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