How do the polycyclic aromatic hydrocarbons approach infinity?

Various topological factors governing the electronic properties of infinitely large periodic polycyclic benzenoids are analyzed graph-theoretically by drawing the density of states (DS). The existence or non-existence of NBMO's in the hypothetical cyclic dimer of the network are shown to be crucial for the profile of DS. CATAHEXES AND PERIHEXES Owing to the rapid progress in organic synthesis the number of experimentally accessible polycyclic aromatic hydrocarbons is ever growing (1,2). On the other hand, existence of a group of large polycyclic aromatic hydrocarbons is reported both in interstellar materials and soot ( 3 ) . Existence of ballshaped C60 molecule and its analogues is also reported and postulated (4,5). In principle, a given polycyclic aromatic compound belongs to a certain series of molecules which, at least mathematically, grow and converge to an infinitely large network with varying properties. In this talk let us confine ourselves to benzenoid hydrocarbons. Several series of those hydrocarbons are known to have interesting electronic and thermodynamic properties potentially applied to new materials, e.g., of highly conductive, semiconductive, or ferro-magnetic property. The relationship between the stability and topological structure of benzenoid hydrocarbons seems to be well analyzed by many didfferent theoretical techniques, e.g., molecular orbital (MO), valence bond (VB), resonance theory (RT), graph theory (GT), etc. ( 6 11). The graph-theoretical MO (GTMO) analysis has.clarified the mathematical secret and thus the limitation of the so-called "Huckel's 4n+2 rule", and succeeded in extending this concept to polycyclic sysmtems (12). However, if we apply this methodology toward large 2-dimensional network converging to graphite, we shall soon be overwhelmed by the "combinatorial explosion". Benzenoid hydrocarbons are classified into catacondensed and pericondensed ones, and may be called catahexes and perihexes, respectively (6). From GT consideration a great gap is found to exist between these two types of graphs. Although for catahexes the structure-activity relationship is satisfactorily analyzed mathematically, a large number of interesting properties are left unsolved for perihexes ( 1 3 ) . The number of possible isomers of catahexes can be derived from the group-theoretical treatment by Polya and expressed in terms of a set of recursion formulas (14). However, for the counting of perihexes we are forced to choose computer searching (15). The useful concept of the aromatic sextet proposed by Clar needs to be modified as to propose the "super-sextet" when "fat" benzenoids are considered (1 6). The purpose of the present talk is to survey the effect of the topological structure, especially the peripheral structure, of infinitely large perihexes on their r-electronic stabilities through GTMO treatment. Before going into the main theme of this talk preliminary remark should be exposed on the topological dependency of the K(G) number and stability of catahexes. It is well known that K(G) of a benzenoid hydrocarbon increases with the number of kinks and branches. This feature is dramatically shown by quite the different mathematical expressions for the K(G) number of the linear (1) and zigzag ( 2 ) polyacenes.