682 FLACK AND BERNARDINELII In a prese view provides und ential growth in a peok like those of a singlengacnice illustrative examp resent the model crysta spects which are e ca sace group.and crystal XRD to dete mine absc onfiguration are point.T macros opi ay be represented as C a le tant topic ofch the mole f ions d and of the two and ind s no go into inin the。3bot s as we have re ntly cont uted detailed inf tior nt in the stal in the rtion 70%of whon de nain are. but du pure nor race ates(scalemates). ay l a little cally und in the correct absolut a value of the Fl that the Flack a no cent a crystal struct configuration of the chiral molecules forming the crystal. e co the crystal by making the im Wha arra ion iso of the growpsorcentersl are reproduced in the glos These sary to this eaction an ena It is iois a chemist's tem and refers to chiral molecu that tal structure versus mole ation detern and the sy purity of the re ugh a int and rotoin are ent.Both terms concem the com hat chemical reaction the sample under investigation by some other physi SINGLE-CRYS which Flack Para by single-crysta XRD of of th n the phe non of ng (se less than 0.04 but this 0.10f sured by the Flack 2The ndly the value of the Flack meter its model u rlying the K par <0.040 oeing real phenomena ven enantiome of100% and /< h The above eneous an h of x(w)al hat o ersion of picturi structure determina where none An unfor Chirality DOI 10.1002/chir In a presentation of the way that absolute-configuration determination is undertaken by single-crystal XRD, the review provides essential background information and highlights those aspects which are the cause of confusion and error. Techniques to improve the capacity of single￾crystal XRD to determine absolute configuration are reviewed, along with a few examples of other more com￾plex cases. The all-important topic of characterization of bulk and individual single crystals is treated and the con￾cluding remarks contain comments on some current tech￾nical limitations. The current review does not go into any detail concerning the phase diagrams of enantiomeric mix￾tures as we have recently contributed detailed information on the absolute-configuration determination from binary enantiomeric mixtures1 which are neither enantiomerically pure nor racemates (scalemates). SINGLE-CRYSTAL XRD TECHNIQUES USING AN INTERNAL CHIRAL REFERENCE The presence in a crystal structure of enantiomerically pure chiral molecules, groups, or chiral centers of known absolute configuration leads directly to the determination of the absolute configuration of the other constituents of the crystal by making the image of the atomic arrange￾ment correspond to that of the chiral molecules whose absolute configuration is known. The chiral molecules (or groups or centers) thus act as an internal reference. These may be introduced as part of the compound by chemical reaction or as part of the crystal by cocrystallization using an enantiomerically pure sample of the reference sub￾stance. It is important to stress that the correctness of absolute-configuration determination using an internal chi￾ral reference depends crucially on the knowledge of the enantiomeric purity of the reference material and its indi￾cated absolute configuration. It is not sufficient to assume that chemical reaction, crystallization, or operations of mechanochemistry (i.e., grinding) will necessarily con￾serve the chirality of the reference material. SINGLE-CRYSTAL XRD TECHNIQUES EXPLOITING RESONANT SCATTERING The Flack Parameter The distinction by single-crystal XRD of inversion￾related models of a noncentrosymmetric crystal structure relies on the phenomenon of resonant scattering (see sec￾tion Resonant scattering and its effect on the diffraction inten￾sities) and is measured by the Flack parameter.2 The phys￾ical model underlying the Flack parameter is that of a crys￾tal twinned by inversion and composed of distinguishable domains, all of these being real phenomena well estab￾lished in the fields of mineralogy, crystal growth, crystal physics, and solid-state physics.3 The macroscopic crystal is formed of two types of homogeneous and perfectly-ori￾ented domains, the relationship between the two domain types being that of inversion. A simple way of picturing the crystal twinned by inversion is to imagine a racemic conglomerate in which the crystals have stuck together at growth in a perfectly oriented manner giving diffraction patterns that look like those of a single crystal. For a nice illustrative example see.4 Let X represent the model crystal structure as given by its cell dimensions, space group, and atomic coordinates and X its image inverted through a point. The macroscopic crystal may be represented as C 5 (1 2 x) X 1 x X for which the Flack parameter x measures respectively the mole fractions (12x) and x of the two types of domain X and X. When x 5 0, there is only one domain in the crystal which is that of the model X. When x 5 1, there is only one domain in the crystal which is that of the inverted model X. When x 5 0.3 both types of do￾main are present in the crystal in the proportion 70% of X to 30% of X. The physically meaningful values of x are 0
x
1, but due to statistical fluctuations and systematic errors, experimental values may lie a little outside of this range by a few standard uncertainties. A crystal of an enantiomerically pure compound in the correct absolute configuration has a value of the Flack parameter of zero. In crystallographic jargon one says that the Flack parame￾ter measures the absolute structure of a noncentrosym￾metric crystal and from this one may deduce the absolute configuration of the chiral molecules forming the crystal. What are Absolute Structure and Absolute Configuration? For convenience, the formal definitions of these quanti￾ties are reproduced5 in the glossary to this review. Abso￾lute structure is a crystallographer’s term and applies to noncentrosymmetric crystal structures. Absolute configu￾ration is a chemist’s term and refers to chiral molecules. Note particularly that both the entity under consideration, viz. crystal structure versus molecule, and the symmetry restrictions, viz. noncentrosymmetric versus lack of mirror reflection, inversion through a point, and rotoinversions, are different. Both terms concern the complete specifica￾tion of the spatial arrangement of atoms with respect to inversion and require that the sample under investigation be characterized by some other physical measurement. Absolute-Structure Determination There are conditions under which one may say that the absolute structure of the crystal has been determined sat￾isfactorily.6 Firstly one wants to know whether the abso￾lute-structure determination is sufficiently precise by look￾ing to see whether the standard uncertainty u of the Flack parameter x(u) is sufficiently small: in general u should be less than 0.04 but this value may be relaxed to 0.10 for a compound proven by other means to be enantiomerically pure. Secondly the value of the Flack parameter itself should be close to zero within a region of three standard uncertainties i.e. u < 0.04 (or u < 0.10 for a chemically proven enantiomeric excess of 100%) and |x|/u < 3.0. Moreover the crystal and bulk need to be characterized. The above criteria have been established by way of statisti￾cal reasoning6 to ensure that the structure analyst, by an examination of x(u) alone, does not claim an absolute￾structure determination where none is valid. An unfortu￾nate consequence of such a conservative or safe approach is that some borderline but valid absolute-structure deter- 682 FLACK AND BERNARDINELLI Chirality DOI 10.1002/chir