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

Abstract

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 IFobsI from an inversiontwinned crystal are given. These corrected Fobs 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 DurbinWatson 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 IFI 2 than IF[. A weighting scheme is presented and tested that increases the sensitivity of a refinement to absolute structure.0108-7673/85/050500-12501.50

Synthesis and Structure of Ba12F19Cl5

“…For comparison a reflection at a similar angle and with similar intensity (hkl = 421, 2H = 28.4°, F obs = 9.1(.9)) was clearly identified experimentally. In the noncentrosymmetric space group P6 the calculated absolute structure factor [11] did not deviate from 0.5 for all refinements. For all these reasons we chose the highest symmetry space group, P6 3 /m, for the disordered variant of Ba 7 F 12 Cl 2 .…”

Structure of RNi3Al9 (R = Y, Gd, Dy, Er) with either ordered or partly disordered arrangement of Al-atom triangles and rare-earth-metal atoms

“…The values obtained for the enantiomorph-polarity parameter x (Bernardinelli & Flack, 1985), see Table 1, show that for each of the four crystals, it was clearly possible to distinguish between the coordinate set of Table 2 and the corresponding inverted set. The crystals of GdNi3AI9 and YNi3AI9 have the same arrangement as (x = 0), and the crystals of ErNi3A19 and DyNi3A19 have the inverted arrangement to (x = 1), the coordinates of Table 2.…”

Mono- and bimetallic lanthanide(III) phenolic cryptates obtained by template reaction: solid state structure, photophysical properties and relaxivity †

“…39 For 7, the absolute configuration has been established by refinement of the enantiomorph polarity parameter [x = Ϫ0.011 (14)]. 40 The structure of 7 presents a slight disorder on an oxygen of one co-ordinated nitrate and another of the ionic nitrate. These disorders have been resolved and two atomic sites have been observed and refined with isotropic atomic displacement parameters in each case.…”