A SYSTEMATIC STUDY OF COORDINATE PRECISION IN X-RAY STRUCTURE ANALYSES .2. PREDICTIVE ESTIMATES OF ESDS FOR THE GENERAL-ATOM CASE

The relationship between the mean isotropic e.s.d. ($) over bar sigma(A)(o) of any element type A in a crystal structure and the R factor and atomic constitution of that structure is explored for 124905 element-type occurrences calculated from 33955 entries in the Cambridge Structural Database. On the basis of the work of Cruickshank [Acta Cryst. (1960), 13, 774-777], it is shown that ($) over bar sigma(A)(p) values can be estimated by equations of the form ($) over bar sigma(A)(p) = KRN(c)(1/2)/Z(A) where N-c is taken as Sigma Z(i)(2)/Z(C)(2), the Z(i) are atomic numbers and the summation is over all atoms in the asymmetric unit. Values of K were obtained by regression techniques using the ($) over bar sigma(A)(o) as basis. The constant K-nc for noncentrosymmetric structures is found to be larger than K-c for centrosymmetric structures by a factor of similar to 2(1/2), as predicted by Cruickshank (1960). Two predictive equations are generated, one for first-row elements and the second for elements with Z(A) > 10. The relationship between the different constants K that arise in these two situations is linked to shape differentials in scattering-factor (f(i)) curves for light and heavy atoms. It is found that predictive equations in which the Z(i) are selectively replaced by f(i) at a constant sin theta/lambda of 0.30 Angstrom(-1) generate closely similar values of K for the light-atom and heavy-atom subsets. The overall analysis indicates that atomic e.s.d.'s may be seriously underestimated in the more precise structure determinations, that e.s.d.'s for the heaviest atoms may be less reliable than those for lighter atoms and that e.s.d.'s in noncentrosymmetric structures may be less accurate than those in centrosymmetric structures.