Least-squares absolute-structure refinement. Practical experience and ancillary calculations

The least-squares refinement of non-centrosymmetric crystal structures as inversion twins is presented. It is shown that the absolute-structure (twin) parameter x may be used to define the chirality or polarity of untwinned crystals. The method has been applied to 20 compounds. The least-squares refinement of the absolute-structure parameter is rapid and stable. The value of x generally falls within three e.s.d.'s of the physically meaningful range 0 ≤ x ≤ 1 and the e.s.d.'s increase as f” becomes smaller. New residual and goodness-of-fit values are defined to judge the efficiency of the method. The estimated standard deviation of x, taken with a pseudo Durbin-Watson d statistic, provides an excellent criterion for the reliability of the absolute-structure determination. Refinements on data sets including very accurately measured Friedel pairs of reflections have also been tested. The determination of the free direction(s) of origin-free space groups and an efficient algorithm for the inversion of a crystal structure that refines to x ≃ 1 are given in detail. The data and procedural structures necessary for an efficient computer implementation of absolute-structure refinement are also considered. The formulae giving the correction for the effects of anomalous dispersion on |F_obs| from an inversion-twinned crystal are given. These corrected |F_obs| are the ones to be used in an electron-density calculation. The correlation of residuals following least-squares refinement is quantified by using a pseudo Durbin-Watson d statistic. The causes of the correlation, its effect on the value of x and its e.s.d., and ways of avoiding the correlation are considered. It is shown that in using x it is more suitable to refine on |F|² than |F|. A weighting scheme is presented and tested that increases the sensitivity of a refinement to absolute structure.