FRACTAL RELATIONS - MODELING FRACTALS AND ANALOGY WITH FEIGENBAUM PERIOD-DOUBLING DIAGRAM

The equivalence relation, l, defined previously on Kekule spaces of be nzenoid hydrocarbons (S. El-Basil, J. Chem. Soc., Faraday Trans., 89 ( 1993) 909; J. Mol. Struct. (Theochem), 288 (1993) 67; J. Math Chem., 1 4 (1993) 305) is used to map Kekule spaces of benzenoid systems onto v arious 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 Kekule spaces of quasicrystalline benzenoids (defined by S. El-Bas il in J. Chem. Soc., Faraday Trans., 89 (1993) 909) exhibit a period-d oubling pattern which manifests itself through a full analogy with Fei genbaum'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 ''p ercolation'' process where larger clustering (of the Kekule space) occ urs (D. Stauffer, Introduction to Percolation Theory, Taylor and Franc is, London, 1985). Cases in which l percolates the Kekule space corres pond to benzenoid systems which are energetically more stable than tho se 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 l imit of the ratio of Clar counts to Kekule counts in a given homologou s series of benzenoid hydrocarbons as the molecule's length approaches infinity. This limit shows that with quasicrystalline benzenoids (bra nched and/or unbranched), the extent of data reduction using Clar stru ctures 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 large r, the computational scheme of Herndon and Hosoya becomes progressivel y less efficient. This scale factor has certain arithmetical propertie s which are in harmony with the ultraviolet spectra of the correspondi ng benzenoid systems.