International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 724-725
Section 8.7.3.8. Uncertainties in experimental electron densitiesa 732 NSM Building, Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14260-3000, USA,bDigital Equipment Co., 129 Parker Street, PKO1/C22, Maynard, MA 01754-2122, USA, and cEcole Centrale Paris, Centre de Recherche, Grand Voie des Vignes, F-92295 Châtenay Malabry CEDEX, France |
It is often important to obtain an estimate of the uncertainty in the deformation densities in Table 8.7.3.1. If it is assumed that the density of the static atoms or fragments that are subtracted out are precisely known, three sources of uncertainty affect the deformation densities: (1) the uncertainties in the experimental structure factors; (2) the uncertainties in the positional and thermal parameters of the density functions that influence ρcalc; and (3) the uncertainty in the scale factor k.
If we assume that the uncertainties in the observed structure factors are not correlated with the uncertainties in the refined parameters, the variance of the electron density is given by where
, up is a positional or thermal parameter, and the γ(up, k) are correlation coefficients between the scale factor and the other parameters (Rees, 1976
, 1978
; Stevens & Coppens, 1976
).
Similarly, for the covariance between the deformation densities at points A and B, where it is implied that the second term includes the effect of the scale factor/parameter correlation.
Following Rees (1976), we may derive a simplified expression for the covariance valid for the space group
. Since the structure factors are not correlated with each other,
where the latter equality is specific for
, and
indicates summation over a hemisphere in reciprocal space. In general, the second term rapidly averages to zero as
increases, while the first term may be replaced by its average
with u = 2π |rA − rB|hmax, or cov(ρobs, Aρobs, B)
(2/V2)C(u) ×
and σ2(ρobs)
, a relation derived earlier by Cruickshank (1949
). A discussion of the applicability of this expression in other centrosymmetric space groups is given by Rees (1976
).
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