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Chirality & Polyhedra theSymmetry EyesSecond of Edition athrough Chemist andBudapestIstván HungarianBudapest,Technical Academy Hargittal Hungary University of Sciences MagdolnaHungarianBudapest,Academy HungaryHargittai of Sciences PlenumPress • New York and London 64 qoo Chapter 2 2.7Simple CHIRALITYand Combined Symmetries 65 pairdecoration,whosesymmetry ofThere crystals.mirror planesaare detail images Themany butfrom simplest which objects,cannot Bach's chiraloccur be Theboth superposed. moleculesArtin animate pairs of the related Fugue,areandFigure those inanimate, bya 2-45pairina symmetrywhich of shows molecules,which a carbon a plane buildinghave andatom andno a a exceptverticesis surrounded glycine. of a tetrahedron. by four different All the ligands—atoms naturallyoccurringor groups amino of acids atoms—at are chiral, the a Contrapunetus XVIII rrrrTirr (a) The centrosymmetric arrangement of acetanilide molecules in the crystal, withmoleculesCopyrightresultingFigure permission 2-44. inin(1981) athe centrosymmetric fromcrystal, Americal Ref. resulting [2-321. Chemical crystalin Copyright the Society; occurrence habit. (1981) Reprinted(b) ofThe American a polarhead-to-tail with axis Chemical permission in alignmentthe crystal Society. from ofhabit. p-acetanilideRef. Reprinted [2-321. b behavior. In solid/gas reactions, for example, crystal polarity may be a source CHO!rtCHO of considerable anisotropy. centerscellsuch of Thereas a pyroelectricityofpyroelectric theare positivealso important crystal andand piezoelectricitynegativepossesses physical chargespropertiesa dipole and changesmoment. otherscharacterizing [2-341.upon The separation heating. Thepoiar primitive crystals, In of this the HZ\OHCH2OH HOZ\.HCH2OH practicalcompressiontricityprocess is the theuses. two separation of chargesthe crystal. of migrate the Both positive to pyroelectricity the andtwo negative ends of and thecharges piezoelectricitypolar upon axis. expansion!Piezoelec- have dersymmetry)Figure Fuge,2-45. Contrapunctus are eachIllustrations other's XVIII, mirror of chiral detail; images; pairs: (c) photographs (a)glyceraldehyde Decorations by the whosemolecules;authors; motifs (b)(d)(of J.quartz S.fourfold Bach, crystals. Dierotational Kunstd 66mirror,figureW or ideallyH. group Thomson, realized,of points Lord cannot'chiral,' Kelvin,beand brought saywrote it has into[2-35]: ëhirality, coincidence"I callif its any imagewith geometricalitself." inChapter aplane He 2 Simple and Combined Symmetries 67 2-46Indeed,heterochiral.called Aand forms chiralthe 2-47 word ofTheobject show the chirality most same andsome common its senseitselfheterochiral mirror comeshomochiralexampleimagefrom and areiIiII theofhomochiraland enantiomorphous, a Greek heterochiralforms IIIIItII word of pairs the for 11offormopposite hand. hands.and is theyFigures hands. sense are lJ*j1.J.1t(hI aminoFigure acid2-46. moleculese(Continued) [2-36]; (e) reproduced heterochiral by pair permission of hands from and R.models N. Bracewell. of a heterochiral pair of ii i. '1 '''II'.'P'I.f . knd\'olunreera hand ..__. a I 7. - b -'I . 2. •...5: I4 US\2Ocb / - .—.. C Heterochiral pairs of hands: (a) Tombstone in theJewish cemetery, Prague; a C Figurephotographmagazine 2-46.Der byby thetheSpiegel, authors;authors; June (d) (b) United 15,Albrecht1992; Nations Durer's reproduced stamp.Praying (Continuedby permission;Hands onon thenext (c) cover page)Buddha of thein German Tokyo; railwayFigure18, 1992; 2-47. station; reproducedHomochiral photograph by permission; pairs by the of authors.hands: (b) U.S. (a) stamp;Cover of(c) the logo German with SOS magazine distress Der sign Spiegel, at a Swiss May Chapter 2 Simple and Combined Symmetries 68 69 nonenantiomorphous symmetry classes of crystals which may exhibit optical activity. Whyte [2-37] extended the definition of chirality as follows: "Three- dimensional forms (point arrangements, structures, displacements, and other processes) which possess non-superposable mirror images are called 'chiral'." A chiral process consists of successive states, all of which are chiral. The two main classes of chiral forms are screws and skews. Screws may be conical or cylindrical and are ordered with respect to a line. Examples of the latter are the left-handed and right-handed helices in Figure 2-50. The skews, on the other hand, are ordered around their center. Examples are chiral molecules having point-group symmetry. From the point of view of molecules, or crystals, left and right are intrinsically equivalent. An interesting overview of the left/right problem in science has been given by Gardner [2-391. Distinguishing between left and Figure 2-48.Louis Pasteur's bust in front of the Pasteur Institute, Paris.Photograph by the authors. right has also considerable social, political, and psychological connotations. For example, left-handedness in children is viewed with varying degrees of tolerance in different parts of the world. Figure 2-51a shows a classroom at the University of Connecticut with different (homochiral and heterochiral) chairs each other's enantiomorphs. Louis Pasteur (Figure 2-48)first suggested that molecules can be chiral. In his famous experiment in 1848,he recrystallized a salt of tartaric acid and obtained two kinds of smallcrystals which were mirror images of each other, as shown by Pasteur's models inFigure 2-49. The two kinds of crystals had the same chemical compositionbut differed in their optical activity. One was levo-active (L), and the other wasdextro-actiVe (D). Since the true absolute configuration of moleculescould not be determined at the time, an arbitrary convention was applied, which,luckily, proved to coincide with reality. If a molecule or a crystalis chiral, it is necessarily optically active. The converse is, however, not true.There are, in fact L— Figure 2-50.Left-handed and right-handed Figure 2-49.Pasteur's models of enantiomeric crystals in the Pasteur Institute, Paris.Photo- helix decorations from Zagorsk, Russia [2-381. graphs by the authors. Photograph by the authors. Chapter 2 70 Simple and Combined Symmetries 71 a#A+ £ a b b* Figure 2-52.Examples of symmetry operations of the first kind (a) and of the second kind(b), Figure 2-51.Classrooms with heterochiral and homochiral chairs: (a) Chairs for both the right- after Shubnikov [2-40]. handed andleft-handedstudents; (b) Older chairs, all for right-handed students only. Photo- graphs by the authors. Pasteur used "dissymmetry" for the first timeas he designated the absence of elements of symmetry of the second kind ina figure. Accordingly, dissymme- to accommodate both the right-handed and the left-handed students. Older try did not exclude elements of symmetry of the first kind. PierreCurie classrooms at the same university have chairs for the right-handed only (Figure suggested an even broader application of thisterm. He called a crystal 2-5ib). dissymmetric in the case of the absence of those elements ofsymmetry upon which depends the existence of oneor another physical property in that crystal. In Pierre Curie's original words [2-41], "Dissymmetry 2.7.1Asymmetry and Dissymmetry creates the phenome- non." Namely a phenomenon exists and is observable dueto dissymmetry, Symmetryoperations of the first kind and of the second kind are i.e., due to the absence of some symmetry elements from thesystem. Finally, sometimes distinguished in the literature (cf. Ref. [2-40}). Operations of the Shubnikov 12-40] called dissymmetry the fallingout of one or another element first kind are sometimes also called even-numbered operations. For example, of symmetry from a givengroup. He argued that to speak of the absence of the identity operation is equivalent to two consecutive reflections from a elements of symmetry makes sense only when thesesymmetry elements are symmetry plane. It is an even-numbered operation, an operation of the first present in some other structures. kind. Simple rotations are also operations of the first kind. Mirror rotation Thus, from the point of view of chirality any asymmetric figure ischiral, leads to figures consisting of right-handed and left-handed components and but asymmetry is not a necessary condition for chirality. All dissymmetricfig- therefore is an operation of the second kind. Simple reflection is also an ures are also chiral if dissymmetry means the absence of symmetry elements of operation of the second kind as it may be considered as a mirror rotation about the second kind. In thissense, dissymmetry is synonymous with chirality. a onefold axis. A simple reflection is related to the existence of two enan- An assembly of molecules may be achiral forone of two reasons. Either tiomorphic components in a figure. Figure 2-52illustratesthese distinctions by all the molecules present are achiralor the two kinds of enantiomorphs are a series of simple sketches after Shubnikov [2-401. In accordance with the present in equal amounts. Chemical reactions between achiral molecules lead above description, chirality is sometimes defined as the absence of symmetry to achiral products. Either all product molecules will be achiralor the two kinds elements of the second kind. of chiral molecules will be produced in equalamounts. Chiral crystals may Sometimes, the terms asymmetry, dissymmetry, and antisymmetry are sometimes be obtained from achiral solutions. When this happens,the two confused in the literature although the scientific meaning of these terms is in enantiomorphs will be obtained in (roughly) equal numbers,as was observed complete conformity with the etymology of
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