Strongly Subspectral Conjugated Molecular Systems. From Small Molecules to Infinitely Large π-Electronic Networks

The analysis of π-electronic structure of polymer networks continues to attract the interest of theoretical1 and synthetic2-7 researchers. One of the major objectives of these studies is to gain insight on the design of potential organic conductors and ferromagnets. Extended pπ-conjugated systems represent the simplest models for molecular wires. The reduction of the HOMO-LUMO bandgap in conjugated systems will enhance the thermal population of the conduction band and thus increase the number of charge carriers, and minimizing the degree of bond length alternation in a conjugated path which is the origin of Peierls instability will help preserve the proximity or degeneracy between the HOMO-LUMO electronic levels. Thus, researchers need to learn what structural variables control the HOMO-LUMO energy gap and bond length alternation in pπ-electronic systems before organic conductors can be designed. There is no such thing as a ferromagnetic molecule. However, if parallel spin alignment can be induced in molecules, like polymethyleneacetylene, while it is in the condensed phase, a paramagnetic (and possibly a ferromagnetic) material would be produced.2 Since electron spins tend to align antiparallel to one another, currently no organic ferromagnetic material has yet been designed and synthesized. Three general theoretical approaches can be identified: study of infinitely long strips, belt-shaped rings, and a series of strips that are progressively incremented (i.e., a homologous series). This latter approach follows more closely the way experimentalists are capable of studying very large molecules and is analogous to the standard methodology in which molecular systems are partitioned into smaller elementary substructures. For example, infinitely long polyacene strips and cyclic polyacene rings have been frequent objects of theoretical study, but are experimentally unknown, whereas members of the homologous acene series from naphthalene to heptacene are known to progressively decrease in stability and ease of experimental manipulation. Nevertheless, these experimental results for smaller homologues along with theoretical molecular modeling studies1 suggest that polyacenes will be conductive and reactive materials but not ferromagnetic.8 By assuming infinitely long polyacenes can be modeled as infinite belt-shaped rings ([∞]cyclacenes), its irreducible substructure can be obtained.9,10 Hosoya and co-workers have shown that the density of states of cyclic and linear polyene systems having common repetitive units are identical in the infinite limit. Also, they showed that the singular points of the density of states of infinitely large polyenes correspond to the energy levels present in the cyclic dimer (or cyclic monomer in some cases).10 By a totally different approach, we showed8 that successive incremental embedding of infinite strips frequently can be used to recognize the existence of a valence band to conduction band energy gap (HOMO-LUMO gap) after just a few iterations. Successive incremental embedding on an infinitely long conjugated polymer strip consists of embedding with successively larger homologous fragments. These approaches suggest that infinite polyacene should have a zero bandgap. In the model analysis of large finite polyenes by infinite analogue systems, a number of ambiguities arise, particularly in regard to bond alternation. Consider the use of the cyclic boundary condition in the analysis of linear polymeric polyenes. For finite, planar, monocyclic 4n+2 polyenes, like benzene, there is no tendency toward bond alternation, whereas for finite linear polyenes, like butadiene and hexatriene, significant bond alternation occurs. From resonance theory, monocyclic polyenes with 4n+2 pπ-electrons have two Kekulé structures (K ) 2) and linear polyenes have one (K ) 1). In other words, bond alternation is a linear polyene “end effect” which is assumed to disappear at infinite length. Let us now consider the use of cyclic acene rings in the analysis of linear acenes. Regardless of the number hexagonal rings in the cyclic acene ring K ) 4, whereas for linear acenes K ) #rings + 1. The number of Kekulé structures in the former can be visualized as arising from two cyclic polyenes joined together via every other carbon, and the number of Kekulé structures in the latter can increase without limit. Given this aforementioned, it appears somewhat amazing that both infinite polyene ring systems and linear systems made of the same repeating unit converge to the same density of states. This result suggests that aromaticity disappears in extended π-electronic systems.11 In this paper, the analysis of numerous pairs of series composed of strongly subspectral molecular graphs is presented and some of the aspects discussed above will be further augmented and clarified. An analytical framework for graphitic polymers is established by the pairwise construction of infinite two-dimensional arrays whereby almost-isospectral pairs of molecular graphs are correlated.