684 FLACK AND BERNARDINELII tering (apart perhaps from the commonly-used names ship allo dkanprionstiteofthestendhduncetainy An important tool in understan cattering ding ing efects nt出 Bijvoet ratio and t s has ning of expen ndard uncertainty on d e Flack pa very rece ently tructures.Unfortuna as this review goes to press.we ric crystal structure with a centro mmetric substructur nding to our limiting values of ratio give eet application availab vith the Least-Squares Refinement the e ntal cor Early results" 7and subsequent experi alues for ator its final v quares refin ment.How if it now some of the prin- physical model of a stal structur the values of which insights tha it i tial n all par The Bijvoet ratio ers be eously.If thi ges ed cene zero ms are mmetrically )the value of the Flack paramete all atoms the criterion and its standard uncertainty eoptinizatiot rectly netr h i matedmost斤 stal structre of elemental se in the form of a helix. aetesvead Rather in an imporant aspect of ntr (heavy)che To oms of one nical tdiminish her ur sity ratio an oh con cal techni fen applie autom manually.A the Bii t rat sses of Bragg reflecti ns ha particul ters stay close to their starting or targe values with stand (wrong)para neter estimates and underestimated standard uncertainties are the result on average Calculation of the Bijvoet ratio at differen lengths enables an optimal cho ce length to be made of the crys o be s than 0.5. ral thi is obta ned by small for absolt de nto =r vmme len e may en for abs aving a highe rati As has been showr here this simple change of coordinates is centros tric arr hi ent in the crystal group belongin nate of the centro ym 2.the they ace groups Chirality DOI 10.1002/chir tering (apart perhaps from the commonly-used names anomalous dispersion and anomalous scattering). An important tool11 in understanding resonant-scatter￾ing effects in XRD and of use in the planning of experi￾mentation and the evaluation of results has been provided very recently in an analytical expression for the mean￾square Friedel intensity difference for a noncentrosymmet￾ric crystal structure with a centrosymmetric substructure. A related Bijvoet intensity ratio v gives a measure of Frie￾del differences relative to the average intensity of Friedel opposites. A spreadsheet application available with the publication11 undertakes the necessary calculations from the elemental composition of the compound for some com￾mon X-ray wavelengths. Values of 104 v called Friedif11 are calculated both for the case of all atoms arranged non￾centrosymmetrically and also allowing for atoms arranged on a centrosymmetric substructure if it is possible to iden￾tify these. We now rapidly pass in review some of the prin￾cipal insights that this work has provided: • The Bijvoet ratio is largest when all atoms are arranged noncentrosymmetrically and zero when all atoms are arranged centrosymmetrically. • The Bijvoet ratio is zero when all atoms are of the same chemical element regardless of whether the structure is noncentrosymmetric or centrosymmetric. Such is the case, in the spherical atom approximation, for the chiral crystal structure of elemental Se in the form of a helix.12 • The Bijvoet ratio quantifies a contrast and needs both resonant and nonresonant atoms to attain large values. • Rather surprisingly the presence in an otherwise non￾centrosymmetric structure of a centrosymmetric arrangement of resonant atoms of one (heavy) chemical element does not diminish the value of the Bijvoet inten￾sity ratio, an observation which had already been con- firmed experimentally.13,14 • The analytical form of the Bijvoet ratio shows that there are no classes of Bragg reflections having particularly large or small values. Consequently, in the absence of a model of the crystal structure, no particular reflections nor any specific regions of reciprocal space on average are established as showing large Friedel differences. Calculation of the Bijvoet ratio at different wavelengths enables an optimal choice of X-ray wavelength to be made prior to experimentation. Further it allows the molecular composition of the crystal to be optimized. Suppose for example that a compound is found to have a Bijvoet ratio that is too small for absolute-configuration determination. One may envisage the synthesis of a suitable derivative or the fabrication of a solvate or cocrystal of the compound having a higher Bijvoet ratio. As has been shown above it is unimportant if the solvent or cocrystal molecule takes an essentially centrosymmetric arrangement in the crystal as this does not tend to diminish the Bijvoet ratio. More￾over we have found,14 using an approximate form of the Bijvoet ratio15 and a small set of pseudocentrosymmetric structures, a relationship between the Bijvoet ratio and the standard uncertainty on the Flack parameter. This relation￾ship allows a priori estimates of the standard uncertainty of the Flack parameter. Work is currently in progress to establish the corresponding relationship between the full Bijvoet ratio and the standard uncertainty on the Flack pa￾rameter for a much larger set of non-pseudosymmetric structures. Unfortunately as this review goes to press, we have not yet completed the data analysis to determine the values of Friedif corresponding to our limiting values of u of 0.04 and 0.10, and to investigate in more detail the influ￾ence of pseudosymmetry. Least-Squares Refinement Early results16,17 and subsequent experience from many crystal-structure determinations have shown that the Flack parameter is robust and converges in only a few cycles to its final value during least-squares refinement. However, as the Flack parameter is one of many parameters of the physical model of a crystal structure, the values of which are to be found by optimization based on some general cri￾terion, it is essential in the final cycles of optimization that all parameters be varied jointly and simultaneously. If this prescription is not followed, two effects may occur sepa￾rately or together: (a) the value of the Flack parameter may not correspond to the best value for the optimization criterion and (b) its standard uncertainty may be incor￾rectly estimated, most frequently underestimated. In the case of least-squares minimization, the final refinement needs to be undertaken by full-matrix least-squares (all pa￾rameters varied jointly and simultaneously) and needs to have converged. Another important aspect of least-squares minimization which needs some words of explanation is that of stabiliza￾tion and damping.18 To avoid a least-squares refinement becoming unstable and failing to converge, certain numeri￾cal techniques, grouped together under the general term damping, are often applied automatically or manually. A side effect of these techniques is that stabilized parame￾ters stay close to their starting or target values with stand￾ard uncertainties that are systematically underestimated. Biased (wrong) parameter estimates and underestimated standard uncertainties are the result. Inverting a Model Structure It sometimes happens that a model crystal structure yields a value of the Flack parameter larger than 0.5. To represent the majority component in the crystal, the model needs to be inverted so the Flack parameter takes a value less than 0.5. In general this inversion is obtained by inver￾sion in the origin by just changing all atomic coordinates x, y, z into 2x, 2y, 2z or some point symmetry-equivalent to it. However, for the chiral crystal structures which are necessary for absolute-configuration determination, there are some cases where this simple change of coordinates is insufficient or inappropriate. So in the case of a space group belonging to one of the 11 pairs of enantiomorphic space groups (P41–P43; P4122–P4322; P41212–P43212; P31– P32; P3121–P3221; P3112–P3212; P61–P65; P62–P64; P6122– P6522; P6222–P6422; P4132–P4332), the space group should also be changed into the other member of the pair. As they occur in enantiomorphic pairs, these 22 space groups 684 FLACK AND BERNARDINELLI Chirality DOI 10.1002/chir