International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 79-82
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Most crystallographic Fourier syntheses are real-valued and originate from Hermitian-symmetric collections of Fourier coefficients. Hermitian symmetry is closely related to the action of a centre of inversion in reciprocal space, and thus interacts strongly with all other genuinely crystallographic symmetry elements of order 2. All these symmetry properties are best treated by factoring by 2 and reducing the computation of the initial transform to that of a collection of smaller transforms with less symmetry or none at all.
The computation of a DFT with Hermitian-symmetric or real-valued data can be carried out at a cost of half that of an ordinary transform, essentially by `multiplexing' pairs of special partial transforms into general complex transforms, and then `demultiplexing' the results on the basis of their symmetry properties. The treatment given below is for general dimension n; a subset of cases for was treated by Ten Eyck (1973).
A vector is said to be Hermitian-antisymmetric if Its transform then satisfies i.e. is purely imaginary.
If X is Hermitian-antisymmetric, then is Hermitian-symmetric, with real-valued. The treatment of Section 1.3.4.3.5.1 may therefore be adapted, with trivial factors of i or , or used as such in conjunction with changes of variable by multiplication by .
The matrix is its own contragredient, and hence (Section 1.3.2.4.2.2) the transform of a symmetric (respectively antisymmetric) function is symmetric (respectively antisymmetric). In this case the group acts in both real and reciprocal space as . If with both factors diagonal, then acts by i.e.
The symmetry or antisymmetry properties of X may be written with for symmetry and for antisymmetry.
The computation will be summarized as with the same indexing as that used for structure-factor calculation. In both cases it will be shown that a transform with and M diagonal can be computed using only partial transforms instead of .
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Conjugate symmetric (Section 1.3.2.4.2.3) implies that if the data X are real and symmetric [i.e. and ], then so are the results . Thus if contains a centre of symmetry, F is real symmetric. There is no distinction (other than notation) between structure-factor and electron-density calculation; the algorithms will be described in terms of the former. It will be shown that if , a real symmetric transform can be computed with only partial transforms instead of .
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If X is real antisymmetric, then its transform is purely imaginary and antisymmetric. The double-multiplexing techniques used for real symmetric transforms may therefore be adapted with only minor changes involving signs and factors of i.
So far the multiplexing technique has been applied to pairs of vectors with similar types of parity-related and/or conjugate symmetry properties, in particular the same value of ɛ.
It can be generalized so as to accommodate mixtures of vectors with different symmetry characteristics. For example if is Hermitian-symmetric and is Hermitian-antisymmetric, so that is real-valued while has purely imaginary values, the multiplexing process should obviously form (instead of if both had the same type of symmetry), and demultiplexing consists in separating
The general multiplexing formula for pairs of vectors may therefore be written where ω is a phase factor (e.g. 1 or i) chosen in such a way that all non-exceptional components of and (or and ) be embedded in the complex plane along linearly independent directions, thus making multiplexing possible.
It is possible to develop a more general form of multiplexing/demultiplexing for more than two vectors, which can be used to deal with symmetry elements of order 3, 4 or 6. It is based on the theory of group characters (Ledermann, 1987).
References
Ledermann, W. (1987). Introduction to group characters, 2nd ed. Cambridge University Press.Google ScholarTen Eyck, L. F. (1973). Crystallographic fast Fourier transforms. Acta Cryst. A29, 183–191.Google Scholar