《有机化学》课程教学资源（文献资料）The use of X-ray crystallography to determine absolute configuration

interScience CHRAITY-00 Review Article The Use of X-ray Crystallography to Determine Absolute Configuration ABSTRACT Estal backgroud the deterination fc lute configuratior n are defined and olute-str ant scattering dand the insights obtained fro of a Bijvoet intensity ratio op s,XRD intens sity me ment oftware of the comp on to and econfiguration determination using combined XRD and CD measur mination from light-atom structures.Chirality 20:681-690.20082007 Wiley-Liss.Ine. KEY WORDS:absolute structure;crystal structure:resonant scattering INTRODUCTION ten for the person who has sufficient knowledge of x-ray X-ray diffraction (RD)of single crystals has the mination are the following questions: the model cr he eometry,bond dis crystal stn ave be s a fine Absolute- grntiot nation,which dep nds on being able small dif raction intens represent the enceisnot guaran hae mall i ulk and山 icient measured crystal been su experimentalist to the material studied to claim that an absolute com try,atomic positions,interatomic cement parh 2007 Wiley-Liss.Ine

Review Article The Use of X-ray Crystallography to Determine Absolute Configuration H. D. FLACK* AND G. BERNARDINELLI Laboratoire de Cristallographie, University of Geneva, Switzerland ABSTRACT Essential background on the determination of absolute configuration by way of single-crystal X-ray diffraction (XRD) is presented. The use and limitations of an internal chiral reference are described. The physical model underlying the Flack pa￾rameter is explained. Absolute structure and absolute configuration are defined and their similarities and differences are highlighted. The necessary conditions on the Flack parameter for satisfactory absolute-structure determination are detailed. The symmetry and purity conditions for absolute-configuration determination are discussed. The phys￾ical basis of resonant scattering is briefly presented and the insights obtained from a complete derivation of a Bijvoet intensity ratio by way of the mean-square Friedel differ￾ence are exposed. The requirements on least-squares refinement are emphasized. The topics of right-handed axes, XRD intensity measurement, software, crystal-structure evaluation, errors in crystal structures, and compatibility of data in their relation to abso￾lute-configuration determination are described. Characterization of the compounds and crystals by the physicochemical measurement of optical rotation, CD spectra, and enan￾tioselective chromatography are presented. Some simple and some complex examples of absolute-configuration determination using combined XRD and CD measurements, using XRD and enantioselective chromatography, and in multiply-twinned crystals clar￾ify the technique. The review concludes with comments on absolute-configuration deter￾mination from light-atom structures. Chirality 20:681–690, 2008. VC 2007 Wiley-Liss, Inc. KEY WORDS: absolute structure; crystal structure; resonant scattering INTRODUCTION X-ray diffraction (XRD) of single crystals has the capacity to distinguish between the enantiomorphs of a chiral crystal structure and the enantiomers of a chiral molecule. The technique may be applied to compounds of a vast range of chemical composition. Essential chemical information such as the molecular geometry, bond dis￾tances and angles, and the packing of the molecules in the crystal are part and parcel of the results of the analysis. However there are limitations. Absolute-configuration determination is a fine detail of crystal-structure determi￾nation, which depends on being able to identify small dif￾fraction intensity differences between two crystal-structure models of opposite chirality. With compounds containing only light atoms a significant difference is not guaranteed. The physical reason that these differences are small is described in the section Resonant scattering and its effect on the diffraction intensities. Clearly it makes no sense to claim that an absolute con- figuration has been determined unless the gross features of the structure and its determination, such as intensity measurements, symmetry, atomic positions, interatomic distances, and atomic displacement parameters have been evaluated and shown not to be in error. This review is writ￾ten for the person who has sufficient knowledge of X-ray crystallography to accomplish this essential step. Of particular relevance in absolute-configuration deter￾mination are the following questions: • Does the model crystal structure properly represent the crystal structure inside the crystal(s) that have been measured? Is the crystal structure chiral? Is the model that of the real crystal structure and not its enantio￾morph? Is the compound enantiomerically pure? Is the assumed space-group symmetry neither too low nor too high? • Does the model crystal structure properly represent the bulk product from which the crystal was grown? • Have the bulk and the measured single crystal been suf- ficiently characterized or fingerprinted to enable another experimentalist to correctly identify the material studied? *Correspondence to: H. D. Flack, Laboratoire de Cristallographie, 24 quai Ernest-Ansermet, CH-1211 Gene`ve 4, Switzerland. E-mail: howard.flack@unige.ch Received for publication 14 May 2007; Accepted 16 July 2007 DOI: 10.1002/chir.20473 Published online 8 October 2007 in Wiley InterScience (www.interscience.wiley.com). CHIRALITY 20:681–690 (2008) VC 2007 Wiley-Liss, Inc

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

USE OF X-RAY CRYSTALLOGRAPHY 683 minations are deemed to be unacd Some of the t partic o h ymetn onta al molecul com ounds onjunction with othe need and expert this needs exceedingly careful.individual and tion dete ation from a bull evaluation. diaster Absolute-Configu of the diaster som her in an ordered or diso show the c on of ome of the el thing can be said about the a afaonigurationofis st the dia hohtestnictereS neces ad to e sitionsansing de t it ha to h ce group bac und h trosymmetric symmetr cryst st In fac structures exist and the absol Space-group restriction simples restnction i their ions of the d kind rotoinver above.As w cularly pure(chiral) molecules,or of chi al mol that th for o th which centr he first ecules are ac and in ch unds are found to crystallize exclu gro sively in c rations of t rst nd pureo ge crystal classes:1.2.222. d432 Chiral molec es own right and lef and it e in the ttern of the tions hereas their are identica rat anion stablis d but the The sym etry group of achira or chirality distinctio ens fo thes may or may not be part of the spa ing he re ions in the Fra nhofe ction nent of a candidate molecule for non graphi of the 1 will be proportion c f thes ee ctrons bohaeconieuraioncanmotbediemimed.Fo (Friedel opns plane is part of the metry or not. hich are very i experime Solid-sta more or no less than the response of a forced damp nature.There is nothing anomalous about resonant sca Chirality DOI 10.1002/chi

minations are deemed to be unacceptable. Some of the above criteria may be relaxed somewhat by taking account of a broader spectrum of knowledge over a class of compounds in conjunction with other nondiffraction data or accumulated knowledge of a particular instrument but this needs exceedingly careful, individual and expert evaluation. Absolute-Configuration Determination Once the absolute structure has been determined satis￾factorily, it is only then the moment to see whether some￾thing can be said about the absolute configuration of its constituent molecules, as not all valid determinations of absolute structure can necessarily lead to the assignment of an absolute configuration. Although the following description of restrictions7 is self-sufficient, it has to be admitted that more background knowledge on chiral and achiral crystal structures5 helps in its understanding. The weakest and most easily-applicable restriction is given first and the strongest one is given last. In fact the third restric￾tion is sufficient in itself, the other two not so. Space-group restriction. The simplest restriction is one of space-group symmetry. If the space group contains symmetry operations of the second kind (i.e., rotoinver￾sions or rotoreflections, glide reflections), it must occur that these operate either intramolecularly, forcing the indi￾vidual molecules to be achiral, or intermolecularly, forcing an arrangement of pairs of opposite enantiomers. Thus, in the first case, the molecules are achiral and in the second a racemate is present. Consequently it is only in crystals displaying space groups containing exclusively symmetry operations of the first kind (i.e., pure or proper rotations, screw rotations) that the determination of absolute config￾uration is possible (geometric crystal classes: 1, 2, 222, 4, 422, 3, 32, 6, 622, 23, and 432). Chiral molecular entity restriction. To comply with the definition of absolute configuration,5 one needs to identify a chiral molecular entity and its spatial arrange￾ment in the crystal structure. For example, in a couple of alkali tartrate salts,8,9 the absolute configuration of the tar￾trate anion (a chiral molecule) was established but cor￾rectly no claims to have done so for the sodium or rubid￾ium atoms were made as these are achiral cations and not molecules. The symmetry group of an achiral molecule contains rotoinversion or rotoreflection operations, and these may or may not be part of the space-group symme￾try of the crystal. So one must always examine the spatial arrangement of a candidate molecule for noncrystallo￾graphic rotoinversion or rotoreflection symmetry opera￾tions and if any are found, the molecule is achiral and its absolute configuration cannot be determined. For example any planar molecule has mirror symmetry and is achiral whether the mirror plane is part of the space-group sym￾metry or not. Solid-state enantiomeric purity restriction. One needs to verify that all occurrences of the chiral molecular entity in the crystal structure are the same enantiomer for an absolute-configuration assignment to be valid. This is of particular concern when the asymmetric unit contains more than one occurrence of the chiral molecule (Z 0 > 1). Some of the above criteria may be relaxed, but such studies need exceedingly careful, individual and expert evaluation as described in the section Absolute-configura￾tion determination from a bulk racemate by combined CD and XRD. 10 Bulk samples which are mixtures of diaster￾eoisomers may give rise to crystals which contain several of the diastereoisomers either in an ordered or disordered arrangement. The crystal structure may clearly and dis￾tinctly show the configuration of some of the elements of chirality common to all of the diastereoisomers whereas those elements which vary amongst the diastereoisomers may be in doubt due to disordered atomic positions arising from the superposition of more than one diastereoisomer. The space-group restriction mentioned earlier implies that absolute configuration may only be undertaken from chiral crystal structures. The latter are necessarily noncen￾trosymmetric, but not all noncentrosymmetric crystal structures are chiral. Achiral noncentrosymmetric crystal structures exist and the absolute structure of their crystals may be determined. However, it is not possible to deduce the absolute configuration of their constituent molecules for the reasons given above. As we have explained previ￾ously,5 chiral crystal structures are found formed either of enantiomerically pure (chiral) molecules, or of chiral mole￾cules as a racemate, or of achiral molecules. Achiral crys￾tal structures, which may be either centrosymmetric or noncentrosymmetric, are found formed either of chiral molecules as a racemate or of achiral molecules. Enantio￾merically pure compounds are found to crystallize exclu￾sively in chiral crystal structures. Resonant Scattering and Its Effect on the Diffraction Intensities Optical systems working with visible light distinguish objects of opposite chirality without difficulty i.e., one can easily see the difference between ones own right and left hand. The major difference in the diffraction pattern of the right and left hand occurs in the phases of the inversion￾related reflections whereas their amplitudes are identical in the absence of any resonant effects. Differences in intensities of the latter occur only if a resonant frequency of the diffracting object is near to that of the incident elec￾tromagnetic radiation. Therein lies the essential difficulty for chirality distinction using X-rays. As no lens exists for focusing X-rays, one has to rely only on the intensity of the reflections in the Fraunhofer diffraction pattern. More￾over, if the frequency of the incident X-rays is close to that of some of the atomic electrons which cause diffraction, it will be only a small proportion of these electrons which are resonant, and the intensity difference between inver￾sion-related reflections (Friedel opposites in crystallo￾graphic jargon) is small. A further complication is that the resonant frequencies of light atoms occur at long X-ray wavelengths which are very difficult to access experimen￾tally. It helps to remember that resonant scattering is no more or no less than the response of a forced damped har￾monic oscillator of which there are numerous examples in nature. There is nothing anomalous about resonant scat￾USE OF X-RAY CRYSTALLOGRAPHY 683 Chirality DOI 10.1002/chir

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

USE OF X-RAY CRYSTALLOGRAPHY 685 are the only ones that are correctly described as being chi- simulated from the results of a single-crystal study allows the presence in the bulk of ne dia in erted point other tha n the ing racemic omerates.i the columns sih the tables any metho of these isomers sh single diffraction studies CHARACTERIZATION OF COMPOUNDS AND CRYSTALS EXPERIMENTATION AND ANALYSIS NEEDS The phase diagrams of enantiomeric mixtures can be IMPECCABLE TECHNIQUE Right-Handed Axe roe in It is for these rea that fo As emphasized and discussed previously. ric pu rity not only of analysis of absolute structure.Of particular danger for the tion o mmended Ther methods of characterization: any ba ran mation matrix mus OR:The sne ivity ive determin As the measureme n is a single-w e OR not ng the number of pos and negative ogram on the di must ha a positive me the after a hardware or soft of ac CD:The al st 他a nt of t scatter In fav orable cir Cture ution t give a flack pa v clos nents taken into so nfiguratio of the acid he the bull compound and p XRD Intensity Measurements pow red singl It has b man,to be hich ingly in the (EC sen estimates of the neri are result It hen imental con ns that the two ure both sepa members of each Friedel ay o nd is At the data age it is essential y (DSC)by vnthetic chemists and ructure ana stal r semi ing a pha se diagram.DSC measurements may be applied In pas ing,we also mention that powder diffraction can any Bra Chirality DOI 10.1002/chi

are the only ones that are correctly described as being chi￾ral.5 Moreover, there are cases where for the standard set￾ting of the space group19 that the coordinates need to be inverted in some point other than the origin. The coordi￾nates of the appropriate inversion point can be found in the columns Inversion through a centre at of the tables of Euclidean normalizers of space groups.20 For nonstandard settings an algorithmic solution to this problem has been provided.16 CHARACTERIZATION OF COMPOUNDS AND CRYSTALS The phase diagrams of enantiomeric mixtures can be complicated,21,22 giving rise to solid and liquid phases of different composition. Also kinetic effects play an impor￾tant role in crystallization. It is for these reasons that for absolute-configuration determination, some characteriza￾tion or measurement of the enantiomeric purity not only of the bulk but also of the single crystal used for the diffrac￾tion studies is recommended. There are three principal methods of characterization: OR: The specific rotation of the optical activity in solution. As the measurement of specific rotation is a single-wave￾length technique, the presence and effect of impurities can easily go undetected. Moreover, OR can not be applied to microgram quantities (i.e., a single crystal used for diffraction studies). Also OR can only provide a measure of enantiomeric excess if the specific rotation of the enantiomerically pure compound is sufficiently strong and has been determined previously. CD: The visible and near-UV circular dichroism spectrum in solution. The presence and effect of impurities may be readily recognized in a CD spectrum. In favorable cir￾cumstances, CD may be applied to the single crystal used for the diffraction measurements taken into solu￾tion. For compounds that racemize rapidly in solution, solid-state CD in a KBr disk may be applied to the bulk compound and perhaps even to a powdered single crys￾tal.23–25 One may expect vibrational CD, either IR or Raman, to be used increasingly in the future. Enantioselective chromatography (EC): This sensitive technique is applicable to microgram quantities and pro￾vides estimates of the enantiomeric excess. It is of course necessary to establish that under the chosen ex￾perimental conditions that the two enantiomers are clearly separated. The retention times provide a satisfac￾tory characterization of the two enantiomers. Regrettably little use is made of differential scanning cal￾orimetry (DSC) by synthetic chemists and structure ana￾lysts. Nevertheless, the measurement of melting tempera￾tures and enthalpies is a valuable technique for establish￾ing a phase diagram. DSC measurements may be applied to the bulk. In passing, we also mention that powder diffraction can be useful. The simple expedient of comparing the X-ray powder diffraction pattern of the bulk product with that simulated from the results of a single-crystal study allows the presence in the bulk of polymorphs and crystalline dia￾stereoisomers to be revealed. Clearly although this tech￾nique is of no help for detecting racemic conglomerates, it is very helpful for other solid mixtures.26 As the presence of diastereoisomers has the capability of invalidating the determination of absolute configuration, any method which establishes the number and relative concentration of these isomers should be used to characterize the bulk compound and if possible the single crystal used for the diffraction studies. EXPERIMENTATION AND ANALYSIS NEEDS IMPECCABLE TECHNIQUE Right-Handed Axes As emphasized and discussed previously,27 right￾handed sets of axes must be used at every stage of an analysis of absolute structure. Of particular danger for the structure analyst are basis transformations performed to bring the unit cell into a standard setting. To maintain right-handed axes, any basis transformation matrix must have a positive determinant. A transformation matrix with a negative determinant will transform a right-handed set of axes into a left-handed set of axes, and conversely. The sign of the determinant cannot be spotted simply by count￾ing the number of positive and negative elements in the transformation matrix. The orientation matrix (UB) of the crystal on the diffractometer must have a positive determi￾nant. It is standard practice in our laboratory to calibrate every diffractometer after a hardware or software modifica￾tion with a well-defined reference material of a chiral crys￾tal structure and containing a sufficient amount of reso￾nant scattering. We use enantiomerically pure potassium hydrogen (2R, 3R) tartrate. With such a test material, structure solution must give a Flack parameter very close to zero for the (2R, 3R) configuration of the acid tartrate anion. XRD Intensity Measurements It has been established,13,14 under certain particular con￾ditions, which it is not necessary to detail here, that unless intensity measurements of both members, hkl and h k l of each pair of Friedel opposites are made and used sepa￾rately in the least-squares refinement, false values of the Flack parameter may result. It hence seems prudent, whether these particular conditions apply or not, to always measure both members of each Friedel pair. Fortunately, with modern-day equipment using area detectors this cri￾terion is easy to achieve and is often the default mode of operation. At the data-reduction stage it is essential for absolute-configuration determination not to average the intensities of Friedel opposites,6 to transform reflection indices only according to the symmetry operations of the crystal point group6 and not to use any semiempirical absorption correction which applies a different correction to the intensities of hkl and h k l. 1 In writing about pairs of Friedel opposites in this review, h k l should be taken in a general sense to mean h k l or any Bragg reflection symmetry-equivalent to it under the USE OF X-RAY CRYSTALLOGRAPHY 685 Chirality DOI 10.1002/chir

686 FLACK AND BERNARDINELII symmetric the t ere ol el the ing into this trap.Ofeqal rel equivale nt to it,has b achiral crystal structure of a racemate (often disordered) lent to it has not been measured then the Friedel coverage has err is 0%. Getting the best out of you ent of al in t cycl un re ing an abs de n is h uses by default a spars rom an qu PLATON mic coordinates and cell pa ter.SHELXL pace group of too low syr has been chosen at it may be poss to add refin to th to invert the structur tion .sho ould be rejec if the evid mestandard uncertaintyis e in stalline state the same value of x(). of the real On The o Framework mated ourrecent study f 135 publsne ing the data me nts and the In general.for these incom mination.Syst s can be d ade n on close to an to.contrad and s in the in. ues of the Flack parameter close to 0.5.with a low stand Vith the aid ard saoehohnedtb nay be completed and justified.An W dete minations that either the are hac opp ha s orded to liminate intensity dif operated by the One should ear in mind that SOME EXAMPLES TO STRETCH ONE'S roneous.Ar with which or should be familia Experimental values of the flack parameter has been toa noncentros mme We owe a debt of gratitude to the referees of this article space group w he crys ng t of theg the inte crys e,the conditions for ly yet aga that eter.is to scrutinize the output of a system such as check Chirality DOI 10.1002/chir

point group of the crystal. For a noncentrosymmetric crys￾tal hkl and h k l are not symmetry-equivalent under the point group of the crystal. Later on in this review the term Friedel coverage is also used. If in the intensity data, for each reflection hkl the reflection h k l, or one symmetry￾equivalent to it, has been measured then the Friedel cover￾age is 100%. If for each hkl, h k l or one symmetry-equiva￾lent to it has not been measured then the Friedel coverage is 0%. Getting the Best Out of Your Software In the section Least-squares refinement above it was pointed out that full-matrix simultaneous refinement of all variables should be used in the final cycles to obtain reli￾able results. One widely-used least-squares refinement programme, SHELXL93/97,28 uses by default a sparse-ma￾trix technique,6 called hole-in-one, for the refinement of the Flack parameter. SHELXL may nevertheless be coaxed into doing the appropriate full-matrix calculation.13,29 One may make one or two simple checks of any refine￾ment software. The first is to invert the structure and check that a value of the Flack parameter of 1 2 x with the same standard uncertainty is obtained. A second check is to undertake the refinement using a different starting value of the Flack parameter which should lead to exactly the same value of x(u). Crystal-Structure Evaluation The results of a crystal-structure determination are transmitted nowadays by means of computer-readable files viz. a Crystallographic Information Framework30 (CIF) file which may be used for display, analysis, evaluation, and archiving. A great deal may be achieved by the automated evaluation of the information contained in a CIF file con￾cerning the data measurements and the crystal-structure determination. Systems can be designed to make use of a considerable amount of general and specific crystallo￾graphic knowledge and know-how in the evaluation of a structure determination, and to alert the structure analyst to ambiguities, contradictions, and shortcomings in the in￾formation encapsulated in a CIF file. With the aid of these alerts, the data-measurement and structure-refinement procedures may be improved, completed and justified. An essential element is the examination of a graphical repre￾sentation of the atomic displacement parameters.31 The most elaborate system currently in operation for the evalu￾ation of crystal-structure determinations is the free-of￾charge online checkCIF/PLATON32,33 operated by the International Union of Crystallography. Erroneous Crystal Structures One should bear in mind that a structure analysis may be erroneous. An error with which one should be familiar is the one in which the symmetry of a crystal structure has been incorrectly assigned to a noncentrosymmetric space group whereas the crystal structure itself is really centrosymmetric. In an erroneous noncentrosymmetric description of a crystal structure, the conditions for abso￾lute-configuration determination may apparently be achieved which do not of course apply in the true centro￾symmetric description.13,14 The measurement of a whole sphere of reflection intensities (see section XRD intensity measurements) is a prudent approach to help avoiding fall￾ing into this trap. Of equal relevance to absolute-configura￾tion determination are those cases of analysis in which an achiral crystal structure of a racemate (often disordered) with a space group containing rotoinversion operations has erroneously been assigned to a crystal which has in fact a chiral crystal structure of an enantiomerically pure compound with a space group containing only rotation and screw rotation operations. This latter case may arise when a bulk racemate crystallizes by spontaneous resolution and the structure analyst force-feeds the structure solution with a racemate. The possibility of undertaking an abso￾lute-configuration is hence lost. From an analysis of the atomic coordinates and cell pa￾rameters, checkCIF/PLATON may provide an alert that a space group of too low symmetry has been chosen. The most common situation is that it may be possible to add a center of symmetry to the chosen space group. This prop￾osition should be rejected if there is strong evidence to show that the compound is enantiomerically pure in the crystalline state. Partial-polar ambiguities34 have the capacity to falsify an absolute-configuration determination. In a crystal-structure solution suffering from a partial-polar ambiguity, some of the atoms are correctly located but the others are images of the real atoms inverted in a point. One may be able to recognize this type of error by a study of interatomic dis￾tances and angles. Compatibility of Chemical and Crystallographic Data In our recent study1 of 135 published crystal structures of metallacycles an appalling 26% had incompatible chemi￾cal and crystallographic data. In general, for these incom￾patible crystal structures, the chemical evidence was adequate to convince us that the bulk products had a com￾position close to an enantiomeric excess of 100%. The crys￾tal structures were determined as being chiral but with val￾ues of the Flack parameter close to 0.5, with a low stand￾ard uncertainty, indicative of a crystal twinned by inversion with an overall composition near to that of the racemate, in contradiction to the chemical evidence. We hypothesized that for the incompatible crystal-structure determinations that either the data-reduction software had averaged Friedel opposites or that an empirical absorption correction procedure had tended to eliminate intensity dif￾ferences between Friedel opposites. Other considerations have also been highlighted.14 SOME EXAMPLES TO STRETCH ONE’S UNDERSTANDING Experimental Values of the Flack Parameter We owe a debt of gratitude to the referees of this article for suggesting the inclusion of this section giving the inter￾pretation of some typical values of the Flack parameter. We cannot stress sufficiently yet again that a necessary step, prior to examining and interpreting the Flack param￾eter, is to scrutinize the output of a system such as check- 686 FLACK AND BERNARDINELLI Chirality DOI 10.1002/chir

USE OF X-RAY CRYSTALLOGRAPHY 687 CIF/PLATON to ensure the overall validity of the radiation of different wavelength.synthesis of a derivative structure determination.For bsote-contion deter containing resonant-sc attering atoms.and synthesis of a ins 0.00:No stan on the Flack pa the no hic ch ical e ssess the to be plete rubbish as with this sac c)Thisis that of() .1②but ith n o ter un taint implying that b stru 9 del sh and the ref m ted is the =0.05(2).space group Pra2:Absolute-configura- in the form of the glide ort th rystal FI for the beer en the rac mate Absolute the crystal is p ed by inye sion contai Du sible in this cas and 19%0 tedoa0 en( =0052,spa and XRD.Ho ver() paran er is less tha 0.10.and ing from a partial-pola achieved. Moreo er,as the spac tains onl hic evid ity of centrosy obta ained for an th he m as impurity nation has achie he meling tempe compour tography is neces y to make the raphers curse is bad softwa oupC2.good Friedel covera tion co diatel sib ed by which For me surements on a SDO tals would ex(u) ≈0.05② and the othe 50% che CIF/PLATON that a center of s needs to b to the insure tha matography the single crystal used for the diffraction dat ement er ha -O.1②：The value0.2 of the standard uncer trosymm etric ure mmetri group.No the ap n det n are n tal in ndicat is within thre dard und ertainties o ckCIF/PLATON that a centerof nmetry needs Possi. re This is a ver mds to achieve smaller sta dard uncertainty or ments,me ement a low rate is either e are indicatio PLAT or achiral:If ther del Friedel inter to he added to the Chirality DOI 10.1002/chi

CIF/PLATON31–33 to ensure the overall validity of the structure determination. For absolute-configuration deter￾mination one pays particular attention to any indications of pseudosymmetry, incorrect space group, and insufficient number of intensity measurements of Friedel opposites. Moreover, the noncrystallographic chemical evidence con￾cerning the enantiopurity both of the bulk and of the indi￾vidual single crystal used for the diffraction studies needs to be reviewed. x(u) 5 0.05(2), space group P21/c: This report is com￾plete rubbish as a crystal structure with this space group is centrosymmetric implying that both absolute structure and the Flack parameter are meaningless. If there are chi￾ral molecules in this structure, they are present as the rac￾emate. x(u) 5 0.05(2), space group Pna21: Absolute-configura￾tion determination is not possible as this space group con￾tains rotoinversion operations in the form of the glide reflections n and a. The crystal structure is noncentrosym￾metric and achiral. If there are chiral molecules in this structure, they are present as the racemate. Absolute￾structure determination, but not absolute-configuration determination, may have been possible in this case. x(u) 5 0.05(2), space group C2, good Friedel coverage, enantiomerically-pure bulk compound: The standard uncertainty on the Flack parameter is less than 0.10, and the Flack parameter itself is within three standard uncer￾tainties of zero. Absolute-structure determination has been achieved. Moreover, as the space group contains only pure rotations and screw rotations, the crystal structure is noncentrosymmetric and chiral. The enantiopurity of the bulk compound ensures the enantiopurity of the single crystal used for the diffraction studies and consequently absolute-configuration determination has been achieved. Characterization of the bulk compound by OR, CD or enantioselective chromatography is necessary to make the absolute-configuration determination complete. x(u) 5 0.05(2), space group C2, good Friedel coverage, racemic bulk compound: Similar to the case immediately above but the compound has crystallized by spontaneous resolution. For measurements on a series of crystals all refined using the same structure model, about 50% of the crystals would give x(u) 0.05(2) and the other 50% would give x(u) 0.95(2). Absolute configuration is achieved if it is possible to characterize by either CD or enantioselective chromatography the single crystal used for the diffraction study. x(u) 5 20.1(2): The value 0.2 of the standard uncer￾tainty is larger than either of the limiting values 0.04 or 0.1, and consequently absolute-structure and absolute-con- figuration determination are not possible from the experi￾mental intensity measurements. The negative value of the Flack parameter is within three standard uncertainties of zero and entirely compatible with the statistical fluctua￾tions inherent in the experimental measurements. Possi￾ble remedies to achieve a smaller standard uncertainty on the Flack parameter are more accurate intensity measure￾ments, measurement at a lower temperature, accurate measurement of a selected set of Bragg reflections having the largest model Friedel intensity differences, use of a radiation of different wavelength, synthesis of a derivative containing resonant-scattering atoms, and synthesis of a cocrystal or solvate containing resonant scatterers. x(u) 5 0.00: No standard uncertainty on the Flack pa￾rameter has been reported and the Flack parameter may not even have been refined. As it is hence impossible to assess the accuracy of the Flack parameter, absolute-struc￾ture and absolute-configuration determination have not been achieved. x(u) 5 0.0(3): This is a situation similar to that of x(u) 5 20.1(2) but with an even greater uncertainty. x(u) 5 0.81(12): For values of the Flack parameter greater than 0.5, the structure model should be inverted and the refinement restarted. x(u) 5 0.19(12): When using SHELXL28 there is the danger that positive values of the Flack parameter notice￾ably different from zero may not have converged. One must use the TWIN/BASF commands13,29 and report the value of BASF1 for the Flack parameter. If this has been done in the present case and there is good Friedel cover￾age, the crystal is probably twinned by inversion contain￾ing 81% of the model and 19% of the inverted model. See the section Absolute-configuration determination from a bulk racemate by combined CD and XRD. However x(u) 5 0.19(12) is also rather typical of structure solutions suffer￾ing from a partial-polar ambiguity.34 x(u) 5 0.49(2), good Friedel coverage, enantiomerically pure bulk compound: The chemical and the crystallo￾graphic evidence are contradictory. One needs to examine both very critically. A similar situation pertains when a centrosymmetric crystal structure is obtained for an enan￾tiomerically pure bulk compound. The chemists’ curse is the opposite enantiomer as impurity in a binary system for which the melting temperature of the racemic compound is much higher than that of the enantiomerically pure com￾pounds. The crystallographers’ curse is bad software im￾plementing inappropriate averaging algorithms or absorp￾tion corrections. No absolute-configuration determination is possible. Watch out as well for disordered racemates which may be enantiomerically-pure crystals produced by spontaneous resolution. x(u) 5 0.042(8), poor Friedel coverage, indications from checkCIF/PLATON that a center of symmetry needs to be added to the space group, no chemical data indicating that the compound is enantiomerically pure: Due to insufficient intensity data, the refinement of the Flack parameter has probably stuck at its starting value of zero13,14 for this cen￾trosymmetric structure refined as noncentrosymmetric. The analyst should undertake refinement in the appropri￾ate centrosymmetric space group. No absolute-configura￾tion determination is possible. x(u) 5 0.042(8), good Friedel coverage, indications from checkCIF/PLATON that a center of symmetry needs to be added to the space group, strong chemical evidence that the compound is enantiomerically pure: This is a very nice absolute-configuration determination. x(u) 5 0.49(2), good Friedel coverage, bulk compound is either a racemate or achiral: If there are indications from checkCIF/PLATON that a center of symmetry needs to be added to the space group, this is most likely the USE OF X-RAY CRYSTALLOGRAPHY 687 Chirality DOI 10.1002/chir

688 FLACK AND BERNARDINELII e-configuratio twinned ystal may by on D structur ach component in this two attri A chiral chrom crystals were grown from a sol suspect thus that the 1.The value may be 0.903)ee 6%.Bo山c ystals are thus t nment t but no nvers one neces the crysta The crystal poin ale manner opp the the first but contains a majority of the enantiomer opp in which th site as ma put ed and 222.4223.266245s16e or the sble to p the determination of ration fo the he ratio of the d peak heights at 350 nm s a CD spectrum of soluti of the on th le hat the lute configuratior bee dete that ha ve a deter racemate in solution. of the analys s of absolute ot rotated-o nly structure,ma Absoluto-Co ion De ation Relying on The to twin 43 nant alent of th 3 me el one semipreparative tal structure.Let u from ar 0.5 ml/min givin retention times of 15.3 and 163 min 中eontmotcoRonentCreteoion lack pa 100 10.00-1 and 9(13) lightly lar er than our upp r safe to give k= 0.5694）fo matri 10,the rim 1f0 and us solute structur mination has re urren sis,one has 1刀.and 0329131. givin In thi it would not lent to nave been ible to ow an the valu for clearly shows that of the crystaine sample contains Chirality DOI 10.1002/chir

case. If there are no such indications, the crystal is most likely a 50:50 inversion twin. No absolute-configuration determination is possible. Absolute-Configuration Determination from a Bulk Racemate by Combined CD and XRD A chiral chromium complex35 was synthesized, and crystals were grown from a solution of the racemate. The crystal structure is chiral displaying the space group P212121. One would suspect thus that the crystallization had proceeded by spontaneous resolution giving rise to a racemic conglomerate. Two different crystals were mea￾sured by XRD and gave values for the Flack parameter2 x of 0.36(4) [ee (i.e., enantiomeric excess) 5 28(8)%] and 0.90(3) [ee 5 280(6)%]. Both crystals are thus twinned by inversion, being in effect oriented agglomerates of enantio￾merically pure domains containing molecules of opposite chirality in the manner of hexahelicene.4 Moreover, the second crystal shows a higher enantiomeric excess than the first but contains a majority of the enantiomer opposite to that present as majority component in the first crystal. The two crystals were put into separate solutions and the CD-spectra of these were measured and normalized to equal crystal volume. The CD-spectrum of the solution from crystal 1 is indeed weaker and in form the mirror image of that from crystal 2. The ratio of the enantiomeric excesses from the XRD gives a value of 20.35(10) whereas the ratio of the normalized peak heights at 350 nm of the CD spectra is 20.42. The agreement is very good indeed. So long as a CD spectrum of a solution of the crystal used for the diffraction experiment is published with the results of the structure analysis, it will be justifiable to claim that the absolute configuration has been determined. This is very satisfactory considering that one is working from a racemate in solution. Absolute-Configuration Determination Relying on Enantioselective Chromatography The synthesis of an N-sulphonated aziridine, resulted in an enantiomeric mixture which was found to have an enan￾tiomeric excess of 43% of the (1R, 3R, 6S) enantiomer.36,37 The enantiomers were separated by semipreparative HPLC on Chiracel OD H using hexane/isopropanol 9:1 at 0.5 ml/min giving retention times of 15.3 and 16.3 min. The product from the minority component (retention time 15.3 min) was used to make crystals. Their crystal struc￾ture is chiral displaying space group P21 giving a Flack pa￾rameter2 x(u) 5 20.03(12). Although the standard uncer￾tainty on x, 0.12, is very slightly larger than our upper safe limit of 0.10, the conditions of experimentation, experience with other similar compounds, the small value of x con￾vince us that absolute-structure determination has been achieved. The absolute configuration was determined to be (1S, 3S, 6R). The retention time and experimental con￾ditions provide a sufficient characterization of the enan￾tiomer in the absolute-configuration determination. In this case, it would not have been possible to use optical activity or CD as these effects are far too weak: [aD] 5 0.78 for ee 5 43% and the CD spectrum is flat. Determination of Absolute Configuration from Multiply-Twinned Crystals A twinned crystal may be viewed as a solid-state agglomerated mixture of rotated and/or inverted copies of the untwinned crystal structure. Each component in this mixture is specifed by two attributes. 1. The volume fraction xi of the ith component in the mac￾roscopic crystal. This value may be established during structure refinement. 2. The isometry relating the orientation of the component to that of the basic one. This twin symmetry operation may be established by arguments of symmetry3,38,39 and is not unique. It comes from a group G of isome￾tries which leave the crystal lattice invariant but not necessarily the crystal structure. The crystal point group P is a subgroup of G, G ) P. As we are dealing here solely with cases in which the crystal structure is chiral, so that P is one of the point groups containing only rotations (geometric crystal classes: 1, 2, 222, 4, 422, 3, 32, 6, 622, 23, 432). So long as the criteria given in the subsection absolute-configuration determination are obeyed, it is still possible to proceed to the determination of absolute configuration for the multi￾ply-twinned crystal. Full details of the group-theoretical analysis with the related restrictions to its application are given in7 but here it suffices to point out that twin-symme￾try operations that have a determinant of 11 are pure rota￾tions and do not change the chirality of the molecules in the crystalline domain upon which they act. On the other hand twin-symmetry operations that have a determinant of 21 are rotoinversions and change the chirality of the mol￾ecules. For the purposes of the analysis of absolute config￾uration, the total amount of rotated-only structure, x 1, may be deduced by summing the volume fractions correspond￾ing to twin laws of determinant 11, x 1 5 S xþ i , and that of rotated-and-inverted structure, x 2, may be deduced by summing the volume fractions corresponding to twin laws of determinant 21, x 2 5 S x i .x 2 is the equivalent of the Flack x parameter for multiply-twinned crystals possessing a chiral crystal structure. Let us make this clear from an example.40 The space group is P31 which belongs to geo￾metric crystal class 3 and thus the crystal structure is chi￾ral. The structure was refined as a four-component twin: k2 5 0.064(13) for matrix 010,100,00-1, k3 5 0.038(17) for matrix -100,0-10,00-1, and k4 5 0.329(13) for matrix 0-10,-100,001.40 k1 may be obtained from the relationship k1 5 1 2 k2 2 k3 2 k4 to give k1 5 0.569(14) for matrix 100,010,001. The twin symmetry operations are of determi￾nant 11 for matrices 1 and 2, and 21 for matrices 3 and 4. In the nomenclature of the current analysis, one has xþ 1 (5k1) 5 0.569(14), xþ 2 (5k2) 5 0.064(13), x 1 (5k3) 5 0.038(17), and x 2 (5k4) 5 0.329(13), giving x + (5xþ 1 1 xþ 2 ) 5 0.633(17) and x 2 (5x 1 1 x 2 ) 5 0.367(17). x 2 is equiva￾lent to the Flack x parameter for this multiply-twinned crystal. Its standard uncertainty is low and hence the value of the Flack parameter is significant. The experiment clearly shows that 63% of the crystalline sample contains 688 FLACK AND BERNARDINELLI Chirality DOI 10.1002/chir

USE OF X-RAY CRYSTALLOGRAPHY 689 the structure as determined in space group P3,and 37% LITERATURE CITED naking . A.Flack HD.Be ic) it e to s and char n Soc Rev 2007 HD.On enan CONCLUDING REMARKS n of tainties o the Flack parameter which are too large fo rag 148 m the having lare lute configura I e dai tio pted by way of alterative B19515416-1 Me 1.F of opposite of the flack p ter or the abs structure.However opmeter and all 14.Flack HD. escapes fro iuauthe ent -70 15.G Kahn lexes to Acta Cryst D 2003 914-192 GLOSSARY 17 G.Flack HD.Leas hemical description (e.g.,(R)or iption by way of unitcelld ensions oordinates of a ton Th.edit or cry ing equatic c=-x+ p时 ting of an oriented domai 20. In reciprocal space.the Flack pa Tables 5213md152.14,p883 (h,,x)= 21. en S.Ena s and resolution of a chiral c al structu an equivalent Flack 5481498、 Chirality DOI 10.1002/chir

the structure as determined in space group P31 and 37% contains the inverted structure in space group P32 making the enantiomeric excess of the crystalline sample 26%. From these measurements, it is clearly not possible to establish the absolute configuration for this compound. CONCLUDING REMARKS Compounds composed only of light atoms, i.e., those having a low value of Friedif,11 give rise to standard uncer￾tainties on the Flack parameter which are too large for absolute-configuration determination. On the experimental side it may help to measure at a longer wavelength although synchrotron radiation does not seem to provide the easy answer that one might at first imagine.14 Higher precision intensity measurements on a small set of Bragg reflections selected from the crystal-structure model as having large Friedel differences may be undertaken.41 In various published41 and unpublished works, improvement of the uncertainty of the absolute-configuration determina￾tion has been attempted by way of alternative statistical procedures to that of least squares. In our view, all the pro￾cedures we have examined suffer from the same problem as sparse-matrix least squares by assuming invariance of parameters of the model which should be variable. It is perfectly correct that the average of Friedel opposites11 calculated from the crystal-structure model is independent of the Flack parameter or the absolute structure. However, the corresponding difference of Friedel opposites (of the model) depends both on the Flack parameter and all the other atomic parameters of the model. It is the latter dependence which is assumed to be invariant in the pro￾cedures we have studied. Only the procedure of Par￾sons42 escapes from this pitfall but needs further devel￾opment. GLOSSARY5 Absolute configuration5 : The spatial arrangement of the atoms of a physically identified chiral molecular entity (or group) and its stereochemical description (e.g., (R) or (S), (P) or (M), D or L, etc). Absolute structure5 : The spatial arrangement of the atoms of a physically identified noncentrosymmetric crys￾tal and its description by way of unit-cell dimensions, space group, and representative coordinates of all atoms. Flack parameter5 : The Flack parameter2 is the molar fraction x in the defining equation C 5 (1 2 x) X 1 x X, where C represents an oriented two-domain-structure crys￾tal, twinned by inversion, consisting of an oriented domain structure X and an oriented inverted domain structure X. In reciprocal space, the Flack parameter2 x is defined by the structure-amplitude equation G2 (h, k, l, x) 5 (1 2 x) |F(hkl)|2 1 x |F(h k l)|2 . For a multidomain-structure twin of a chiral crystal structure, an equivalent Flack parameter may be calculated according to the method of Flack and Bernardinelli.7 LITERATURE CITED 1. Djukic JP, Hijazi A, Flack HD, Bernardinelli G. Non-racemic (sca￾lemic) planar-chiral five-membered metallacycles: routes, means, and pitfalls in their synthesis and characterization. Chem Soc Rev 2007; DOI: 10.1039/B618557F. 2. Flack HD. On enantiomorph-polarity estimation. Acta Cryst A 1983; 39:876–881. 3. Hahn Th, Janovec V, Klapper H, Privratska ,J. Twinning and domain structures. In: Authier A, editor. International tables for crystallogra￾phy, volume D: physical properties of crystals. Dordrecht: Interna￾tional Union of Crystallography and Kluwer Academic Publishers; 2003. p 377–505. 4. Green BS, Knossow M. Lamellar twinning explains the nearly racemic composition of chiral, single crystals of hexahelicene. Science 1981; 214:795–797. 5. Flack HD. Chiral and achiral crystal structures. Helv Chim Acta 2003;86:905–921. 6. Flack HD, Bernardinelli G. Reporting and evaluating absolute-struc￾ture and absolute-configuration determinations. J Appl Cryst 2000; 33:1143–1148. 7. Flack HD, Bernardinelli G. Absolute structure and absolute configura￾tion. Acta Cryst B 1999;55:908–915. 8. Bijvoet JM. Phase determination in direct Fourier-synthesis of crystal structures. Proc K Ned Akad Wet Ser B 1949;52:313–314. 9. Peerdeman AF, van Bommel AJ, Bijvoet JM. Determination of abso￾lute configuration of optical active compounds by means of X-rays. Proc K Ned Akad Wet Ser B 1951;54:16–19. 10. Flack HD, Bernardinelli G. The Mirror of Galadriel: looking at chiral and achiral crystal structures. Cryst Eng 2003;6:213–223. 11. Flack HD, Shmueli U. The mean-square Friedel intensity difference in P1 with a centrosymmetric substructure. Acta Cryst A 2007;63:257– 265. 12. McIntyre G. A prediction of Bijvoet intensity differences in the non￾centrosymmetric structures of selenium and tellurium. Acta Cryst A 1978;34:936–939. 13. Flack HD, Bernardinelli G. Centrosymmetric crystal structures described as non-centrosymmetric; an analysis of reports in Inorgan￾ica Chimica Acta. Inorg Chim Acta 2006;359:383–387. 14. Flack HD, Bernardinelli G, Clemente DA, Linden A, Spek AL. Centro￾symmetric and pseudo-centrosymmetric structures refined as non￾centrosymmetric. Acta Cryst B 2006;62:695–701. 15. Girard E, Stelter M, Vicat J, Kahn R. 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