Variance and covariance in experimental electron density studies, and the use of chemical equivalence

Expressions are derived for the estimation of the variance and the covariance in the deformation electron density Q_o/k - Q_c, for a centrosymmetric crystal structure. The uncertainty in the calculated density Q_c depends strongly on the distance to the nuclei. The uncertainty in the observed density Q_o is fairly constant, except in the vicinity of a symmetry element, where it depends on the same correlation function as the covariance between observed densities. This correlation function is itself dependent on the resolution λ/(2 sin θ)_max, the choice of which is of basic importance in any electron density study. The variance-covariance matrix may be used in an averaging procedure of electron densities at chemically, but not crystallographically, equivalent points. The method is applied to the results of X-ray and neutron diffraction studies of chromium hexacarbonyl Cr(CO)₆.