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), 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 , but also to the polynomials which determine the type of differential synthesis being calculated. This leads to convoluting with the same ordinary weighted difference Fourier synthesis, rather than 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 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. Letbe the inverse-variance weighted difference map, and let us assume that parameter belongs to atom j. Then the Agarwal form for the pth component of the right-hand side of the normal equations iswhile the Lifchitz form is

### 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.