Clar Theory for Molecular Benzenoids Anirban Misra,* ,†,‡ D. J. Klein, † and T. Morikawa § Texas A&M UniVersity at GalVeston, MARS, 5007 AVenue U, Texas 77551, Department of Chemistry, UniVersity of North Bengal, Darjeeling, PIN 734 013, West Bengal, India, and Department of Chemistry, Joetsu UniVersity of Education, Joetsu, Japan ReceiVed: May 2, 2008; ReVised Manuscript ReceiVed: December 1, 2008 Eric Clar’s ideas concerning “aromatic sextets” are extended to a quantitative format in terms of a polynomial called the “Clar 2-nomial”, along with related derivative quantities. The quantification is successfully tested to make correlations with a selection of numerical data, including resonance energies, bond lengths, and NICS ring-aromaticity values. Introduction About 70 years ago, Pauling and Wheland started a funda- mental development of resonance theory with a quantitative formulation that entailed diagonalization of ever larger matrices. Still, especially Pauling managed to cast the theory in a more convenient, more chemically manifest, less computational qualitative form as summarized in his masterwork 1 on The Nature of the Chemical Bond. Then also in a modestly more quantitative more purely organic focus, Wheland 2 wrote his Resonance in Organic Chemistry. However, starting about 60 years ago, there built up a shift to focus ever more toward molecular-orbital theory especially if quantitative conclusions were to be drawn. Still, almost 50 years ago Eric Clar began what may be viewed as a refinement of the simplified qualitative resonance theory, to focus on the efficacy of considering “aromatic sextets”, the accompanying notation, and its use to qualitatively correlate a wide range of molecular properties. In 1970, Clar brought this work to a conclusion with the publication of his short charming book 3 The Aromatic Sextet, where his ideas were illustrated for the range of benzenoids then available. Since then there have been dramatic developments of related novel conjugated species: benzenoid polymers (such as poly- para-phenylene), carbon nanotubes, fullerenes, and miscel- laneous other nanostructures (nanocones, nanotori, and hypo- thetical “negatively-curved” structures); and at the same time more focus on decorated (or defected) graphite has emerged, along with synthetic work on ever larger benzenoids, especially with Mu ¨llen’s group. 4 It has become not uncommon to note correlations of Clar’s formulations with various properties - but almost solely in a purely qualitative manner. Notably, Randic’s extensive review 5 of “aromaticity” emphasizes Clar’s qualitative ideas, where the author calls attention to the fact that Clar’s theory is an easy model that allows understanding and predicting many properties of benzenoid species without the need of complicated calculations. Here, then a quantitative formulation of Clar theory is sought, to apply not only to benzenoids considered by Clar but also to these various more recent novel conjugated nanostructures. Rather strangely over the decades since Clar’s, book surprisingly little quantitative work deriving from his ideas has been reported. Part of the reason for this may be found in Clar’s style of exposition - by example, with qualitative rationalizations, and without formal definitions or formal statements of principles. Thence to more plausibly discern Clar’s overall idea and intent, one needs to carefully look at his book as a whole. One type of approach to quantification is to seek to parallel Pauling and Wheland’s early work in setting up Hamiltonian matrices on a basis of the structures Clar used (rather than the Kekule structures of Pauling-Wheland resonance theory). Indeed this has been nicely done by Herndon and Hosoya, 6 to find quite favorable quantitative correlations with resonance energies otherwise elaborately computed via quantum chemical packages. Moreover, it has been emphasized 7 how Clar’s ideas can be seen to directly motivate the (quantitative) “conjugated-circuits” resonance-energy formalism, which however is based on Kekule structures. But in fact such quantum-theoretic rationalization seems to be fairly foreign to Clar’s style. Nevertheless, one would surmise that a more direct quantification of Clar’s ideas should certainly be successful because of this thoroughly founded empirical basis of Clar’s development. Here, we seek to use “Clar structures,” and a few invariants derived directly therefrom to correlate to molecular properties, without any explicit intervention of Hamiltonian matrices or Kekule structures. Moreover, beyond resonance energies we look at bond lengths as well as local aromaticity indices as given from the nuclear independent chemical shift (NICS) values. Indeed, a favorable comparison of NICS aromaticities with Clar’s ideas has been recently noted by Ruiz-Morales 8 and Gutman and Ruiz-Morales 9 in terms of their Y-rule. The focus here on conventional benzenoids is to lay the appropriate quantitative groundwork for next dealing with the great variety of related novel nanostructures. A conjugated polyhex or benzenoid network B is viewed as a graph, consisting of its sites and edges between the pairs of (σ-bonded) sites. A Clar structure C of B may be viewed as a condensation of Kekule structures into aromatic sextets insofar as possible. More formally one considers C to be a substructure of B such that every site is included in C such as to be either paired to exactly one other site or else included in an aromatic sextet (a 6-cycle of B). The set of pairings (or double bonds) and aromatic sextets are all to be disjointed from one another, and done so as to saturate all sites such that among the pairings no triple occurs around a hexagon of B, this rather being recast * Corresponding author fax: +913532581212; e-mail: anirbanmisra@ yahoo.com. † Texas A&M University at Galveston. ‡ University of North Bengal. § Joetsu University of Education. J. Phys. Chem. A 2009, 113, 1151–1158 1151 10.1021/jp8038797 CCC: $40.75  2009 American Chemical Society Published on Web 01/08/2009