Fuzzy symmetries of two classes of linear polyacene molecules

The fuzzy symmetries of two kinds of linear polyacene molecules are probed into in the paper. In these molecules, any one of the benzene rings abreast connects at most other two rings in two ways: either its two opposite C–C bonds combine with two other rings, respectively, or its two meta-position C–C bonds connect two rings in cis- and trans-form, respectively. The former is called p-polyacenes (or straight polyacenes), and the latter is m-polyacenes (or kinked ones). It can be thought as the planar molecule with approximate one-dimensional space periodic transformation (parallel translation) symmetry, namely, group G₁²{{\rm G}_{1}^{2}} symmetry, when the number of its benzene ring is very large; on the other hand, it can be considered as the fuzzy group G₁²{{\rm G}_{1}^{2}} symmetry, if the benzene ring number is not large enough. The p-polyacene and m-polyacene with 20 benzene rings are analyzed as typical examples, and the energies of the π-molecular orbital (MO) and the fuzzy symmetry characters related to the space symmetry transformations are carefully examined. Moreover, the π-MOs of the p-polyacenes and m-polyacenes with different numbers of benzene ring are investigated to obtain the related rules.