International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 8991

All the necessary ingredients are now available for calculating the CDFT for any given space group.
Space group P1 is dealt with by the methods of Section 1.3.4.3.5.1 and by those of Section 1.3.4.3.5.4.
A general monoclinic transformation is of the formwith 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 thusThe symmetry relations obeyed by and F are as follows: for electron densitiesor, after factoring by 2,while for structure factorswith its Friedel counterpartor, after factoring by 2,with Friedel counterpart
When calculating electron densities, two methods may be used.
Almost all orthorhombic space groups are generated by two monoclinic transformations and of the type described in Section 1.3.4.3.6.2, with the addition of a centre of inversion for centrosymmetric groups. The only exceptions are Fdd2 and Fddd which contain diamond glides, in which some nonprimitive translations are `square roots' not of primitive lattice translations, but of centring translations. The generic case will be examined first.
To calculate electron densities, the unique octant of data may first be transformed on (respectively ) as in Section 1.3.4.3.6.2 using the symmetry pertaining to generator . These intermediate results may then be expanded by generator by the formula of Section 1.3.4.3.3 prior to the final transform on (respectively ). To calculate structure factors, the reverse operations are applied in the reverse order.
The two exceptional groups Fdd2 and Fddd only require a small modification. The Fcentring causes the systematic absence of parity classes with mixed parities, leaving only (000) and (111). For the former, the phase factors in the symmetry relations of Section 1.3.4.3.6.2 become powers of (−1) so that one is back to the generic case. For the latter, these phase factors are odd powers of i which it is a simple matter to incorporate into a modified multiplexing/demultiplexing procedure.
All the symmetries in this class of groups can be handled by the generalized Rader/Winograd algorithms of Section 1.3.4.3.4.3, but no implementation of these is yet available.
In groups containing axes of the form with g.c.d. along the c direction, the following procedure may be used (Ten Eyck, 1973):
These are usually treated as their orthorhombic or tetragonal subgroups, as the bodydiagonal threefold axis cannot be handled by ordinary methods of decomposition.
The threedimensional factorization technique of Section 1.3.4.3.4.1 allows a complete treatment of cubic symmetry. Factoring by 2 along all three dimensions gives four types (i.e. orbits) of parity classes:Orbit exchange using the threefold axis thus allows one to reduce the number of partial transforms from 8 to 4 (one per orbit). Factoring by 3 leads to a reduction from 27 to 11 (in this case, further reduction to 9 can be gained by multiplexing the three diagonal classes with residual threefold symmetry into a single class; see Section 1.3.4.3.5.6). More generally, factoring by q leads to a reduction from to . Each of the remaining transforms then has a symmetry induced from the orthorhombic or tetragonal subgroup, which can be treated as above.
No implementation of this procedure is yet available.
Lattice centring is an instance of the duality between periodization and decimation: the extra translational periodicity of ρ induces a decimation of described by the `reflection conditions' on h. As was pointed out in Section 1.3.4.2.2.3, nonprimitive lattices are introduced in order to retain the same matrix representation for a given geometric symmetry operation in all the arithmetic classes in which it occurs. From the computational point of view, therefore, the main advantage in using centred lattices is that it maximizes decomposability (Section 1.3.4.2.2.4); reindexing to a primitive lattice would for instance often destroy the diagonal character of the matrix representing a dyad.
In the usual procedure involving three successive onedimensional transforms, the loss of efficiency caused by the duplication of densities or the systematic vanishing of certain classes of structure factors may be avoided by using a multiplexing/demultiplexing technique (Ten Eyck, 1973):
The threedimensional factorization technique of Section 1.3.4.3.4.1 is particularly well suited to the treatment of centred lattices: if the decimation matrix of N contains as a factor a matrix which `integerizes' all the nonprimitive lattice vectors, then centring is reflected by the systematic vanishing of certain classes of vectors of decimated data or results, which can simply be omitted from the calculation. An alternative possibly is to reindex on a primitive lattice and use different representative matrices for the symmetry operations: the loss of decomposability is of little consequence in this threedimensional scheme, although it substantially complicates the definition of the cocycles and .
The preceding sections have been devoted to showing how the raw computational efficiency of a crystallographic Fourier transform algorithm can be maximized. This section will briefly discuss another characteristic (besides speed) which a crystallographic Fourier transform program may be required to possess if it is to be useful in various applications: a convenient and versatile mode of presentation of input data or output results.
The standard crystallographic FFT programs (Ten Eyck, 1973, 1985) are rather rigid in this respect, and use rather rudimentary data structures (lists of structurefactor values, and twodimensional arrays containing successive sections of electrondensity maps). It is frequently the case that considerable reformatting of these data or results must be carried out before they can be used in other computations; for instance, maps have to be converted from 2D sections to 3D `bricks' before they can be inspected on a computer graphics display.
The explicitly threedimensional approach to the factorization of the DFT and the use of symmetry offers the possibility of richer and more versatile data structures. For instance, the use of `decimation in frequency' in real space and of `decimation in time' in reciprocal space leads to data structures in which realspace coordinates are handled by blocks (thus preserving, at least locally, the threedimensional topological connectivity of the maps) while reciprocalspace indices are handled by parity classes or their generalizations for factors other than 2 (thus making the treatment of centred lattices extremely easy). This global threedimensional indexing also makes it possible to carry symmetry and multiplicity characteristics for each subvector of intermediate results for the purpose of automating the use of the orbit exchange mechanism.
Brünger (1989) has described the use of a similar threedimensional factoring technique in the context of structurefactor calculations for the refinement of macromolecular structures.
References
Brünger, A. T. (1989). A memoryefficient fast Fourier transformation algorithm for crystallographic refinement on supercomputers. Acta Cryst. A45, 42–50.Ten Eyck, L. F. (1973). Crystallographic fast Fourier transforms. Acta Cryst. A29, 183–191.
Ten Eyck, L. F. (1985). Fast Fourier transform calculation of electron density maps. In Diffraction Methods for Biological Macromolecules (Methods in Enzymology, Vol. 115), edited by H. Wyckoff, C. H. W. Hirs & S. N. Timasheff, pp. 324–337. New York: Academic Press.