Recursion Equations in Polycyclic Hydrocarbon Series and the Number of Dewar Long-Bond Resonance Structures with Applications

Basic recursion equations for characteristic and matching polynomials and the number of Dewar resonance structures for polycyclic polyene series are derived. Fibonacci-like series of numbers frequently appear in recursion and analytical expressions for determining resonance energy terms for polycyclic polyene series successively built up by given Aufbau units. Linear polycyclic polyene series built up from given Aufbau units usually give analytical expressions for determining their resonance energy terms. Polycyclic polyene series with all fixed pπ bonds (<b><i>K</i></b> = 1) have some aromatic stabilization energy as measured by topological resonance energy (TRE), which is explained by the appearance of sextet and larger aromatic circuits in some of their Dewar resonance structures (<i>DS</i>); note that <i>DS</i> can be read as singular (Dewar structure) or plural (Dewar structures) depending on the context. It is demonstrated that a finer evaluation of relative resonance energy and aromaticity requires the inclusion of both Dewar structures (<i>DS</i>) and Kekulé structures (<b><i>K</i></b>). In the valence-bond determination of bond lengths and aromaticity of polycyclic conjugated systems with fixed single and double bonds, the inclusion of Dewar resonance structures is required. Topological conjugation energy (TCE) for all series, whether they have all fixed pπ double bonds (<b><i>K</i></b> = 1) or numerous Kekulé resonance structures (<b><i>K</i></b> > 1), is very similar.