Journal of Mathematical Chemistry

, Volume 56, Issue 8, pp 2226–2266 | Cite as

Hierarchical enumeration of Kekulé’s benzenes, Ladenburg’s benzenes, Dewar’s benzenes, and benzvalenes by using combined-permutation representations

  • Shinsaku Fujita
Original Paper


The group hierarchy for each skeleton of ligancy 6 is formulated to be: point group (PG \({\varvec{G}}_{\sigma }\)) \(\subseteq \) RS-stereoisomeric group (RS-SIG \({\varvec{G}}_{\sigma \widetilde{\sigma }\widehat{I}}\)) \(\subseteq \) stereoisomeric group (SIG \(\widetilde{{\varvec{G}}}_{\sigma \widetilde{\sigma }\widehat{I}}\)) \(\subseteq \) isoskeletomeric group (ISG \(\widetilde{\widetilde{{\varvec{G}}}}_{\sigma \widetilde{\sigma }\widehat{I}}\) = \({\varvec{S}}^{[6]}_{\sigma \widehat{I}}\)), where we start from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{D}}_{6h}\) for the Kekulé benzene skeleton, from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{D}}_{3h}\) for the Ladenburg benzene skeleton, from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{C}}_{2v}\) for the Dewar benzene skeleton, or from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{C}}_{2v}\) for the benzvalene skeleton. After these groups are constructed as combined-permutation representations, the calculation of the respective cycle indices with chirality fittingness (CI-CFs) and the introduction of ligand-inventory functions are conducted to give generation functions for 3D-based enumerations (for PGs and RS-SIGs) and 2D-based enumerations (for SIGs and ISGs). The enumeration results are discussed by means of isomer-classification diagrams, in which equivalence classes under enantiomerism (for PGs), RS-stereoisomerism (for RS-SIGs), stereoisomerism (for SIGs), and isoskeletomerism (for ISGs) are illustrated schematically. The implicit connotations of the conventional terms “skeletal isomerism”, “positional isomerism”, and “constitutional isomerism” are discussed, where the effects of the concept of isoskeletomerism are emphasized.


Group hierarchy Benzene Prismane Enumeration RS-Stereoisomeric group 


  1. 1.
    A.J. Ihde, The Development of Modern Chemistry (Dover, New York, 1984)Google Scholar
  2. 2.
    A.J. Rocke, It began with a daydream: the 150th anniversary of the Kekulé benzene structure. Angew. Chem. Int. Ed. 54, 46–50 (2015)CrossRefGoogle Scholar
  3. 3.
    A. Kekulé, Ueber einige Condensationsproducte des Aldehyds. Liebigs Ann. Chem. 162, 77–124 (1872)CrossRefGoogle Scholar
  4. 4.
    A. Ladenburg, Bemerkungen zur aromatischen Theorie. Chem. Ber. 2, 140–142 (1869)CrossRefGoogle Scholar
  5. 5.
    T.J. Katz, N. Acton, Synthesis of prismane. J. Am. Chem. Soc. 95, 2736–2739 (1973)CrossRefGoogle Scholar
  6. 6.
    J. Dewar, On the oxidation of phenyl alcohol and a mechanical arrangement adapted to illustrate structure in the non-saturated hydrocarbons. Proc. Royal Soc. Edinb. 6, 82–86 (1867)CrossRefGoogle Scholar
  7. 7.
    T.J. Katz, E.J. Wang, N. Acton, benzvalene synthesis. J. Am. Chem. Soc. 93, 3782–3783 (1971)CrossRefGoogle Scholar
  8. 8.
    M. Christl, Benzvalene-properties and synthetic potential. Angew. Chem. Int. Ed. 20, 529–546 (1981)CrossRefGoogle Scholar
  9. 9.
    F.A. Kekulé, Untersuchungen über aromatische Verbindungen. Liebigs Ann. Chem. 137, 129–196 (1866)CrossRefGoogle Scholar
  10. 10.
    W. Koerner, Studj sull’ismeria delle considette sostanze aromatiche sei atomi di carbonio. Gazz. Chim. Ital. 4, 305–446 (1874)Google Scholar
  11. 11.
    G. Pólya, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Math. 68, 145–254 (1937)CrossRefGoogle Scholar
  12. 12.
    G. Pólya, R.C. Read, Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds (Springer, New York, 1987)CrossRefGoogle Scholar
  13. 13.
    G. Pólya, Un problème combinatoire général sur les groupes de permutations et le calcul du nombre des isomères des composés organiques. Compt. Rend. 201, 1167–1169 (1935)Google Scholar
  14. 14.
    G. Pólya, Tabelle der Isomerenzahlen für die einfacheren Derivate einiger cyclischen Stammkörper. Helv. Chim. Acta 19, 22–24 (1936)CrossRefGoogle Scholar
  15. 15.
    G. Pólya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen. Z. Kristal. (A) 93, 415–443 (1936)Google Scholar
  16. 16.
    G. Pólya, R.E. Tarjan, D.R. Woods, Notes on Introductory Combinatorics (Birkhäuser, Boston, 1983)CrossRefGoogle Scholar
  17. 17.
    S. Fujita, Sphericities of cycles. What Pólya’s theorem is deficient in for stereoisomer enumeration. Croat. Chem. Acta 79, 411–427 (2006)Google Scholar
  18. 18.
    S.J. Cyvin, I. Gutman, Kekulé Structures in Benzenoid Hydrocarbons (Springer, Berlin, 1988). Lecture Note in Chemistry Vol. 46CrossRefGoogle Scholar
  19. 19.
    I. Gutman, D. Vukicević, A. Graovac, M. Randić, Algebraic Kekulé structures of benzenoid hydrocarbons. J. Chemİnf. Comput. Sci. 44, 296–299 (2004)CrossRefGoogle Scholar
  20. 20.
    S. Fujita, The restricted-subduced-cycle-index (RSCI) method for counting matchings of graphs and its application to Z-counting polynomials and the Hosoya index as well as to matching polynomials. Bull. Chem. Soc. Jpn. 85, 439–449 (2012)CrossRefGoogle Scholar
  21. 21.
    F. Cataldo, A. Graovac, O. Ori (eds.), The Mathematics and Topology of Fullerenes, vol. 4 (Springer, Amsterdam, 2011)Google Scholar
  22. 22.
    M.V. Diudea, Energetics of multi-shell clusters, in Multi-shell Polyhedral Clusters, vol. 10, Carbon Materials: Chemistry and Physics, ed. by M.V. Diudea (Springer, Berlin, 2018), pp. 385–438CrossRefGoogle Scholar
  23. 23.
    S. Fujita, The restricted-subduced-cycle-index (RSCI) method for symmetry-itemized enumeration of Kekulé structures and its application to fullerene C\(_{60}\). Bull. Chem. Soc. Jpn. 85, 282–304 (2012)CrossRefGoogle Scholar
  24. 24.
    S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry (Springer, Berlin, 1991)CrossRefGoogle Scholar
  25. 25.
    S. Fujita, Diagrammatical Approach to Molecular Symmetry and Enumeration of Stereoisomers (University of Kragujevac, Faculty of Science, Kragujevac, 2007)Google Scholar
  26. 26.
    S. Fujita, Benzene derivatives with achiral and chiral substituents and relevant derivatives derived from \({ D}_{6h}\)-skeletons. Symmetry-itemized enumeration and symmetry characterization by the unit-subduced-cycle-index approach. J. Chem. Inf. Comput. Sci. 39, 151–163 (1999)CrossRefGoogle Scholar
  27. 27.
    S. Fujita, Chirality fittingness of an orbit governed by a coset representation. Integration of point-group and permutation-group theories to treat local chirality and prochirality. J. Am. Chem. Soc. 112, 3390–3397 (1990)CrossRefGoogle Scholar
  28. 28.
    S. Fujita, Diagrammatical Approach to Molecular Symmetry and Enumeration of Stereoisomers (University of Kragujevac, Faculty of Science, Kragujevac, 2013)Google Scholar
  29. 29.
    S. Fujita, Mathematical Stereochemistry (De Gruyter, Berlin, 2015)CrossRefGoogle Scholar
  30. 30.
    S. Fujita, Stereogenicity revisited. Proposal of holantimers for comprehending the relationship between stereogenicity and chirality. J. Org. Chem. 69, 3158–3165 (2004)CrossRefPubMedGoogle Scholar
  31. 31.
    S. Fujita, Pseudoasymmetry, stereogenicity, and the RS-nomenclature comprehended by the concepts of holantimers and stereoisograms. Tetrahedron 60, 11629–11638 (2004)CrossRefGoogle Scholar
  32. 32.
    S. Fujita, Chirality and RS-stereogenicity as two kinds of handedness. Their aufheben by Fujita’s stereoisogram approach for giving new insights into classification of isomers. Bull. Chem. Soc. Jpn. 89, 987–1017 (2016)CrossRefGoogle Scholar
  33. 33.
    S. Fujita, Conceptual defects of modern stereochemistry. Comprehensive remedy by stereoisograms based on RS-stereoisomerism for mediating between enantiomerism and stereoisomerism. Tetrahedron: Asymmetry 28, 1–33 (2017)CrossRefGoogle Scholar
  34. 34.
    S. Fujita, Misleading classification of isomers and stereoisomers in organic chemistry. Bull. Chem. Soc. Jpn. 87, 1367–1378 (2014)CrossRefGoogle Scholar
  35. 35.
    S. Fujita, Classification of stereoisomers. Flowcharts without and with the intermediate concept of RS-stereoisomers for mediating between enantiomers and stereoisomers. Tetrahedron: Asymmetry 27, 43–62 (2016)CrossRefGoogle Scholar
  36. 36.
    S. Fujita, Computer-oriented representations of point groups and cycle indices with chirality fittingness (CI-CFs) calculated by the GAP system. Enumeration of three-dimensional structures of ligancy 4 by Fujita’s proligand method. MATCH Commun. Math. Comput. Chem. 76, 379–400 (2016)Google Scholar
  37. 37.
    S. Fujita, Computer-oriented representations of O\(_{h}\)-skeletons for supporting combinatorial enumeration by Fujita’s proligand method. Gap calculation of cycle indices with chirality fittingness (CI-CFs). MATCH Commun. Math. Comput. Chem. 77, 409–442 (2017)Google Scholar
  38. 38.
    S. Fujita, Computer-oriented representations of RS-stereoisomeric groups and cycle indices with chirality fittingness (CI-CFs) calculated by the GAP system. Enumeration of RS-stereoisomers by Fujita’s proligand method. MATCH Commun. Math. Comput. Chem. 77, 443–478 (2017)Google Scholar
  39. 39.
    S. Fujita, Graphs to chemical structures 1. Sphericity indices of cycles for stereochemical extension of Pólya’s theorem. Theor. Chem. Acc. 113, 73–79 (2005)CrossRefGoogle Scholar
  40. 40.
    S. Fujita, Graphs to chemical structures 2. Extended sphericity indices of cycles for stereochemical extension of Pólya’s coronas. Theor. Chem. Acc. 113, 80–86 (2005)CrossRefGoogle Scholar
  41. 41.
    S. Fujita, Graphs to chemical structures 3. General theorems with the use of different sets of sphericity indices for combinatorial enumeration of nonrigid stereoisomers. Theor. Chem. Acc. 115, 37–53 (2006)CrossRefGoogle Scholar
  42. 42.
    E.L. Eliel, Stereochemical non-equivalence of ligands and faces (heterotopicity). J. Chem. Educ. 57, 52–55 (1980)CrossRefGoogle Scholar
  43. 43.
    H. Hirschmann, K.R. Hanson, The differentiation of stereoheterotopic groups. Eur. J. Biochem. 22, 301–309 (1971)CrossRefPubMedGoogle Scholar
  44. 44.
    E.L. Eliel, Prostereoisomerism (prochirality). Top. Curr. Chem. 105, 1–76 (1982)CrossRefGoogle Scholar
  45. 45.
    A. von Zelewsky, Stereochemistry of Coordination Compounds (Wiley, Chichester, 1996)Google Scholar
  46. 46.
    G. Gunawardena, The Elements of Organic Chemistry: A Compendium of Terminology, Definitions, and Concepts for the Biginner, Linus Learning (New York, 2015); Website: OChemPal,
  47. 47.
    M. Ooki, T. Oosawa, M. Tanaka, H. Chihara (eds.), Kagaku Dai-Jiten (Encyclopedic Dictionary of Chemistry) (Tokyo Kagaku Dojin, Tokyo, 1989)Google Scholar
  48. 48.
    I.U.P.A.C., Organic chemistry division, basic terminology of stereochemistry (IUPAC recommendations). Pure Appl. Chem. 68(1996), 2193–2222 (1996)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Shonan Institute of Chemoinformatics and Mathematical ChemistryAshigara-Kami-GunJapan

Personalised recommendations