Journal of Mathematical Chemistry

, Volume 23, Issue 1–2, pp 197–227 | Cite as

Some aspects of the symmetry and topology of possible carbon allotrope structures

  • R.B. King


Elemental carbon has recently been shown to form molecular polyhedral allotropes known as fullerenes in addition to the familiar graphite and diamond known since antiquity. Such fullerenes contain polyhedral carbon cages in which all vertices have degree 3 and all faces are either pentagons or hexagons. All known fullerenes are found to satisfy the isolated pentagon rule (IPR) in which all pentagonal faces are completely surrounded by hexagons so that no two pentagonal faces share an edge. The smallest fullerene structures satisfying the IPR are the known truncated icosahedral C60 of I h symmetry and ellipsoidal C70 of D 5h symmetry. The multiple IPR isomers of families of larger fullerenes such as C76, C78, C82 and C84 can be classified into families related by the so-called pyracylene transformation based on the motion of two carbon atoms in a pyracylene unit containing two linked pentagons separated by two hexagons. Larger fullerenes with 3ν vertices can be generated from smaller fullerenes with ν vertices through a so‐called leapfrog transformation consisting of omnicapping followed by dualization. The energy levels of the bonding molecular orbitals of fullerenes having icosahedral symmetry and 60n 2 carbon atoms can be approximated by spherical harmonics. If fullerenes are regarded as constructed from carbon networks of positive curvature, the corresponding carbon allotropes constructed from carbon networks of negative curvature are the polymeric schwarzites. The negative curvature in schwarzites is introduced through heptagons or octagons of carbon atoms and the schwarzites are constructed by placing such carbon networks on minimal surfaces with negative Gaussian curvature, particularly the so-called P and D surfaces with local cubic symmetry. The smallest unit cell of a viable schwarzite structure having only hexagons and heptagons contains 168 carbon atoms and is constructed by applying a leapfrog transformation to a genus 3 figure containing 24 heptagons and 56 vertices described by the German mathematician Klein in the 19th century analogous to the construction of the C60 fullerene truncated icosahedron by applying a leapfrog transformation to the regular dodecahedron. Although this C168 schwarzite unit cell has local O h point group symmetry based on the cubic lattice of the D or P surface, its larger permutational symmetry group is the PSL(2,7) group of order 168 analogous to the icosahedral pure rotation group, I, of order 60 of the C60 fullerene considered as the isomorphous PSL(2,5) group. The schwarzites, which are still unknown experimentally, are predicted to be unusually low density forms of elemental carbon because of the pores generated by the infinite periodicity in three dimensions of the underlying minimal surfaces.


Fullerene High Occupied Molecular Orbital Carbon Allotrope Icosahedral Symmetry Transitive Permutation Group 
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  1. [1]
    S. Andersson, S.T. Hyde, K. Larsson and S. Lidin, Chem. Rev. 88 (1988) 221.CrossRefGoogle Scholar
  2. [2]
    S. Andersson, S.T. H yde and H.G. von Schnering, Z. Kristallograph. 168 (1984) 1.CrossRefGoogle Scholar
  3. [3]
    D. Bakowies, M. Bühl and W. Thiel, J. Am. Chem. Soc. 117 (1995) 10 113.CrossRefGoogle Scholar
  4. [4]
    W.E. Billups and M.A. Ciufolini, eds., Buckminsterfullerenes (VCH Publishers, New York, 1993).Google Scholar
  5. [5]
    D.A. Bochvar and E.G. Gal’pern, Dokl. Akad. Nauk SSSR 209 (1973) 610; Proc. Acad. Sci. USSR 209 (1973) 239.Google Scholar
  6. [6]
    J. Bokowski and J.M. Wills, Math. Intelligencer 10(1) (1988) 27.CrossRefGoogle Scholar
  7. [7]
    E. Bruton, Diamonds (NAG Press, London, 1970).Google Scholar
  8. [8]
    F. Chung, B. Kostant and S. Sternberg, in: Lie Theory and Geometry, eds. J.-L. Brylinski, R. Brylinski, V. Guillemin and V. Kac (Birkhäuser, Boston, 1994).Google Scholar
  9. [9]
    J. Cioslowski, Electronic Structure Calculations on Fullerenes and Their Derivatives (Oxford University Press, New York, 1995).Google Scholar
  10. [10]
    M.L. Cohen and V.H. Crespi, in: Buckminsterfullerenes, eds. W.E. Billups and M.A. Ciufolini (VCH Publishers, New York, 1993) pp. 197–200.Google Scholar
  11. [11]
    R.A. Davidson, Theoret. Chim. Acta 58 (1981) 193.CrossRefGoogle Scholar
  12. [12]
    L.E. Dickson, Modern Algebraic Theories (Sanborn, Chicago, 1930) chapter XIII.Google Scholar
  13. [13]
    F. Diederich and R.L. Whetten, Acc. Chem. Res. 25 (1992) 119.CrossRefGoogle Scholar
  14. [14]
    F. Diederich, R.L. Whetten, C. Thilgen, R. Ettl, I. Chao and M. Alvarez, Science 254 (1991) 1768.Google Scholar
  15. [15]
    V. Elser and R.C. Haddon, Nature 325 (1987) 792.CrossRefGoogle Scholar
  16. [16]
    W. Fischer and E. Koch, Acta Cryst. A 45 (1989) 166, 169, 485, 558, 726.CrossRefGoogle Scholar
  17. [17]
    P.W. Fowler, Chem. Phys. Lett. 131 (1986) 444.CrossRefGoogle Scholar
  18. [18]
    P.W. Fowler, J. Chem. Soc. Perkin II (1992) 145.Google Scholar
  19. [19]
    P.W. Fowler, D.E. Manolopoulos and R.P. Ryan, Chem. Comm. (1992) 408.Google Scholar
  20. [20]
    P.W. F owler and D.B. Redmond, Theoret. Chim. Acta 83 (1992) 367.CrossRefGoogle Scholar
  21. [21]
    T.L. Gilchrist and R.C. Storr, Organic Chemical Reactions and Orbital Symmetry (Cambridge University Press, Cambridge, 1972) p. 38.Google Scholar
  22. [22]
    M. Goldberg, Tohoku Math. J. 43 (1937) 104.Google Scholar
  23. [23]
    J. Gray, Math. Intelligencer 4 (1982) 59.Google Scholar
  24. [24]
    B. Grünbaum and T.S. Motzkin, Can. J. Math. 15 (1963) 744.Google Scholar
  25. [25]
    R.C. Haddon, Acc. Chem. Res. 25 (1992) 127.CrossRefGoogle Scholar
  26. [26]
    J.M. Hawkins, A. Meyer, T.A. Lewis, S.D. Loren and F.J. Hollander, Science 252 (1991) 312.Google Scholar
  27. [27]
    D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination (Chelsea Publishing, New York, 1952) pp. 183–204.Google Scholar
  28. [28]
    S.T. Hyde, Z. Kristallograph. 179 (1987) 53.CrossRefGoogle Scholar
  29. [29]
    S.T. Hyde, Acta Chem. Scand. 45 (1991) 860.CrossRefGoogle Scholar
  30. [30]
    S.T. Hyde and S. Andersson, Z. Kristallograph. 170 (1985) 225.CrossRefGoogle Scholar
  31. [31]
    S.T. Hyde and S. Andersson, Z. Kristallograph. 174 (1986) 225.CrossRefGoogle Scholar
  32. [32]
    D.E.H. Jones, New Scientist 35(519) (1966) 245.Google Scholar
  33. [33]
    R.B. King, Beyond the Quartic Equation (Birkhäuser, Boston, 1996).Google Scholar
  34. [34]
    R.B. King, J. Chem. Educ. 73 (1996) 993.CrossRefGoogle Scholar
  35. [35]
    R.B. King, J. Phys. Chem. 100 (1996) 15 096.Google Scholar
  36. [36]
    R.B. King, Mol. Phys. 92 (1997) 293.CrossRefGoogle Scholar
  37. [37]
    R.B. King and E.R. Canfield, Comput. Math. Appl. 24 (1992) 13.CrossRefGoogle Scholar
  38. [38]
    R.B. King and D.H. Rouvray, Theoret. Chim. Acta 69 (1986) 1.CrossRefGoogle Scholar
  39. [39]
    F. Klein, Math. Ann. 14 (1879) 428.CrossRefGoogle Scholar
  40. [40]
    F. Klein, Vorlesungen über das Ikosaeder (Teubner, Leipzig, 1884) part I, chapter 2.Google Scholar
  41. [41]
    F. Klein, Gesammelte Mathematischen Abhandlungen, Vol. 3 (Springer, Berlin, 1923) pp. 90–136.Google Scholar
  42. [42]
    D.J. Klein and X. Liu, J. Math. Chem. 11 (1992) 199.CrossRefGoogle Scholar
  43. [43]
    D.J. Klein and X. Liu, Int. J. Quantum Chem. Symp. 28 (1994) 501.CrossRefGoogle Scholar
  44. [44]
    B. Kostant, Proc. Natl. Acad. Sci. USA 91 (1994) 11 714.CrossRefGoogle Scholar
  45. [45]
    B. Kostant, Notices Amer. Math. Soc. 42 (1995) 959.Google Scholar
  46. [46]
    W. Krätschmer, L.D. Lamb, K. Fostiropoulos and D.R. Huffman, Nature 347 (1990) 35.CrossRefGoogle Scholar
  47. [47]
    H.W. Kroto, Angew. Chem. Int. Ed. 31 (1992) 111.CrossRefGoogle Scholar
  48. [48]
    H.W. Kroto, A.W. Allaf and S.P. Balm, Chem. Rev. 91 (1991) 1212.CrossRefGoogle Scholar
  49. [49]
    H.W. Kroto, J.R. Heath, S.C. O'Brien, R.F. Curl and R.E. Smalley, Nature 318 (1985) 62.CrossRefGoogle Scholar
  50. [50]
    T. Lenosky, X. Gonze, M. Teter and V. Elser, Nature 355 (1992) 333.CrossRefGoogle Scholar
  51. [51]
    X. Liu, T.G. Schmalz and D.J. Klein, Chem. Phys. Lett. 188 (1992) 550.CrossRefGoogle Scholar
  52. [52]
    A.L. Mackay and H. Terrones, Nature 352 (1991) 762.CrossRefGoogle Scholar
  53. [53]
    A.L. Mackay and H. Terrones, Phil. Trans. Roy. Soc. Lond. Ser. A 343 (1993) 113.Google Scholar
  54. [54]
    D.E. Manolopoulos, D.R. Woodall and P.W. Fowler, J. Chem. Soc. Faraday 88 (1992) 2427.CrossRefGoogle Scholar
  55. [55]
    C.L. Mantell, Carbon and Graphite Handbook (Interscience, New York, 1968).Google Scholar
  56. [56]
    E.R. Neovius, Bestimmung Zweier Spezieller Periodische Minimalflächen (J.C. Frenkel & Sohn, Helsinki, 1883).Google Scholar
  57. [57]
    M. O’Keeffe, G.B. Adams and O.F. Sankey, Phys. Rev. Lett. 68 (1992) 2325.CrossRefGoogle Scholar
  58. [58]
    E. Osawa, Kagaku (Kyoto) 25 (1970) 854; Chem. Abstr. 74 (1971) 75 698v.Google Scholar
  59. [59]
    W.N. Reynolds, Physical Properties of Graphite (Elsevier, Amsterdam, 1968).Google Scholar
  60. [60]
    T.G. Schmalz, W.A. Seitz, D.J. Klein and G.E. Hite, J. Am. Chem. Soc. 110 (1988) 1113.CrossRefGoogle Scholar
  61. [61]
    H.A. Schwarz, Gesammelte Mathematische Abhandlungen (Springer, Berlin, 1890).Google Scholar
  62. [62]
    P.W. Stephens, L. Mihaly, P.L. Lee, R.L. Whetten, S.-M. Huang, R.F. Kaner, F. Diederich and K. Holczer, Nature 351 (1991) 632.CrossRefGoogle Scholar
  63. [63]
    A.J. Stone, Mol. Phys. 41 (1980) 1339.CrossRefGoogle Scholar
  64. [64]
    A.J. Stone, Inorg. Chem. 20 (1981) 563.CrossRefGoogle Scholar
  65. [65]
    A.J. Stone, Polyhedron 3 (1984) 1299.CrossRefGoogle Scholar
  66. [66]
    A.J. Stone and M.J. Alderton, Inorg. Chem. 21 (1982) 2297.CrossRefGoogle Scholar
  67. [67]
    A.J. Stone and D.J. Wales, Chem. Phys. Lett. 128 (1986) 501.CrossRefGoogle Scholar
  68. [68]
    A.C. Tang, F.Q. Huang, Q.S. Li and R.Z. Liu, Chem. Phys. Lett. 227 (1994) 579.CrossRefGoogle Scholar
  69. [69]
    D. Vanderbilt and J. Tersoff, Phys. Rev. Lett. 68 (1992) 511.CrossRefGoogle Scholar
  70. [70]
    Z. Yoshida and E. Osawa, Aromaticity (Kagakudojin, Kyoto, 1971) pp. 174–178.Google Scholar

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© Kluwer Academic Publishers 1998

Authors and Affiliations

  • R.B. King
    • 1
  1. 1.Department of ChemistryUniversity of GeorgiaAthensUSA E-mail:

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