International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

International Tables for Crystallography (2006). Vol. F. ch. 18.5, p. 404   | 1 | 2 |

Section 18.5.2.2. Weights

D. W. J. Cruickshanka*

aChemistry Department, UMIST, Manchester M60 1QD, England
Correspondence e-mail: dwj_cruickshank@email.msn.com

18.5.2.2. Weights

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In the early stages of refinement, artificial weights may be chosen to accelerate refinement. In the final stages, the weights must be related to the precision of the structure factors if parameter variances are being sought. There are two distinct ways, covering two ranges of error, in which this may be done.

  • (1) The weights for [R_{1}], say, may reflect the precision of the [|F_{o}|], so that [w(hkl) = 1 / \sigma^{2} (|F_{hkl}|)], where [\sigma^{2}] is the estimated variance of [|F_{o}|] due to a specific class of experimental uncertainties. These absolute weights are derived from an analysis of the experiment. Weights chosen in this way lead to estimated parameter variances [\sigma_{i}^{2} = (a^{-1})_{ii}], (18.5.2.11)[link], which cover only the specific class of experimental uncertainties.

  • (2) The weights may reflect the trends in the [|\Delta| \equiv \|F_{o}| - |F_{c}\|]. A weighting function with a small number of parameters is chosen so that the averages of [w \Delta^{2}] are constant when the set of [w\Delta^{2}] values is analysed in any pertinent fashion (e.g. in bins of increasing [|F_{o}|] and [2\sin \theta/\lambda]). Weights chosen in this way are relative weights, and the expression for the parameter variances needs a scaling factor, [S^{2} = \left(\textstyle\sum\limits_{3}\displaystyle w\Delta^{2}\right) \bigg/ (n_{\rm obs} - n_{\rm params}). \eqno(18.5.2.12)] Hence, in the full-matrix case, [\sigma_{i}^{2} = \left[\left(\textstyle\sum\limits_{3}\displaystyle w\Delta^{2}\right)\bigg / (n_{\rm obs} - n_{\rm params})\right] (a^{-1})_{ii}, \eqno(18.5.2.13)] which allows for all random experimental errors, such systematic experimental errors as cannot be simulated in the [|F_{c}|] and imperfections in the calculated model.








































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