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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 89   | 1 | 2 |

## Section 1.3.4.3.6.2. Monoclinic groups

G. Bricognea

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#### 1.3.4.3.6.2. Monoclinic groups

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A general monoclinic transformation is of the form with a diagonal matrix whose entries are or , and a vector whose entries are 0 or . We may thus decompose both real and reciprocal space into a direct sum of a subspace where acts as the identity, and a subspace where acts as minus the identity, with . All usual entities may be correspondingly written as direct sums, for instance: 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 with , . The nonprimitive translation vector then belongs to , and thus The symmetry relations obeyed by and F are as follows: for electron densities or, after factoring by 2, while for structure factors with its Friedel counterpart or, after factoring by 2, with Friedel counterpart When calculating electron densities, two methods may be used.

 (i) Transform on first. The partial vectors defined by obey symmetry relations of the form with independent of . This is the same relation as for the same parity class of data for a (complex or real) symmetric or antisymmetric transform. The same techniques can be used to decrease the number of by multiplexing pairs of such vectors and demultiplexing their transforms. Partial vectors with different values of may be mixed in this way (Section 1.3.4.3.5.6 ). Once is completed, its results have Hermitian symmetry with respect to , and the methods of Section 1.3.4.3.5.1 may be used to obtain the unique electron densities. (ii) Transform on first. The partial vectors defined by obey symmetry relations of the form with independent of . This is the same relation as for the same parity class of data for a Hermitian symmetric or antisymmetric transform. The same techniques may be used to decrease the number of . This generalizes the procedure described by Ten Eyck (1973) for treating dyad axes, i.e. for the case , and (simple dyad) or (screw dyad). Once is completed, its results have Hermitian symmetry properties with respect to which can be used to obtain the unique electron densities. Structure factors may be computed by applying the reverse procedures in the reverse order.

### References

Ten Eyck, L. F. (1973). Crystallographic fast Fourier transforms. Acta Cryst. A29, 183–191.