A SYSTEMATIC STUDY OF COORDINATE PRECISION IN X-RAY STRUCTURE ANALYSES .1. DESCRIPTIVE STATISTICS AND PREDICTIVE ESTIMATES OF ESDS FOR C ATOMS

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 ($) over bar sigma(C-C), the mean e.s.d. of a C-C bond length in a structure, or ($) over bar sigma(C), the mean isotropic e.s.d. of a C atom, can be estimated by expressions of the form ($) over bar sigma = kappa RN(c)(1/2). Here, N-c is taken as Sigma Z(i)(2)/Z(C)(2), With the atomic numbers Z(i) summed over all atoms in the asymmetric unit and Z(C) = 6. It is also shown that ($) over bar sigma(E), the mean isotropic e.s.d. of a non-C atom, can be estimated by ($) over bar sigma(E) = kappa RN(c)(1/2)/Z(E). Values of kappa were determined by regression analyses based on subsets of 25 984 and 20 334 entries in the Cambridge Structural Database (CSD) that contain atomic coordinate e.s.d.'s. 95% of coordinate e.s.d.'s for C atoms can be estimated to within 0.005 Angstrom of their published value and 78% to within 0.0025 Angstrom. These predicted ($) over bar sigma values provide useful estimates of precision for those 39 000 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 N-r (the number of reflections) and N-p (the number of parameters) used in refinement were added manually for 817 entries, and the variation of ($) over bar sigma(C-C) with decreasing N-r/N-p ratios is examined: there is a rapid increase in ($) over bar sigma(C-C) as N-r/N-p decreases below circa 6.0. A method for approximating ($) over bar s, the r.m.s. reciprocal radius for the reflections observed, is presented, but it is found that a function of the form ($) over bar sigma(C-C) = kappa RN(c)(1/2)/($) over bar s(N-r-N-p)(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 N-c(1/2) alone. Possible reasons for this apparent anomaly are discussed.