On Quantifying Chirality

Since Pasteur's epochal discoveries a century and a half ago, the concept of chirality has continued to play a central role in chemistry and biochemistry. Can chirality be measured? It has long been known that molecular chirality can be given a quantitative meaning through functions specifically parametrized to match the magnitude of pseudoscalar observables. However, chirality is a property that is independent of its physical and chemical manifestations : for a system to be chiral, all that is required is the absence of improper rotations in the symmetry group of the system. This being the case, how can chirality be measured if the “system” is an abstract geometric figure, for example, a scalene triangle in the plane or an asymmetric tetrahedron in three-dimensional space? How does chirality vary as a function of pure shape? In this review we describe recent efforts designed to answer these and related questions.

[1]  A. Bauder,et al.  The isometric group of nonrigid molecules , 1979 .

[2]  E. Flapan.  Rigid and nonrigid achirality , 1987 .

[3]  A. Buda,et al.  On a measure of axiality for triangular domains. , 1991 .

[4]  S. Schröder,et al.  N‐Aryl‐O‐acylhydroxylamine: Herstellung durch O‐Acylierung oder N→O‐Umacylierung und Umsetzung mit Aminen; Modellreaktionen für Schlüsselschritte im Zusammenhang mit der Carcinogenität aromatischer Amine , 1988 .

[5]  J. Sauvage,et al.  Ein topologisch chiraler [2]Catenand , 1988 .

[6]  E. Baer,et al.  STUDIES ON ACETONE-GLYCERALDEHYDE IV. PREPARATION OF d(+)-ACETONE GLYCEROL , 1939 .

[7]  L. Schulman,et al.  Chiral interaction, magnitude of the effects and application to natural selection of L-enantiomer , 1985 .

[8]  David M. Walba,et al.  Total synthesis of the first molecular Moebius strip , 1982 .

[9]  Jean-Pierre Sauvage,et al.  Interlacing molecular threads on transition metals: catenands, catenates, and knots , 1990 .

[10]  R. Woolley.  Natural optical activity and the molecular hypothesis , 1982 .

[11]  Kurt Mislow,et al.  On geometric measures of chirality , 1991 .

[12]  L. Barron.  Fundamental symmetry aspects of optical activity , 1981 .

[13]  A. Buda,et al.  Geometric chirality products , 1991 .

[14]  Jean-Pierre Sauvage,et al.  A Synthetic Molecular Trefoil Knot , 1989 .

[15]  C. G. Broyden.  The Convergence of a Class of Double-rank Minimization Algorithms 1. General Considerations , 1970 .

[16]  J. Hougen.  Symmetry beyond point groups in molecular spectroscopy , 1986 .

[17]  R. G. Woolley,et al.  Quantum theory and molecular structure , 1976 .

[18]  G. Geison,et al.  Pasteur and the Process of Discovery: The Case of Optical Isomerism , 1988, Isis.

[19]  I. Ugi.  Stereospezifische Synthesen I , 1965 .

[20]  L. Paquette,et al.  Threefold transannular epoxide cyclization. Synthesis of a heterocyclic C17-hexaquinane , 1981 .

[21]  C. Dietrich-Buchecker,et al.  Structure of a Synthetic Trefoil Knot Coordinated to Two Copper(I) Centers , 1990 .

[22]  J. Sauvage,et al.  Struktur einer an zwei Kupfer(I)‐Zentren koordinierten Kleeblattknoten‐ Verbindung , 1990 .

[23]  K. Mislow,et al.  An Epistemological Note on Chirality , 1976 .

[24]  K. Mislow,et al.  Intrinsic asymmetry and diastereotopism , 1978 .

[25]  Stabilization of planar tetracoordinate carbon , 1976 .

[26]  G. Gilat.  Chiral coefficient-a measure of the amount of structural chirality , 1989 .

[27]  E. Baer,et al.  Preparation and Properties of Optically Active Derivatives of Glycerol. , 1941 .

[28]  D. Walba,et al.  The thyme polyethers: An approach to the synthesis of a molecular knotted ring , 1986 .

[29]  E. Ruch.  Homochiralität als Klassifizierungsprinzip von Molekülen spezieller Molekülklassen , 1968 .

[30]  L. Barron,et al.  Fundamental symmetry aspects of chirality. , 1987, Bio Systems.

[31]  E. BUCHAR.  Motion of the Nodal Line of the Second Russian Earth Satellite (1957β) and Flattening of the Earth , 1958, Nature.

[32]  K. Mislow,et al.  SEPARATION AND IDENTIFICATION OF RESIDUAL DIASTEREOMERS IN A MAXIMALLY LABELED TRIARYLAMINE , 1980 .

[33]  Ivar Ugi,et al.  A Group Theoretical Analysis of Conformational Flexibility , 1980 .

[34]  F. Hund.  Zur Deutung der Molekelspektren. III. , 1927 .

[35]  E. Muetterties.  Intramolecular racemization processes in pentacoordinate structures , 1967 .

[36]  J. Simon.  Topological chirality of certain molecules , 1986 .

[37]  Vladimir Prelog,et al.  Pseudoasymmetrie in der organischen Chemie , 1972 .

[38]  J. Sauvage,et al.  A Topologically Chiral [2]Catenand , 1988 .

[39]  H. C. Longuet-Higgins.  The symmetry groups of non-rigid molecules , 1963 .

[40]  Alberto Guenzi,et al.  Dynamic gearing and residual stereoisomerism in labeled bis(9-triptycyl)methane and related molecules. Synthesis and stereochemistry of bis(2,3-dimethyl-9-triptycyl)methane, bis(2,3-dimethyl-9-triptycyl)carbinol, and bis(1,4-dimethyl-9-triptycyl)methane , 1983 .

[41]  The mathematical tourist , 1991 .

[42]  Jonathan Simon,et al.  Molecular graphs as topological objects in space , 1987 .

[43]  Wilhelm Jr. Schlenk,et al.  Synthesis and analysis of optically active triglycerides , 1965, Journal of the American Oil Chemists' Society.

[44]  P. Mezey.  New developments in molecular chirality , 1991 .

[45]  Frank Harary,et al.  Similarity and complexity of the shapes of square-cell configurations , 1991 .

[46]  S. Mauskopf.  Crystals and compounds : molecular structure and composition in nineteenth-century French science , 1976 .

[47]  Jacopo Tomasi.  Models and modeling in theoretical chemistry , 1988 .

[48]  D. Gust,et al.  Analysis of isomerization in compounds displaying restricted rotation of aryl groups , 1973 .

[49]  Must a molecule have a shape , 1978 .

[50]  Howard E. Simmons,et al.  Synthesis of the first topologically non-planar molecule , 1981 .

[51]  K. Millett.  Stereotopological indices for a family of chemical graphs , 1987 .

[52]  K. Mislow,et al.  MOLECULAR DISSYMMETRY AND OPTICAL INACTIVITY , 1955 .

[53]  R. King.  Experimental tests of chirality algebra , 1991 .

[54]  T. Damhus,et al.  Three reference systems for chirality specification. Application, geometric properties, and mutual relationships , 1983 .

[55]  Desymmetrization and degree of chirality , 1991 .

[56]  The molecular structure conundrum: Can classical chemistry be reduced to quantum chemistry? , 1984 .

[57]  K. Humbel,et al.  Chemical Applications of Topology and Graph Theory, R.B. King (Ed.). Elsevier Science Publishers, Amsterdam (1983), (ISBN 0-444-42244-7). XII + 494 p. Price Dfl. 275.00 , 1985 .

[58]  B. Abel Devalcourt.  Measures of axial symmetry for ovals , 1966 .

[59]  W. Schlenk.  Recent Results of Configurational Research , 1965 .

[60]  Gabor A. Somorjai,et al.  Modern concepts in surface science and heterogeneous catalysis , 1990 .

[61]  Laurence D. Barron,et al.  True and false chirality and absolute asymmetric synthesis , 1986 .

[62]  Ivar Ugi,et al.  Die Vandermondesche Determinante als Näherungsansatz für eine Chiralitätsbeobachtung, ihre Verwendung in der Stereochemie und zur Berechnung der optischen Aktivität , 1967 .

[63]  Jona-Lasinio,et al.  Instability of tunneling and the concept of molecular structure in quantum mechanics: The case of pyramidal molecules and the enantiomer problem. , 1986, Physical review. A, General physics.

[64]  E. Ruch,et al.  Das stereochemische Strukturmodell, ein mathematisches Modell zur gruppentheoretischen Behandlung der dynamischen Stereochemie , 1966 .

[65]  F. Hund.  Zur Deutung der Molekelspektren. IV , 1927 .

[66]  D. Goldfarb.  A family of variable-metric methods derived by variational means , 1970 .

[67]  V. Moy,et al.  Shapes of finite two-dimensional lipid domains , 1988 .

[68]  Robert A. Harris,et al.  On the time dependence of optical activity , 1981 .

[69]  N. Cozzarelli,et al.  Biochemical topology: applications to DNA recombination and replication. , 1986, Science.

[70]  Crum A. Brown.  On the Relation of Optical Activity to the Character of the Radicals united to the Asymmetric Carbon Atom. , 1891 .

[71]  C. G. Broyden.  The Convergence of a Class of Double-rank Minimization Algorithms 2. The New Algorithm , 1970 .

[72]  E. Ruch,et al.  Algebraic aspects of the chirality phenomenon in chemistry , 1972 .

[73]  H. Günthard,et al.  Isometric groups and chirality of nonrigid molecules: a generalization of Kelvin's theorem , 1976 .

[74]  H. Bürgi,et al.  Angular distortions at tetracoordinate carbon: planoid configurations in substituted spiro[4.4]nonanes , 1987 .

[75]  K. Mislow,et al.  Role of pseudorotation in the stereochemistry of nucleophilic displacement reactions , 1970 .

[76]  Chirality as a quasi-classical property of molecular systems , 1990 .

[77]  J. D. Morrison,et al.  Asymmetric Organic Reactions , 1976 .

[78]  Mark A. Johnson.  A review and examination of the mathematical spaces underlying molecular similarity analysis , 1989 .

[79]  O. Ramsay.  van't Hoff-Le Bel Centennial , 1975 .

[80]  D. Shanno.  Conditioning of Quasi-Newton Methods for Function Minimization , 1970 .

[81]  菅野 暁.  マルティン・ガードナー著, 坪井忠二, 小島弘共訳: 自然界における左と右, 紀伊国屋書店, 1971, 340 ページ, 20×14cm, 870円 [Martin Gardner; The Ambidextrous Universe, Basic Books Inc., 1964] , 1972 .

[82]  J. Weyer.  A Hundred Years of Stereochemistry—The Principal Development Phases in Retrospect , 1974 .

[83]  E. Arnett,et al.  Molecular recognition in chiral monolayers of stearoylserine methyl ester , 1989 .

[84]  F. Cotton,et al.  Stereochemically nonrigid organometallic molecules. XII. Temperature dependence of the proton nuclear magnetic resonance spectra of the 1,3,5,7-tetramethylcyclooctatetraene tricarbonyl compounds of chromium, molybdenum, and tungsten , 1968 .

[85]  Laurence D. Barron,et al.  Symmetry and molecular chirality , 1986 .

[86]  K. Mislow,et al.  Stereochemical consequences of dynamic gearing in substituted bis(9-triptycyl) methanes and related molecules. , 1980, Proceedings of the National Academy of Sciences of the United States of America.

[87]  Paul G. Mezey,et al.  The degree of similarity of three-dimensional bodies: Application to molecular shape analysis , 1991 .

[88]  L. Barron.  Optical activity and time reversal , 1981 .

[89]  Shmuel Peleg,et al.  Two-dimensional rotational dynamic chirality and a chirality scale , 1990, Langmuir.

[90]  Ernst Ruch,et al.  Theorie der ChiralitÄtsfunktionen , 1970 .

[91]  E. Ruch,et al.  Chiral Derivatives of Achiral Molecules: Standard Classes and the Problem of a Right‐Left Classification , 1977 .

[92]  Jost Weyer.  Hundert Jahre Stereochemie — Ein Rückblick auf die wichtigsten Entwicklungsphasen , 1974 .

[93]  K. Millett.  Configuration Census. Topological Chirality and the New Combinatorial Invariants , 1986 .

[94]  L. Barron.  True and false chirality and parity violation , 1986 .

[95]  B. Chion,et al.  Structural aspects of solid solutions of enantiomers: the 3-hydroxymethyl- and 3-carboxy-2,2,5,5-tetramethylpyrrolidinyl 1-oxyl systems as examples , 1978 .

[96]  James G. Nourse,et al.  Self-inverse and nonself-inverse degenerate isomerizations , 1980 .

[97]  J. Sauvage,et al.  Eine Kleeblattknoten‐Verbindung , 1989 .

[98]  W. Schlenk.  Neuere Ergebnisse der Konfigurationsforschung , 1965 .

[99]  R. Fletcher,et al.  A New Approach to Variable Metric Algorithms , 1970, Comput. J..

[100]  K Mislow,et al.  Limitations of the Symmetry Criteria for Optical Inactivity and Resolvability. , 1954, Science.

[101]  K. Mislow,et al.  Stereochemical consequences of correlated rotation in molecular propellers , 1976 .

[102]  E. Ruch,et al.  Theory of chirality functions, generalized for molecules with chiral ligands , 1973 .

[103]  P. Curie.  Sur la symétrie dans les phénomènes physiques, symétrie d'un champ électrique et d'un champ magnétique , 1894 .

[104]  Shmuel Peleg,et al.  HOW TO TELL RIGHT FROM LEFT , 1988 .

[105]  W. Graham Richards,et al.  Similarity of molecular shape , 1991, J. Comput. Aided Mol. Des..

[106]  Roald Hoffmann,et al.  Planar tetracoordinate carbon , 1970 .

[107]  Ernst Ruch,et al.  Näherungsformeln für spiegelungsantimetrische Moleküleigenschaften , 1968 .

[108]  D. Walba,et al.  The thyme polyethers , 1986 .

[109]  E. Muetterties.  Stereochemically Nonrigid Structures , 1965 .

[110]  D. Avnir,et al.  Quantifying the degree of molecular shape distortion. a chirality measure , 1991 .

[111]  F. Cotton.  Fluxional organometallic molecules , 1968 .

[112]  G. Erker,et al.  Planar tetracoordinate carbon in zirconocene complex Cp2Zr(.mu.-.eta.1,.eta.2-Me3SiCCPh)(.mu.-Cl)AlMe2 , 1990 .