Fractal relations. Modelling fractals and analogy with Feigenbaum's period-doubling diagram

Abstract

The equivalence relation, l, defined previously on Kekulé spaces of benzenoid hydrocarbons (S. El-Basil, J. Chem. Soc., Faraday Trans., 89 (1993) 909; J. Mol. Struct. (Theochem), 288 (1993) 67; J. Math Chem., 14 (1993) 305) is used to map Kekulé spaces of benzenoid systems onto various stages of deterministic fractals including the Cantor set, the Sierpinski triangle, the Sierpinski carpet, the Koch curve and the box fractal as well as certain stages of cellular automata. Furthermore, the Kekulé spaces of quasicrystalline benzenoids (defined by S. El-Basil in J. Chem. Soc., Faraday Trans., 89 (1993) 909) exhibit a period-doubling pattern which manifests itself through a full analogy with Feigenbaum's scaling theorem (M. Feigenbaum, J. Stat. Phys., 19 (1978) 25) including a bifurcation universality constant, a control parameter (of the logistic equation) and cycle-sizes (which can only be integral powers of two). It was found that in certain instances l is also a “ percolation” process where larger clustering (of the Kekulé space) occurs (D. Stauffer, Introduction to Percolation Theory, Taylor and Francis, London, 1985). Cases in which l percolates the Kekulé space correspond to benzenoid systems which are energetically more stable than those in which the resulting clusters are too small to allow percolation of the space by l. A fractal-like scale factor, s, is defined as the limit of the ratio of Clar counts to Kekulé counts in a given homologous series of benzenoid hydrocarbons as the molecule's length approaches infinity. This limit shows that with quasicrystalline benzenoids (branched and/or unbranched), the extent of data reduction using Clar structures is maximum (i.e. s = 0) and hence the use of such structures as a quantum-mechanical basis set in the method of Herndon and Hosoya (W.C. Herndon and H. Hosoya, Tetrahedron, 40 (1984) 3987) is most suited for this class of benzenoid systems. The linear acenes, however, lead to a value of s = 1 and hence, as the size of the molecule gets larger, the computational scheme of Herndon and Hosoya becomes progressively less efficient. This scale factor has certain arithmetical properties which are in harmony with the ultraviolet spectra of the corresponding benzenoid systems.