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/chirthe 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from the crystal-structure model as having large Friedel differences may be undertaken.41 In various published41 and unpublished works, improvement of the uncertainty of the absolute-configuration determina￾tion has been attempted by way of alternative statistical procedures to that of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. A new class of lanthanide com￾plexes to obtain high-phasing-power heavy-atom derivatives for macro￾molecular crystallography. Acta Cryst D 2003;59:1914–1922. 16. Bernardinelli G, Flack HD. Least-squares absolute-structure refine￾ment. Practical experience and ancillary calculations. Acta Cryst A 1985;41:500–511. 17. Bernardinelli G, Flack HD. Least-squares absolute-structure refine￾ment. A case study of the effect of absorption correction, data region, stability constant and neglect of light atoms. Acta Cryst A 1987;43:75– 78. 18. Schwarzenbach D, Flack HD. High energy spectroscopy: structure refinement (solid state diffraction). In: Lindon J, Tranter G, John Holmes J, editors. Encyclopedia of spectroscopy and spectrometry. London: Academic Press; 1999. p 2271–2278. 19. Hahn Th, editor. International tables for crystallography, volume A: space-group symmetry. Dordrecht: International Union of Crystallog￾raphy and Kluwer Academic Publishers; 2002. 20. Koch E, Fischer W, Mu¨ller U. Normalizers of space groups and their use in crystallography. In: Hahn Th, editor. International tables for crystallography, volume A: space-group symmetry. Dordrecht: Inter￾national Union of Crystallography and Kluwer Academic Publishers; 2002. Tables 15.2.1.3 and 15.2.1.4, p 883–898. 21. Jacques J, Collet A, Wilen S. Enantiomers, racemates and resolutions. New York: Wiley; 1981. 22. Coquerel G. Review on the heterogeneous equilibria between con￾densed phases in binary systems of enantiomers. Enantiomer 2000; 5:481–498. USE OF X-RAY CRYSTALLOGRAPHY 689 Chirality DOI 10.1002/chir