section 1.3.4.3.5.6).

Once $[F({\bf N}^{-})]$ is completed, its results have Hermitian symmetry with respect to $[{\bf h}^{+}]$ , and the methods of Section 1.3.4.3.5.1 may be used to obtain the unique electron densities.
(ii) Transform on $[{\bf h}^{+}]$ first.

The partial vectors defined by $[X_{{\bf h}^{-}, \, {\bf h}_{2}^{+}} = F({\bf h}_{1}^{+}, {\bf h}_{2}^{+}, {\bf h}^{-})]$ obey symmetry relations of the form $[X({\bf h}_{1}^{+} - {\bf h}_{2}^{+}) = \overline{\varepsilon X[{\bf M}^{+}{\boldzeta} ({\bf h}_{1}^{+}) - {\bf h}_{1}^{+}]}]$ with $[\varepsilon = \pm 1]$ independent of $[{\bf h}_{1}^{+}]$ . This is the same relation as for the same parity class of data for a Hermitian symmetric $[(\varepsilon = + 1)]$ or antisymmetric $[(\varepsilon = - 1)]$ transform. The same techniques may be used to decrease the number of $[F({\bf M}^{+})]$ . This generalizes the procedure described by Ten Eyck (1973) for treating dyad axes, i.e. for the case $[n_{+} = 1, {\bf t}_{g}^{(2) -} = {\bf 0}]$ , and $[{\bf t}_{g}^{(2) +} = {\bf 0}]$ (simple dyad) or $[{\bf t}_{g}^{(2) +} {\ne {\bf 0}}]$ (screw dyad).

Once $[F({\bf N}^{+})]$ is completed, its results have Hermitian symmetry properties with respect to $[{\bf h}^{-}]$ which can be used to obtain the unique electron densities.

Structure factors may be computed by applying the reverse procedures in the reverse order.