Symmetry

Diamond, R.

doi:10.1107/97809553602060000561

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

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International Tables for Crystallography (2006). Vol. B. ch. 3.3, p. 373 | 1 | 2 |

Section 3.3.1.3.12. Symmetry

R. Diamond^a ^*

^aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England
Correspondence e-mail: rd10@cam.ac.uk

3.3.1.3.12. Symmetry

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In Section 3.3.1.1.1 it was pointed out that it is usual to express coordinates for graphical purposes in Cartesian coordinates in ångström units or nanometres. Symmetry, however, is best expressed in crystallographic fractional coordinates. If a molecule, with Cartesian coordinates, is being displayed, and a symmetry-related neighbour is also to be displayed, then the data-space coordinates must be multiplied by $[ \pmatrix{{\bi W} &{\bf T}\cr {\bf 0}^{T} &{W}\cr} \pmatrix{{\bi M} &{\bf 0}\cr {\bf 0}^{T} &{1}\cr} {\scr S}\ \pmatrix{{\bi M}^{-1} &{\bf 0}\cr {\bf 0}^{T} &{1}\cr} \pmatrix{{\bi W} &-{\bf T}\cr {\bf 0}^{T} &{W}\cr},]$ where $[\pmatrix{{\bf T}\cr {W}\cr}]$ are the data-space coordinates of the crystallographic origin, M and $[{\bi M}^{-1}]$ are as in Section 3.3.1.1.1 and $[ \hbox{\scr S}]$ is a crystallographic symmetry operator in homogeneous coordinates, expressed relative to the same crystallographic origin.

For example, in $[P2_{1}]$ with the origin on the screw dyad along b, $[ {\scr S}\ = \pmatrix{-1 &0 &0 &0\cr 0 &1 &0 &{1\over 2}\cr 0 &0 &-1 &0\cr 0 &0 &0 &1\cr}]$ and $[ \pmatrix{{\bi M} &{\bf 0}\cr {\bf 0}^{T} &{1}\cr} \ {\scr S}\ \ \pmatrix{{\bi M}^{-1} &{\bf 0}\cr {\bf 0}^{T} &{1}\cr} = \pmatrix{-1 &0 &0 &0\cr 0 &1 &0 &{1\over 2}b\cr 0 &0 &-1 &0\cr 0 &0 &0 &1\cr}.]$

$[ {\scr S}]$ comprises a proper or improper rotational partition, S, in the upper-left $[3 \times 3]$ in the sense that $[{\bi MSM}^{-1}]$ is orthogonal, and with the associated fractional lattice translation in the last column, with the last row always consisting of three zeros and 1 at the 4, 4 position. See IT A (2005, Chapters 5.2 and 8.1 ) for a fuller discussion of symmetry using augmented (i.e. $[4 \times 4]$ ) matrices.

References

International Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by T. Hahn. Heidelberg: Springer.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 3.3, p. 373