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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 98   | 1 | 2 |

Section 1.3.4.4.7.7. Lifchitz's reformulation

G. Bricognea

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1.3.4.4.7.7. Lifchitz's reformulation

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Lifchitz [see Agarwal et al. (1981)[link], Agarwal (1981)[link]] proposed that the idea of treating certain multipliers in Cruickshank's modified differential Fourier syntheses by means of a convolution in real space should be applied not only to [g_{j} ({\bf h})], but also to the polynomials [P_{p} ({\bf h})] which determine the type of differential synthesis being calculated. This leads to convoluting [\partial \sigma\llap{$-$}_{j}/\partial u_{p}] with the same ordinary weighted difference Fourier synthesis, rather than [\sigma\llap{$-$}_{j}] with the differential synthesis of type p. In this way, a single Fourier synthesis, with ordinary (scalar) symmetry properties, needs be computed; the parameter type and atom type both intervene through the function [\partial \sigma\llap{$-$}_{j}/\partial u_{p}] with which it is convoluted. This approach has been used as the basis of an efficient general-purpose least-squares refinement program for macromolecular structures (Tronrud et al., 1987[link]).

This rearrangement amounts to using the fact (Section 1.3.2.3.9.7[link]) that convolution commutes with differentiation. Let[D({\bf x}) = {\textstyle\sum\limits_{{\bf h}}} w_{{\bf h}}(|F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs}) \exp (i\varphi_{{\bf h}}^{\rm calc}) \exp (-2\pi i {\bf h} \cdot {\bf x})]be the inverse-variance weighted difference map, and let us assume that parameter [u_{p}] belongs to atom j. Then the Agarwal form for the pth component of the right-hand side of the normal equations is[\left({\partial D \over \partial u_{p}} * \sigma\llap{$-$}_{j}\right)(x_{j}),]while the Lifchitz form is[\left(D * {\partial \sigma\llap{$-$}_{j} \over \partial u_{p}}\right)({\bf x}_{j}).]

References

Agarwal, R. C. (1981). New results on fast Fourier least-squares refinement technique. In Refinement of Protein Structures, compiled by P. A. Machin, J. W. Campbell & M. Elder (ref. DL/SCI/R16), pp. 24–28. Warrington: SERC Daresbury Laboratory.
Agarwal, R. C., Lifchitz, A. & Dodson, E. J. (1981). Appendix (pp. 36–38) to Dodson (1981).
Tronrud, D. E., Ten Eyck, L. F. & Matthews, B. W. (1987). An efficient general-purpose least-squares refinement program for macromolecular structures. Acta Cryst. A43, 489–501.








































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