Symmetry of Kekulé structures

The symmetry of Kekulé structures for aromatic hydrocarbons is studied by group theory. The general problem of deducing the distribution over irreducible representations (Λ_Kek) or characters of the representation based on the Kekulé structures (χ_Kek) has not been solved. A partial solution is given for two classes of molecules, namely (a) the “straight chain” aromatics (polyacenes): naphthalene, anthracene, naphthacene, etc., and (b) the “zig-zag chain” aromatics: phenanthrene, chrysene, picene, etc. As a part of this solution the number of Kekulé structures (K) in the two cases was found to be (a) K = Q + 1 and (b) K = F_Q+1, respectively. Here Q is the number of benzene rings in the molecule in question, and F_i denotes the i-th member of the Fibonacci series. Symmetrical structures (Λ_Kek) or characters (χ_Kek) are given for a number of additional molecules as examples: benzene (D_6h), tetraphene (C_s), benzoc] phenanthrene (C_2v), pyrene (D_2h), triphenylene (D_3h), perylene (D_2h), pentaphene (C_2v), dibenzophenanthrene (C_2v), heptaphene (C_2v) and coronene (D_6h). Here the appropriate symmetry groups are given in parentheses.