A systematic study of coordinate precision in X-ray structure analyses. I. Descriptive statistics and predictive estimates of e.s.d.'s for C atoms

Abstract

This study examines the relationship of structure precision, as expressed by the e.s.d.'s of atomic coordinates, to the R-factor and chemical constitution of a given crystal structure. On the basis of the work of Cruickshank [Acta Cryst. (1960), 13, 744–777], it is shown that \sigma \bar (C–C), the mean e.s.d. of a C–C bond length in a structure, or \sigma \bar(C), the mean isotropic e.s.d. of a C atom, can be estimated by expressions of the form \sigma \bar = kRNc1/2. Here, Nc is taken as ΣZi2/ZC2, with the atomic numbers Zi summed over all atoms in the asymmetric unit and ZC = 6. It is also shown that \sigma \bar(E), the mean isotropic e.s.d. of a non-C atom, can be estimated by \sigma \bar(E) = kRNc1/2/ZE. Values of k were determined by regression analyses based on subsets of 25984 and 20334 entries in the Cambridge Structural Database (CSD) that contain atomic coordinate e.s.d.'s. 95% of coordinate e.s.d.'s for C atoms can be estimated to within 0.005 Å of their published value and 78% to within 0.0025 Å. These predicted \sigma \bar values provide useful estimates of precision for those 39000 structures for which coordinate e.s.d.'s are not available in the CSD. Details of the diffraction experiment, which might provide an improved estimating function in Cruickshank's (1960) treatment, are not available in any CSD entries. However, values of Nr (the number of reflections) and Np (the number of parameters) used in refinement were added manually for 817 entries, and the variation of \sigma \bar(C–C) with decreasing Nr/Np ratios is examined: there is a rapid increase in \sigma \bar(C–C) as Nr/Np decreases below ca 6.0. A method for approximating s \bar, the r.m.s. reciprocal radius for the reflections observed, is presented, but it is found that a function of the form \sigma \bar(C–C) = kRNc1/2/s \bar(Nr − Np)1/2 [directly analogous to Cruickshank's (1960) equation] had only slightly improved predictive ability for this data set by comparison with functions based upon R and Nc1/2 alone. Possible reasons for this apparent anomaly are discussed.

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