Uncertainties in derived functions

Coppens, P.; Su, Z.; Becker, P. J.

doi:10.1107/97809553602060000615

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

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International Tables for Crystallography (2006). Vol. C. ch. 8.7, p. 725

Section 8.7.3.9. Uncertainties in derived functions

P. Coppens,^a Z. Su^b and P. J. Becker^c

^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

8.7.3.9. Uncertainties in derived functions

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The electrostatic moments are functions of the scale factor, the positional parameters x, y and z of the atoms, and their charge-density parameters κ and $[P_{lmp}]$ . The standard uncertainties in the derived moments are therefore dependent on the variances and covariances of these parameters.

If M_p represents the m × m variance–covariance matrix of the parameters [p_j] , and T is an n × m matrix defined by $[\partial \mu ^l/\partial p_j]$ for the lth moment with n independent elements, the variances and covariances of the elements of m^l are obtained from $[ {\bi M}_\mu ={\bi TM}_p{\bi T}^T. \eqno (8.7.3.96a) ]$ If a moment of an assembly of pseudo-atoms is evaluated, the elements of T include the effects of coordinate rotations required to transfer atomic moments into a common coordinate system.

A frequently occurring case of interest is the evaluation of the magnitude of a molecular dipole moment and its standard deviation. Defining $[ y=\mu ^2={\boldmu}{\bi G}{\boldmu}^T, \eqno (8.7.3.96b)]$ where $[{\boldmu}]$ is the dipole-moment vector and G is the direct-space metric tensor of the appropriate coordinate system. If Y is the 1 × 3 matrix of the derivatives $[\partial y/\partial \mu _{{i}}]$ , $[ \sigma ^2(y) ={\bi YM}_\mu {\bi Y}^T, \eqno (8.7.3.96c)]$ where M_μ is defined by (8.7.3.96a). The standard uncertainty in μ may be obtained from σ(μ) = σ(y)/2μ. Significant contributions often result from uncertainties in the positional and charge-density parameters of the H atoms.

References

International Tables for Crystallography (2006). Vol. C. ch. 8.7, p. 725