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Symmetry-itemized enumeration of RS-stereoisomers of allenes. I. The fixed-point matrix method of the USCI approach combined with the stereoisogram approach

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Abstract

After the RS-stereoisomeric group \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) of order 16 has been defined by starting point group \(\mathbf{D}_{2d}\) of order 8, the isomorphism between \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) and the point group \(\mathbf{D}_{4h}\) of order 16 is thoroughly discussed. The non-redundant set of subgroups (SSG) of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) is obtained by referring to the non-redundant set of subgroups of \(\mathbf{D}_{4h}\). The coset representation for characterizing the orbit of the four positions of an allene skeleton is clarified to be \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{s\widetilde{\sigma }\widehat{I}})\), which is closely related to the \(\mathbf{D}_{4h}(/\mathbf{C}_{2v}^{\prime \prime \prime })\). According to the unit-subduced-cycle-index (USCI) approach (Fujita, Symmetry and combinatorial enumeration of chemistry. Springer, Berlin 1991), the subduction of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{s\widetilde{\sigma }\widehat{I}})\) is examined so as to generate unit subduced cycle indices with chirality fittingness (USCI-CFs). Then, the fixed-point matrix method of the USCI approach is applied to the USCI-CFs. Thereby, the numbers of quadruplets are calculated in an itemized fashion with respect to the subgroups of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\). After the subgroups of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) are categorized into types I–V, type-itemized enumeration of quadruplets is conducted to illustrate the versatility of the stereoisogram approach.

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References

  1. H.H. Jaffé, M. Orchin, Symmetry in Chemistry (Wiley, Chichester, 1965)

    Google Scholar 

  2. D.M. Bishop, Group Theory and Chemistry (Clarendon, Oxford, 1973)

    Google Scholar 

  3. E.L. Eliel, S.H. Wilen, Stereochemistry of Organic Compounds (Wiley, New York, 1994)

    Google Scholar 

  4. F.A. Cotton, Chemical Applications of Group Theory (Wiley-International, New York, 1990)

    Google Scholar 

  5. L.H. Hall, Group Theory and Symmetry in Chemistry (McGraw-Hill, New York, 1969)

    Google Scholar 

  6. S. Fujita, Theor. Chim. Acta 78, 45–63 (1990)

    Article  CAS  Google Scholar 

  7. S. Fujita, Combinatorial Enumeration of Graphs, Three-Dimensional Structures, and Chemical Compounds (University of Kragujevac, Faculty of Science, Kragujevac, 2013)

    Google Scholar 

  8. S. Fujita, Bull. Chem. Soc. Jpn. 63, 315–327 (1990)

    Article  CAS  Google Scholar 

  9. S. Fujita, J. Am. Chem. Soc. 112, 3390–3397 (1990)

    Article  CAS  Google Scholar 

  10. K. Mislow, M. Raban, Top. Stereochem. 1, 1–38 (1967)

    Article  CAS  Google Scholar 

  11. S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry (Springer, Berlin, 1991)

    Book  Google Scholar 

  12. S. Fujita, Tetrahedron 47, 31–46 (1991)

    Article  CAS  Google Scholar 

  13. S. Fujita, J. Chem. Inf. Comput. Sci. 40, 135–146 (2000)

    Article  CAS  Google Scholar 

  14. S. Fujita, Bull. Chem. Soc. Jpn. 73, 329–339 (2000)

    Article  CAS  Google Scholar 

  15. S. Fujita, S. El-Basil, J. Math. Chem. 36, 211–229 (2004)

    Article  CAS  Google Scholar 

  16. S. Fujita, MATCH Commun. Math. Comput. Chem. 54, 251–300 (2005)

    CAS  Google Scholar 

  17. S. Fujita, MATCH Commun. Math. Comput. Chem. 55, 5–38 (2006)

    CAS  Google Scholar 

  18. S. Fujita, MATCH Commun. Math. Comput. Chem. 55, 237–270 (2006)

    CAS  Google Scholar 

  19. S. Fujita, Diagrammatical Approach to Molecular Symmetry and Enumeration of Stereoisomers (University of Kragujevac, Faculty of Science, Kragujevac, 2007)

    Google Scholar 

  20. S. Fujita, Theor. Chem. Acc. 113, 73–79 (2005)

    Article  CAS  Google Scholar 

  21. S. Fujita, MATCH Commun. Math. Comput. Chem. 57, 5–48 (2007)

    Google Scholar 

  22. S. Fujita, J. Math. Chem. 42, 481–534 (2007)

    Article  CAS  Google Scholar 

  23. S. Fujita, Bull. Chem. Soc. Jpn. 75, 1949–1962 (2002)

    Article  CAS  Google Scholar 

  24. S. Fujita, J. Math. Chem. 33, 113–143 (2003)

    Article  CAS  Google Scholar 

  25. S. Fujita, J. Org. Chem. 69, 3158–3165 (2004)

    Article  CAS  Google Scholar 

  26. S. Fujita, J. Math. Chem. 35, 265–287 (2004)

    Article  CAS  Google Scholar 

  27. S. Fujita, Tetrahedron 60, 11629–11638 (2004)

    Article  CAS  Google Scholar 

  28. S. Fujita, MATCH Commun. Math. Comput. Chem. 61, 11–38 (2009)

    CAS  Google Scholar 

  29. S. Fujita, J. Comput. Aided Chem. 10, 16–29 (2009)

    Article  Google Scholar 

  30. S. Fujita, Stereoisograms: a remedy against oversimplified dichotomy between enantiomers and diastereomers in stereochemistry, in Chemical Information and Computational Challenge in the 21st Century, ed. by M.V. Putz (Nova, New York 2012) Chapter 9, pp. 223–242.

  31. S. Fujita, Tetrahedron 62, 691–705 (2006)

    Article  CAS  Google Scholar 

  32. S. Fujita, Yuki Gosei Kagaku Kyokai-Shi/J. Synth. Org. Chem. Jpn. 66, 995–1004 (2008)

  33. S. Fujita, MATCH Commun. Math. Comput. Chem. 61, 39–70 (2009)

    CAS  Google Scholar 

  34. S. Fujita, J. Comput. Aided Chem. 10, 76–95 (2009)

    Article  Google Scholar 

  35. S. Fujita, Prochirality and pro-RS-stereogenicity. Stereoisogram Approach Free from the Conventional “Prochirality” and “Prostereogenicity”, in Carbon Bonding and Structures. Advances in Physics and Chemistry, ed. by M. V. Putz (Springer, Dordrecht, Heidelberg, London 2011) Vol. 5 of Carbon Materials: Chemistry and Physics Chapter 10, pp. 227–271.

  36. S. Fujita, MATCH Commun. Math. Comput. Chem. 52, 3–18 (2004)

    CAS  Google Scholar 

  37. S. Fujita, MATCH Commun. Math. Comput. Chem. 54, 39–52 (2005)

    CAS  Google Scholar 

  38. S. Fujita, Memoirs of the Faculty of Engineering and Design. Kyoto Inst. Technol. 53, 19–38 (2005)

    Google Scholar 

  39. S. Fujita, Helv. Chim. Acta 85, 2440–2457 (2002)

    Article  CAS  Google Scholar 

  40. S. Fujita, MATCH Commun. Math. Comput. Chem. 61, 71–115 (2009)

    CAS  Google Scholar 

  41. G. Helmchen, in Stereoselective Synthesis, 4 edn., ed. by G. Helmchen, R. W. Hoffmann, J. Mulzer, E. Schaumann, (Georg Thieme, Stuttgart New York 1996) Vol. 1 of Methods of Organic Chemistry (Houben-Weyl). Workbench Edition E21 pp 1–74

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  1. Shonan Institute of Chemoinformatics and Mathematical Chemistry, Kaneko 479-7 Ooimachi, Ashigara-Kami-Gun, Kanagawa-Ken , 258-0019, Japan

    Shinsaku Fujita

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  1. Shinsaku Fujita
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Correspondence to Shinsaku Fujita.

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Fujita, S. Symmetry-itemized enumeration of RS-stereoisomers of allenes. I. The fixed-point matrix method of the USCI approach combined with the stereoisogram approach. J Math Chem 52, 1717–1750 (2014). https://doi.org/10.1007/s10910-014-0344-y

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