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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 99-100   | 1 | 2 |

Section 1.3.4.4.7.10. Sampling considerations

G. Bricognea

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1.3.4.4.7.10. Sampling considerations

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The calculation of the inner products [(D, \partial \rho\llap{$-\!$}^{\rm calc}/\partial u_{p})] from a sampled gradient map D requires even more caution than that of structure factors via electron-density maps described in Section 1.3.4.4.5[link], because the functions [\partial \sigma\llap{$-$}_{j}/\partial u_{p}] have transforms which extend even further in reciprocal space than the [\sigma\llap{$-$}_{j}] themselves. Analytically, if the [\sigma\llap{$-$}_{j}] are Gaussians, the [\partial \sigma\llap{$-$}_{j}/\partial u_{p}] are finite sums of multivariate Hermite functions (Section 1.3.2.4.4.2[link]) and hence the same is true of their transforms. The difference map D must therefore be finely sampled and the relation between error and sampling rate may be investigated as in Section 1.3.4.4.5.[link] An examination of the sampling rates commonly used (e.g. one third of the resolution) shows that they are insufficient. Tronrud et al. (1987)[link] propose to relax this requirement by applying an artificial temperature factor to [\sigma\llap{$-$}_{j}] (cf. Section 1.3.4.4.5[link]) and the negative of that temperature factor to D, a procedure of questionable validity because the latter `sharpening' operation is ill defined [the function exp [(\|{\bf x}\|^{2})] does not define a tempered distribution, so the associativity properties of convolution may be lost]. A more robust procedure would be to compute the scalar product by means of a more sophisticated numerical quadrature formula than a mere grid sum.

References

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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