International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 724-725

Section 8.7.3.8. Uncertainties in experimental electron densities

P. Coppens,a Z. Sub and P. J. Beckerc

a 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

8.7.3.8. Uncertainties in experimental electron densities

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It is often important to obtain an estimate of the uncertainty in the deformation densities in Table 8.7.3.1[link]. 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 [\eqalignno{ \sigma ^2 (\Delta\rho) &=\sigma ^2 (\rho _{{\rm obs}}^{\prime }) +\sigma ^2(\rho _{{\rm calc}}) +(\rho _{{\rm obs}}^{\prime }) ^2 \, {\sigma ^2(k) \over k^2} \cr &\quad+\sum \limits _p\displaystyle {\delta \rho _{{\rm calc}} \over \delta u_p}\sigma(u_p) \rho _{{\rm obs}}^{\prime } {\sigma(k) \over k}\gamma (u_p,k), & (8.7.3.93)}]where [\rho_{\rm obs}^\prime =\rho_{\rm obs}/k], 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[link], 1978[link]; Stevens & Coppens, 1976[link]).

Similarly, for the covariance between the deformation densities at points A and B, [\eqalignno{ {\rm cov}(\Delta\rho _A,\Delta\rho _B) &={\rm cov} (\rho _{{\rm obs},A},\rho _{{\rm obs},B}) /k^2 \cr &\quad+{\rm cov}(\rho _{{\rm calc},A},\rho _{{\rm calc},B}) \cr &\quad+\rho _{{\rm obs},A}\rho _{{\rm obs},B} [\sigma (k) /k] ^2, & (8.7.3.94)}]where it is implied that the second term includes the effect of the scale factor/parameter correlation.

Following Rees (1976[link]), we may derive a simplified expression for the covariance valid for the space group [P\bar 1]. Since the structure factors are not correlated with each other, [\eqalignno{ {\rm cov} (\rho _{{\rm obs},A},\rho _{{\rm obs},B}) &\simeq \sum \displaystyle{\partial \rho _{{\rm obs},A} \over\partial F_{{\rm obs}}({\bf h}) }\, {\partial \rho _{{\rm obs},B} \over \partial F_{{\rm obs}}({\bf h}) }\sigma ^2\left [F_{{\rm obs}}({\bf h}) \right] \cr &\simeq { 2\over V^2}\sum _{1/2}\sigma ^2 (F_{{\rm obs}}) \left [\cos 2\pi({\bf r}_A+{\bf r}_B) \cdot {\bf h}\right. \cr &\left. \quad+\cos 2\pi ({\bf r}_A-{\bf r}_B) \cdot {\bf h}\right] , & (8.7.3.95)}]where the latter equality is specific for [P\bar 1], and [\sum _{1/2}] indicates summation over a hemisphere in reciprocal space. In general, the second term rapidly averages to zero as [h_{{\it \max }}] increases, while the first term may be replaced by its average [ \left \langle \cos 2\pi \left ({\bf r}_A-{\bf r}_B\right) \cdot {\bf h}\right \rangle =3\left (\sin u-u\cos u\right) /u^3\equiv {\rm C}\left (u\right), \eqno (8.7.3.96)]with u = 2π |rArB|hmax, or cov(ρobs, Aρobs, B) [\simeq] (2/V2)C(u) × [\sum _{1/2}\sigma ^2(F_0)] and σ2obs) [\simeq (2/{V^2})\sum _{1/2}\sigma^2(F_0)], a relation derived earlier by Cruickshank (1949[link]). A discussion of the applicability of this expression in other centrosymmetric space groups is given by Rees (1976[link]).

References

First citation Cruickshank, D. W. (1949). The accuracy of electron density maps in X-ray analysis with special reference to dibenzyl. Acta Cryst. 2, 65–82.Google Scholar
First citation Rees, B. (1976). Variance and covariance in experimental electron density studies, and the use of chemical equivalence. Acta Cryst. A32, 483–488.Google Scholar
First citation Rees, B. (1978). Errors in deformation-density and valence-density maps: the scale-factor contribution. Acta Cryst. A34, 254–256.Google Scholar
First citation Stevens, E. D. & Coppens, P. (1976). A priori estimates of the errors in experimental electron densities. Acta Cryst. A32, 915–917.Google Scholar








































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