Lifchitz's reformulation

Bricogne, G.

doi:10.1107/97809553602060000551

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 91 | 1 | 2 |

Section 1.3.4.4.7.7. Lifchitz's reformulation

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.4.4.7.7. Lifchitz's reformulation

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Lifchitz [see Agarwal et al. (1981), Agarwal (1981)] 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).

This rearrangement amounts to using the fact (Section 1.3.2.3.9.7) 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.Google Scholar

Agarwal, R. C., Lifchitz, A. & Dodson, E. J. (1981). Appendix (pp. 36–38) to Dodson (1981).Google Scholar

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.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 91