Clar Theory Extended for Polyacenes and Beyond Debojit Bhattacharya,† Anirban Panda,†,‡ Anirban Misra,§ and Douglas J. Klein*,† †

MARS, Texas A&M University at Galveston, Galveston, Texas 77553, United States Department of Chemistry, J. K. College-Purulia, Post & District - Purulia, 723101, West Bengal, India § Department of Chemistry, University of North Bengal, Darjeeling, 734013, West Bengal, India ‡

ABSTRACT: An extension of Clar’s classical sextet ideas is presented to allow resonance-based weak-pairing long bonds. As the prototypic illustration, the theory is developed in the context of polyacenes, where this extension is needed to properly understand what goes on in this sort of polymer, whose radicality increases with chain length. A quantification of these novel Clar-sextic ideas is made, and detailed computational results are reported for the polyacenes even to the limit of arbitrary long chains. Resonance energies, bond lengths, and local (ring) aromaticity indices are addressed. It is emphasized that weak pairing is not at all unique to the polyacenes, but also applies whenever there are suitable boundaries (say of the “zig-zag” type) on general grapheneic structures thereby readily explaining novel features of different boundaries.

INTRODUCTION Clar’s ideas on the “aromatic sextet” have long been recognized, at least in organic chemical circles, with a seminal presentation available in Clar’s charming booklet.1 Clar’s description there was quite successful in rationalizing (reactivities, optical spectra, NMR shifts, and more), but was framed at a qualitative level. The novel mathematical aspects of his “Clar structures” were subsequently widely pursued in the chemical graph theory community, but mostly at a formal level. Also often an experimental result or a computational result has been observed to correlate in a qualitative fashion with Clar’s ideas. However, relatively little was done in the way of chemically attentive quantification of Clar’s ideas, although in the mid-1980s Herndon and Hosoya2 (successfully) quantified these ideas to generate quantitative resonance energies for ordinary benzenoids. This involved a matrix diagonalization over all the Clar structures, whose number diverges (exponentially) for general structures, but there was no follow-up work. Only more recently we have quantified3 Clar’s ideas in a different way (avoiding the matrix diagonalization) to deal with a range of chemical properties: bond lengths, resonance energies, and local (ring) aromaticities. In this work, the close relation of Clar’s ideas to resonating VB theory was noted, so as to make quantifications parallel to what has been proved4,5 successful for Pauling− Wheland resonance theory. Indeed continuing to utilize this resonance-theoretic connection, we went on to extend6 Clar’s ideas to the case of benzenoid radicals, at least of the type where the degree of the radical is given by the mismatch between the number of “starred” and “unstarred” sites. That is, this dealt with systems where the π -centers of the σ -bonded carbon network are alternant, which is to say a system where the sites partition into starred and unstarred subsets such that the members of one set have neighbors only in the other set6 - and it was further © 2014 American Chemical Society

assumed that there was no significant tendency toward unpairing beyond that dictated by any mismatch of these starred and unstarred sites. But in fact, even with an overall balance between “starred” and “unstarred” sites, there are a variety of cases where there is a tendency to weak pairing, with the proto-typic example being the polyacenes, which become increasingly radicaloid with increasing chain length. Another case of recent interest is found with graphene boundaries that are of a subtly unbalanced type (e.g., “zig-zag” boundaries), which is to say the boundaries are locally unbalanced, with all degree-2 sites on one boundary being of the same type (starred or unstarred) and having a deficit of local-pairing possibilities (compared to what is available in the interior of a graphene sheet). In the present work, in tune with our previous works, we further extend the Clar theoretic approach to deal with radicaloid species which manifest more unpaired (or very weakly paired) electrons than is accounted for by the simple mismatch between starred and unstarred sites. As an interesting illustrative case, we seek to treat the polyacenes that have long been a staple of consideration for theorists, with the first treatments starting more than 6 decades ago,7−10 continuing on through11 the 1980s to the present times,12,13 and there is a sizable body of experimental works as well.14 A polyacene oligomer is a quasi-onedimensional molecule composed of a chain of fused benzene rings. However, though manifesting (fully neighbor-paired) Kekule structures, a twist to the story is that the polyacenes with a number of benzene units, N ≥ 7 are experimentally unstable. In the present case, the chosen N-acenes for our theoretical treatment can be considered to extend to a hypothetical infinite Received: March 4, 2014 Revised: May 16, 2014 Published: May 19, 2014 4325 | J. Phys. Chem. A 2014, 118, 4325−4338

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length. For N-acenes with N ≥ 7, there is a notable propensity for these species with increasing length to be radicaloid, highly colored, and very susceptible to oxidation. Thus, we focus on the likelihood of weakened singlet pairing, due to long bonds. Again this is done in terms of a quantification of Clar’s ideas extended in consonance with resonance theory to allow the possibility of this incipient radicaloid behavior−generalizing our earlier results to deal with this type of radical, now with matching numbers of starred and unstarred sites. Incidental to the extensions to deal with weak pairing, a neat computational scheme is described here to deal with the electronic-structure functions (Clar 3-nomials) that arise in our quantification. This scheme is that of the “transfer matrix” originating in explicit form in Montroll’s seminal paper,15 but brought to fame in Onsager’s solution to the two-dimensional Ising model, 16 in which context it has become often described.17,18 More recently, this general technique has been argued19 to be widely applicable in a general chemical graphtheoretic context (such as we have here), and even considered20 in the context of Clar structures. A broad overview of the present work, along with our previous two articles,3,6 is that they comprise steps toward a general quantification of classically based electronic structure ideas so as to enable description of modern conjugated-carbon nanostructures−in terms of an electron-pairing density function P realized as a polynomial over the molecular graph G, with variables representing activities for the different possible pairing substructures on G. That such an approach should be successful is indicated by (1) the qualitative success of Clar’s theory,1,21 as is also viewable22 as a refinement so as to reach accord with different resonance-theoretic ideas;4,5,23−25 (2) the success of our earlier quantifications3,6 for several different properties, for conventional stable benzenoids as well as a subclass of radicals; and (3) some indications26 of broad qualitative consiliences of our proposed Clar-theoretic approach, in the context of a very wide range of conjugated-carbon nanosturctures (including radicals, polymers, graphene, decorations thereon, and even antiaromatic and nonalternant species, such as fullerenes, and beyond). We present our quantitative description for polyacenes, and give tests of it against resonance energies, bond lengths, and NMR chemical shifts. The quantitative methodology is shown to extend to the high polymer limit. And following this we indicate the qualitative relevance of our extended Clar-theoretic ideas for “zig-zag-boundary” graphene or graphene nanoribbons.

Figure 1. Clar structures for benzene, naphthalene, and anthracene.

centage of rings identified to aromatic sextets in a Clar structure, the greater the importance of that structure, and the greater its contribution to the stability of the molecule. Among benzenoids, the polyacenes are rather special in having just one aromatic sextet in any Clar structure - unless any unusual non-neighbor pairing were to occur. Such a single aromatic sextet may occur in any one of the rings. With an increase in the length of a polyacene, the percentage of Clar sextets diminishes and so does the experimentally observed stability of the polyacene. To understand the enhancement of radicaloid behavior with increasing length, it is appropriate to allow nominally unpaired electrons, or better long-bonds between non-neighbor atoms. Especially for singlet biradicals, long-bond structures are relevant. For benzene, then one has the five resonance structures of Figure 2: two Kekule structures and three (long-bond) “Dewar”

Figure 2. Five resonance structures of benzene.

structures. However, in fact it is appropriate to think of the Clar sextet for benzene as shown in Figure 1 to incorporate all these, albeit4 with relatively small fixed optimal contributions from each of the long-bond structures. This notably makes clear strong contact with Pauling−Wheland resonance theory4,5 (though Clar1 himself makes little reference to this). Hence, ordinarily it is as a first refinement sufficient to account just for long bonds only between next−next nearest neighbor sites, as this is what can arise most readily (via quantum chemical “configuration interaction”) from what is otherwise a fully neighbor-paired (Clar) structure, understanding that those involving a rearrangement of bonding entirely within a single hexagon (as in Figure 2) are already implicitly incorporated in a Clar-sextet. That is, the longbond structures we wish to account for typically arise from a short-bond structure as indicated in Figure 3. Then, for instance, for anthracene, we display in Figure 4 the complete collection of such extended Clar-sextet structures with no more than a single long next next neighbor bond, without those as in Figure 2 already understood to be incorporated in the Clar sextet. In the first column, the three Clar structures with no long bonds are shown; while in the later columns those with one long bond appear. The second column consists of structures having long bonds within the first ring. In the next column we represent those with long bonds between the first and second ring. The next two columns consist of long bond only within the second ring and in between second and third ring, while the last column contains structures having long bond only within the last ring of the anthracene. Sometimes a long bond destroys the possibility of a Clar sextet, while in other cases it does not, and yet further in one case the presence of a long bond allows two Clar sextets (as at the top of the fourth column in Figure 4). Thus, at the expense of a

QUALITATIVE PREVIEW Perhaps it is worthwhile first to present a qualitative rationale for the behavior of polyacenes. Clar’s ideas were predicated on valence structures, with conjugated 6-cycles viewed as a unit and indicated by a circle enclosed in the hexagon of the (benzenoid) structure under consideration. Thence a Clar structure is a classical electronic structure with such circles, called aromatic sextets, supplemented with ordinary π -bonds not enclosable in any further such sextet circle. Thus, there is a single Clar structure for benzene, there are two Clar structures for naphthalene, and three for anthracene as indicated in Figure 1. There the arrows are to be viewed to indicate that the sextet may be moved to this adjacent ring, which is to say that the so moved sextet represents another structure with which the parent structure may readily interact. In general Clar emphasized that the higher the per4326 | J. Phys. Chem. A 2014, 118, 4325−4338

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Figure 3. Long-bond structure formed from a more dominant indicated short-bond structure.

Figure 4. Complete collection of such extended Clar-sextet structures with no more than a single long bond, for the anthracene molecule.

sextets, so that such a triplet state manifests a related stabilization, and higher-membered polyacenes tend toward biradicality. A further point is that to properly approach the asymptotic large-N limit, one should allow for multiple long bonds. This is to say that were one to limit to no more than a single long bond, then this would in essence be postulating (an extraordinary) long-range correlation where the presence of a long bond at one location somehow precludes any other long bonds anywhere else. Thus, we should allow multiple long bonds - but with ever decreasing likelihood, if we imagine that their likelihood of occurring at different positions is near independent. Note that with ever more vertical long bonds, ever more Clar sextets can arise. Such double long-bond structures admitting three Clar sextets are shown for pentacene and hexacene in Figure 6. To summarize: our description should be “size consistent”.

long-bond, an extra Clar sextet has been obtained. While this is not extremely important for anthracene, it becomes ever more important for the longer polyacenes, because the number of these double-sextet structures increases rapidly−basically involving more and more places to put the second Clar-sextet. If the two Clar sextets are well separated, there will be more ways to enter a long bond between the two sextets. As a consequence, the long-bond structures become more important as they allow many more Clar structures with more Clar sextets, which if they are numerous enough, provide a resonance stabilization which out-strips the energy cost of the loss of a short-bond (for a longbond). That is, the combined stabilization of the second Clar sextet and the resonance among these double-Clar-sextet structures ends up making these double-Clar-sextet long-bond structures more important−at least for longer polyacenes. At a minimum level, we then surmise a need for vertical long bonds (allowing a second Clar sextet), as shown for tetracene in Figure 5. A nonvertical long bond within a simple ring does not lead to any additional Clar-sextet, and in fact as seen in Figure 4 leads to the elimination of a Clar sextet. Thus, here we neglect nonvertical longbonds. At the same time a long-bond is much easier to break and replace by a triplet spin-pairing, while still allowing multiple Clar

Figure 6. Double long-bond structures admitting three Clar sextets for pentacene and hexacene.

GROUND-STATE SINGLET-PAIRING QUANTIFICATION First, we seek to describe in suitable detail the ground-state electronic structure in terms of singlet-pairing with the inclusion

Figure 5. Tetracene’s extended Clar structures with no more than a single long bond. 4327 | J. Phys. Chem. A 2014, 118, 4325−4338

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of the effects of long bonds in describing our molecular π -network identified by a graph G. For this quantified description, we build on our earlier work6 utilizing a type of Clar 3-nomial:

with A1 and A2 stabilizing parameters and A3 destabilizing, since A1, A2, and A3 respectively represent energetic contributions for Clar sextets for their neighbor interaction and long bonds (so that A3 is opposite in sign to A1 and A2). This energy is understood to be referenced against what the energy would be without resonance. Moreover, we may anticipate that A1 and A2 take values similar to those used in earlier works.3,6


P(G ; x , y , z) ≡

∑ x s (C)ya (C)zl (C) G





Here x, y, and z are parameters indicating activities for different respective substructures: Clar sextets, mode of interaction between related Clar structures, and weakly paired electrons. The sum goes over all our extended Clar structures C, with sG(C) being the number of Clar sextets in C, while aG(C) is the number of rings with 2 π-bonds immediately adjacent to a Clar sextet ring in C (as occurs in each of the ordinary Clar structures of naphthalene), and lG(C) is the number of our next−next neighbor vertical long bonds. An allowance for multiple long bonds in disparate parts of a large molecule are naturally anticipated to approach independence, so that if a single longbond occurs with probability p, the probability for a pair of (separated) long bonds should be ≈ p2, which is very small compared to p if p itself is small (compared to 1). However, for a very large system with ∼N places to put the second long-bond, the collection of double-long-bond structures can exceed in importance the single-long-bond structures in the sense that Np2 ≫ p, which is to say that the double-long-bond structures “dominate” in some sense.27 The arrow count aG(C) is seen to be the number of positions into which a Clar sextet can move with a minimum of rearrangement of bonds; that is, it concerns the “mobility” of Clar sextets, perhaps better phrased to say it involves the extent of immediate interaction with other Clar structures (namely those with the considered Clar sextet moved into one of these neighboring rings). The variables x, y, and z gauge the importance of the different types of substructures; as before3,6 we use x = 2 and y = 1. Our previous Clar-theoretic work6 on radicals was for those where there were strictly unpaired electrons as a consequence of the mismatch between starred and unstarred sites, with z in the 3-nomial there counting such unpaired electrons. Here if we imagine each long-bond has 2 non-neighbor-paired electrons at each end of the bond, these should in some sense be weakly paired or nearly unpaired, so that the current z -weight for each long bond should be z2. As a first guess, then we take our current z = 0.4. We also require that there is no more than 1 long-bond to any of the sites of any hexagonal ring of G. The possible patterns of connection for the first 6 sites of the first ring of a polyacene then are as indicated in Figure 7. The extension

MOLECULAR DESCRIPTORS The computation of many graph-theoretic enumerations and polynomials such as the Clar 3-nomials can be neatly formulated16 for a polymeric species, especially if regular, with the same patterns repeating, as indeed applies with our current polyacenes. Basically, granted a possible subpattern for a Clar 3-nomial in a ring, one looks at the different possible ways it can propagate to the next ring (Figure 8). This recursive process

Figure 8. Propagation pattern of polyacenes from one ring to its adjacent ring. Associated weights are indicated below each substructure.

Table 1. Clar Polynomials for Different N-Acenesa species

Clar polynomials


1 2 3 4

benzene naphthalene anthracene tetracene

x 2xy + 2xz2 + z4 2xy + xy2 + 4xyz2 + x2z2 + 3xz4 + z6 2xy + 2xy2 + 4xyz2 + 2xy2z2 + 4x2yz2 + 6xyz4 + 3x2z4+ 4xz6 + z8

0.942 1.447 1.616 1.700


The bold portions of the Clar polynomials are the portions for molecular benzenoids reported in ref 3. The notations x and y bear the same meanings as used in ref 3. However, z is used to account for the contribution from Dewar benzene structures. bTaken from ref 3.

is explicated in Appendix A. One can correlate Clar-nominal in Table 1 with Figures 4 and 5 (for anthracene and tetracene) taking up to the z2 terms of the series, that is, taking only the vertical Dewar structures and neglecting the nonvertical long bonds in those figures. The Clar 3-nomials for the first few polyacenes are shown in Table 1, and, as can be seen, they rather rapidly increase in complexity, but the transfer-matrix method entails manipulations that, in our present case, involve our simple 4 × 4 transfer matrix, first for our Clar 3-nomial, then subsequently to yield different substructural expectations. From the preceding discussion it should be clear that the polynomial in eq 1 elegantly encodes Clar-structure aspects, identified as important by Clar, via three counting aspects: (1) the number of Clar sextets (using x); (2) sextet “mobilities” (by use of y); and (3) the long bond (next−next nearest neighbor)

Figure 7. Different extended Clar-substructural features for a terminal ring of a benzenoid, with in fact the pattern fully determined just for the first four vertices of this ring (excepting the first structure where the presence of a Clar-sextet dictates the state of the next two sites). Respective initiating weights are x, 1, and z2.

of the patterns to a full polyacene is done via a convenient “transfermatrix” technique as demonstrated in the next section; this allows one to treat a general-length polyacene, just attending to which patterns can follow from one ring to the next. The resonance energy for the associated singlet state is ⎛ ∂ ∂ ∂ ⎞ + A2 + A3 E ≡ ⎜A1 ⎟ln P(x , y , z) ∂ ln y ∂ ln z ⎠ ⎝ ∂ ln x


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propensity (using z). From another perspective, the Clar polynomials, with assigned values for parameters x,y,z, can be viewed as analogous to the Kekule structure count K(G) used in concurrence with ordinary Pauling−Wheland resonance theory.4,5 That is, P(x,y,z) is a weighted count of (extended) Clar-structure. Going into more detail of the “sextetness”, “mobility”, and the “long-bond propensity” of the Clar sextet, one can easily formulate corresponding contents (or expectations) ⟨s⟩G = =

RE in terms of the average counts of conjugated circuits of different size within the collection of Kekule valence structures. In a more recent DFT based computational work Schleyer et al. computed RE of polyacenes using a homodesmatic reaction33 scheme. In conjunction with our previous work,3 we construct a modified scheme (and equation) for Clar resonance energy in parallel with conjugated circuit theory. The equation is given by CRE(G) = A1⟨s⟩G + A 2 ⟨a⟩G + A3⟨l⟩G

∂ ln PG(x , y , z) ∂ ln x G

where A1 and A2 are numerical constants. Granted the parallel of the present scheme to conjugated circuit theory, the typical values of these constants should be less than 1.0 eV. The Clar resonance energy of N-acenes for N = 1, 2, 3, 4, 5 has been determined using eq 4 for x = 2.0, y = 1.0, and z = 0.4. This choice of x, y, z might conceivably be based on the Pauling−Wheland resonance-theoretic argument.4,5 Nevertheless, we retain these values throughout this article, especially with x and y coming from our previous works.3,6 The values of the parameters in eq 4 can be obtained through a least-squares fitting (in kcal/mol unit) as

∑ sG(C)x s (C)ya (C)zl (C)/PG(x , y , z) G




⟨a⟩G =

∂ ln PG(x , y , z) ∂ ln y G


∑ aG(C)x s (C)ya (C)zl (C)/PG(x , y , z) G




⟨l⟩G = =

∂ ln PG(x , y , z) ∂ ln z G

A1 = 21.414, A 2 = 11.897, and A3 = − 0.761, G


The fitting has been done against the calculated RE values reported by Dewar and de Llano (DdLRE)34 for N = 2−5. The correlation between estimated Clar resonance energies following our treatment with the calculated resonance energy is depicted in Figure 9. The agreement is seen to be excellent.


C sG(C)



with σ = 0.695 and r = 0.993

∑ lG(C)x s (C)ya (C)zl (C)/PG(x , y , z) G



The weight factor x y z /PG(x,y,z) can be viewed as the probability for the occurrence of a Clar structure C with sG sextets, aG arrows and lG long-bonds. Thus, ⟨s⟩G can be described as the average number of aromatic sextets, or simply the sextetness. Similarly, ⟨a⟩G is the average number of mobility-indicating arrows, and may be termed Clar mobility. The quantity ⟨l⟩G is the average number of long-bonds or “long-bondedness” (or “weakpairedness”). These and other expectations may be neatly evaluated in the framework of the transfer-matrix method; each expectation entails a further matrix, which typically may be viewed as a derivative of the basic transfer matrix−as explicated in Appendix B. Perhaps a few words on a body of earlier chemical graph theoretic work may be noted. In an earlier work, Hosoya and coworkers28 defined (nonradical) Clar polynomials based on a completely different representation, and their concept was further studied mathematically in later publications by He et al.29 El-Basil30 also studied (nonradical) cata-condensed all-kink chains by taking similar polynomials with y = 1 and later extended his idea for all benzenoids. Also, Zhang et al. have dealt with this idea in a formal manner.31 However, none of these studies (and several others, e.g., mentioned elsewhere3) made quantitative comparisons to experiment, and furthermore, none of them have considered derivative invariants (as in eq 3). The y and z terms, the mean structure contents (ref 14), and our comparisons to experiment (in subsequent sections) make our current work quite different from these previous works.

Figure 9. Comparison of calculated Dewar−de Llano Resonance Energies (DdlRE) (kcal/mol) and Clar resonance energies (CRE) (kcal/mol) estimated from our formalism. The symbols ▲,■, ⧫, and ● represent polyacenes with N = 2, 3, 4, and 5, respectively.

BOND-ORDER AND BOND LENGTH Besides global invariants (for the molecule as a whole), local invariants (bond orders, bond lengths, etc.) for different parts of the molecule are also of interest. Following our previous work,3 one writes the Clar bond order as

RESONANCE ENERGY CORRELATION The resonance energy (RE) of an aromatic compound can be defined in various ways depending on the methods of energy determination involved. According to Pauling−Wheland resonance theory, the resonance energy of a benzenoid compound is a standard measure of the extra stabilization energy of the aromatic system compared to what should occur with the corresponding number of localized double bonds.4,5,15 Alternatively, Randić and Trinajstić32 define aromaticity in terms of presence and absence of conjugated circuits. They also represent


CBOG (e) ≡

∑ C

{ε (C , e) + 12 η (C , e)}x G

/PG(x , y , z)


sG(C) aG(C) lG(C)




where εG(C, e) is 1 or 0 based on whether edge e is or is not a double bond in C. Similarly, ηG(C, e) is 1 or 0 as e is or is not in an aromatic sextet of C. The rationale for this expression is that, for a given edge e of graph G, its occurrence as a double bond of Clar structure contributes fully to a π bond order, whereas its 4329 | J. Phys. Chem. A 2014, 118, 4325−4338

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occurrence in Clar sextet contributes only 1/2 and its presence as long bonds does not contribute at all. Our neat computational scheme for these (local) bond orders is described in Appendix B. The increase of bond length from the carbon−carbon double bond and at the same time decrease of bond length from the carbon−carbon single bond is a measure of aromaticity in aromatic systems. It is known that the Clar bond length (CBL) for a bond e = {i, j} can be expressed as CBL(em) = B1 + B2 {CBO(em)} + B3 / didj

Table 2. Experimental Bond Length, Calculated (Using UB3LYP/6-311+g(d,p)) Bond Length, and Clar Bond Length of Chosen Bonds in Polyacenes




experimental bond length (Å)

calculated bond length (Å)



a0 b0 c0 a′0 a0 b0 c0 a′0 b1 a0 b0 c0 a′0 b1 c1 a1 a0 b0 c0 a′0 b1 c1 a1 b2

1.407 1.371 1.422 1.420 1.418 1.353 1.428 1.432 1.395 1.479a 1.385a 1.431a 1.439a 1.398a 1.409a 1.475a 1.441b 1.358b 1.428b 1.453b 1.381b 1.409b 1.464b 1.396b

1.415 1.374 1.419 1.432 1.425 1.367 1.429 1.443 1.399 1.430 1.364 1.433 1.450 1.390 1.409 1.450 1.432 1.363 1.435 1.454 1.387 1.414 1.455 1.400

1.418 1.369 1.437 1.410 1.424 1.363 1.445 1.430 1.409 1.425 1.362 1.447 1.436 1.400 1.421 1.442 1.425 1.361 1.448 1.438 1.398 1.425 1.446 1.414


where B1, B2, and B3 are parameters, and di and dj are the (π-network) degrees of the two π-centers i and j of the graph G. The third term introduced in eq 7 takes care of the fuctionalities of the carbons involved, which makes it successful in making corrections for a wide variety of properties. As an example for the present study, we have taken naphthalene (N = 2), anthracene (N = 3), tetracene (N = 4), and pentacene (N = 5) having different types of bonds as marked in Figure 10. The experimental bond lengths for naphthalene and




From ref 53. bFrom ref 54.

B1 = 1.511, B2 = − 0.138, and B3 = − 0.098, (8)

with s = 0.018 and r = 0.830

The numerical values of bond lengths calculated using the Gaussian program have been reported in Table 2. A similar correlation with CBL yields (in Å units) Figure 10. Numbering scheme for different polyacene rings followed in our work.

B1 = 1.527, B2 = − 0.118, and B3 = − 0.158, (9)

with s = 0.011 and r = 0.929

The quality of these fits seems acceptable.

anthracene have been collected from Kiralji and Ferreira.35 At the same time, bond lengths were calculated using the Gaussian 09 suite of programs.36 Moreover, our calculations are done using unrestricted B3LYP density functional theory with 6-31G(d,p) basis set. A least-squares fitting of eq 7 with the experimental values yields (in Å unit) is more or less a decent fit (Figure 11) for

LOCAL AROMATICITY INDICES A discussion of local aromaticity is due here. We adopt a similar treatment as in our previous work for quantifying local aromiticity,3 for a six-cycle ring X of our (polyacene) graph G. Our indices are given by G

CAIG(m) ≡

∑ sG(C , m)x s (C)ya (C)zl (C)/PG(x , y , z) G




CAI′G (m) ≡

∑ aG(C , m)x s (C)ya (C)zl (C)/PG(x , y , z), G





where sG(C,m) is 1 or 0 depending on whether or not a Clar sextet C occupies X. Also, aG(C,m) counts the number of naphthalene substructures containing X ⊆ C along with the two other double bonds of naphthalene contained in C. Here CAI′G plays the role of a secondary aromaticity index arising from the migration of the sextet to a neighboring ring. Here aG(C,m) is

Figure 11. Plot of Clar bond length versus experimental bond length in Å units. 4330 | J. Phys. Chem. A 2014, 118, 4325−4338

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either 0 or 1. Also these local invariants are related to the global invariants through ∑GmSG(C,m) = SG(C) and ∑GmaG(C,m) = aG(C), following G

⟨s⟩G =

∑ CAIG(m) X G

⟨a⟩G =

∑ CAI′G (m) X


Hence one can think of expressing local aromaticity of any ring of the polyacene by a linear combination of sextetness (CAIG(m)) and sextet mobility (CAI′G(m)) for that particular ring. Therefore, the local part of the Clar aromaticity (CAG) index of a graph G for a ring m can be written as

Figure 12. Plot of HOMA values versus calculated Clar aromaticity index. The ▲ represents HOMA values obtained from experimental data and ● represents HOMA obtained from “Gaussian” calculated data.

{local part of CA G(m)} = C1 × CAIG(m) + C2 × CAI′G(m)

Table 3. HOMA Values Based on Experimental Structural Data, Optimized Geometry Obtained from “Gaussian” Calculations and HOMA Estimated from Clar Bond Length of Different Ring Fragments of Polyacenes


A third term can be added to eq 12a for nonlocal contributions, and the resulting expression can be used to correlate with the other well-established aromaticity index values, say, the harmonic oscillator model of aromaticity (HOMA) values or the nucleus independent chemical shift (NICS) values. As described in ref 3, this nonlocal (spin-polarization) effect is accounted for with a three-parameter expression

molecule naphthalene anthracene tetracene

1 CA G(m) = C1·CAIG(m) + C2·CAI′G(m) + C3 1 + (nm /6)



where nm is the number of benzene rings which are adjacent to ring m. The relevance for introducing the third term can be understood3 when one fits the aromaticity index against the nonlocal contributions from the neighboring sites. A method for calculating CAI and CAI′ is discussed in Appendix B. The parameters C1, C2, and C3 in eq 12b can be determined by comparing CAs of different rings of polyacenes with HOMA values of that particular site. In this work, we take four polyacenes with N = 2 to N = 5 and compare CAI (eq 12b) with the HOMA values obtained from experimental35 bond lengths. A similar kind of comparison can be done by taking HOMA value obtained from calculated (using Gaussian package program) bond lengths. A least-squares fit with “experimental” HOMA values yields C1 = 1.284, C2 = 0.988, and C3 = −0.297 with σ = 0.098 and r = 0.788, while a fit using “calculated” HOMA values yields C1= 0.616, C2 = 0.601 and C3= 0.211 with σ = 0.073 and r = 0.633. These two are only marginal fits for such a correlation coefficient and is also seen from Figure 12. In both these cases, the C1 term dominates over the C2 term, however, the ratio C1 and C2 is against our anticipated value of 1/2. Moreover, there is a significant contribution from the third term that is indicative of nonlocal contributions from the neighboring rings of the polyacenes. Alternatively, one can estimate the Clar HOMA values of the individual ring fragments of the polyacene rather directly from the Clar bond lengths (obtained using eq 7) associated with the particular ring fragment. Different Clar HOMA values obtained from various methods are reported in Table 3. A similar kind of correlation between the Clar aromaticity index and NICS37 values of the polyacenes can be established. As the nonlocal effect (influence of neighboring rings) on the NICS values are more pronounced than those in the case of HOMA values, it is expected that proposed correlation between Clar and NICS values will be to some extent less convincing.3

ring number (m)

experimental HOMA

calculated HOMA


0 0 1 0 1 0 1 2

0.816 0.636 0.825 0.373 0.517 0.483 0.528 0.493

0.786 0.629 0.719 0.536 0.632 0.486 0.562 0.590

0.703 0.536 0.773 0.485 0.670 0.462 0.622 0.595

A relatively poor fit for aromaticity indices might be attributed to the various methods of defining numerical aromaticity indices, either local or global. In one school of thought,15,38 one describes aromaticity as a “partial ordering” rather than a strict numerical ordering. In general, aromaticity indices are described as the deviations of a physical property (e.g., energy, geometry, magnetic properties, polarizibilities, reactivities, etc.) from those expected of a nonaromatized (bond-localized) reference. The anamolies in these different properties reflect peculiarities in the electronic wave function, and there seems little reason to expect such wave function anomaly to be well described by a single number (an aromaticity index). Then, naturally, the aromaticity ordering based on a certain physical property deviation might not follow the same trend followed by some different physical property. Rather, aromaticity should better be viewed to satisfy general conditions of “partial ordering”. Thus, generally, aromaticity should be represented not as a single number but as a sequence of numbers, the minimum number of which is linearly independent being viewable as a “dimensionality” of the partial ordering. If aromaticity in terms of different physical properties can be expressed as a linear combination of functions determining the different members of the sequence, then the dimensionality devolves to that of a linear space and eventually relates to the concept of multidimensional aromaticity of Katrizky et al.39 The multidimensional nature of aromaticity has been addressed by many authors40 and points to the fact that aromaticity is difficult to be quantified by a single descriptor, which may be also the origin of the problem discussed here.

DISCUSSION The results here enable a better understanding of polyacenes at both a qualitative and a quantitative level based on Clar’s 4331 | J. Phys. Chem. A 2014, 118, 4325−4338

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Figure 13. Two widths w = 6 and w = 7 of zigzag-edge graphene nanoribbons, when a maximum of Clar sextets appear in the interior; hence there is unpairing at the boundaries.

qualitatively illustrated ideas1 (which were primarily for nonradicaloid species). In our previous communication, some notable3 quantification occurred for just the very early (nonradicaloid) polyacenes. There the main focus was on stable molecular benzenoids at first, and then6 it was extended for a particular class of radical benzenoids. All throughout several different molecular properties have been treated quantitatively. Moreover, a broad overview23 has been made especially for large conjugated carbon nanostructures, and the polyacenes were touched upon qualitatively. In the present work, we quantitatively extend the Clar-theoretic treatment, for arbitrarily long linear polyacenes, utilizing a powerful transfer-matrix method, which readily extends to the N → ∞ limit, notably avoiding the matrix diagonalization of exponentially ever-increasing sizes, as involved in the quantification of Herndon and Hosoya.2 This general polyacenic method entails consideration of next−next nearest neighbor long-bond interactions. The fittings we obtain for various physical properties are comparable with some earlier fits, especially remarkably well for the resonance energy, and allow a general global understanding of the polyacenes. Moreover, like our previous treatments, our present work also is based on a unified “Clar-theoretic” viewpoint, long hugely successful in a qualitative fashion. Overall, we followed the interpretation and quantification of Clar’s ideas done in our previous article with a modification of Clar-2-nomial by incorporation of another activity z considering “propensity” for next−next nearest neighbor long-bond formation. As in our previous treatment, the present study is also in consonance with Pauling−Wheland resonance-theoretic ideas. We retain the (0-order) values x = 2 and y = 1 from our earlier work and introduced z = 0.4. This particular value of z was determined using some trial variation of z, keeping other parameters (x and y) fixed. It is expected that our overall idea could be applied to other properties than those discussed here

(e.g., reactivities, UV−visible spectra, magnetic properties, electric properties, etc.). Besides testing our Clar-theoretic quantification against other quantitative results, the readiness of its extension to the highpolymer limit may also be noted. The transfer-matrix technique applied here for polyacenes readily extends to even very large N-acenes. Particularly asymptotic (N → ∞) results are computable in terms of the maximum eigenvalue Λ and associated left (Λ(l)| and right |Λ(r)) eigenvectors−as indicated in the last of the equations (eq A.8 in Appendix A). So doing this, for the limit of very long polyacene chains, we find lim E /N = 3.913 kcal/mol

N →∞


(Here there are different methods of finding (Λ(l)| and |Λ(r)), with the simple power-method very readily implementable). Moreover, as is complicit in our many-body picture, the extent of radicality continually increases with chain length from nonradical, to diradical, on to tetraradical, and ultimately beyond. This can be seen via expectation values for weak-paired longbonds ⟨l⟩N =

∂ N ∂Λ log P(x , y , z) → · ∂log z Λ ∂z


(following the ideas of eq A.3 in Appendix A). One may note that when z = 0, ⟨S⟩N = 1, 1 ≤ ⟨a⟩N ≤ 2 and ⟨l⟩N = 0, where E/N → 0, which is a qualitative difference without the long-bond pairing, the resonance energy per site is →0. That is, for size-extensive expectations, one needs size-consistent structure function P(G;x,y,z), as per the discussion in our argument in the “Qualitative Preview” section. 4332 | J. Phys. Chem. A 2014, 118, 4325−4338

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Figure 14. An “arm-chair-boundary” graphene nanoribbon, with full pairing indicated, when the maximum number of Clar sextets are drawn in the interior of the strip.

Figure 15. A “hairy-boundary” graphene nanoribbon, with (more extensive) unpairing along the two boundaries.

BEYOND POLYACENES TO GRAPHENES A natural question concerns systems beyond the polyacenes. For instance, there has been experimental41,42 and theoretical43−47 work to obtain nanoribbons cut from graphene, and these systems might be considered in view of Clar-theoretic ideas, now with our extension allowing weak pairing. If the ribbons have socalled “zig-zag” boundaries as indicated in Figure 13, then as we see in this figure, a high density of Clar sextets in the interior of a ribbon leads to “unpaired” electrons at the boundaries; that is, each boundary in itself manifests a type of imbalance of “starred” and “unstarred” sites: all the degree-2 sites (with fewer neighborpairing options) are of the same type (starred on one boundary and unstarred on the other), hence this leads to longer (weaker) pairings. Thus, we realize that the unpaired electrons on the two boundaries of the strip are respectively found on “starred” and “unstarred” sites, and they should in fact be weakly paired, indeed very weakly paired, even for strips less wide than those shown. An interesting point implicit in Figure 13 is that the unpairing localization pattern for the two sides of the strip is differently phased relative to one another, depending on whether the strip width w is even or odd. Also it may be seen that there is a net of one unpaired electron for every three unit cells of edge. Further, it may be seen that such ideas extend to the limit of very wide strips, which is to say for boundaries on bulk graphene. Such results have indeed been found43 from detailed quantumchemical computations,43,45 and yet further there is solid experimental evidence42 to match the same effect (though here the extent of unpairing of one electron per three unit cells is not at all precisely determined). It is to be emphasized that this is easily anticipated either from a resonance-theoretic view43 or from our (current) Clar-theoretic viewpoint.26 Without the consideration44 of weak-pairing in either of these views, the “unpaired” (nonbonding) electrons are missed. The result for

very wide strips as a representation of graphene has been made several times,43,45,46 again in agreement with experiment.48,49 It is a nontrivial feature that our extended Clar theory immediately predicts much the same as revealed in detailed quantum computations and recent experiments. The extension to allow weak-pairing is crucial. Yet further, this consonance extends to other related predictions for other types of strips, with different boundaries. For instance, consider the “arm-chair” strips of Figure 14. The prediction is that there are no unpaired electrons, in agreement with detailed computations43,46,50 and experiment.48,49 Moreover, for this arm-chair case of the indicated width, it is seen that the boundary tends to lock in the locationing of the Clar sextets, so that there should be notable bond localization. Further note that this bond localization should not be so severe for arm-chair ribbons of 1 greater or 1 lesser hexagon in width, since then the Clar sextets tend to delocalize. For the “hairy” edges of Figure 15, the prediction is that there are two unpaired electrons per three unit cells of edge, which is twice the density for the “zig-zag” boundary, and also is in agreement with quantum computations,43,51 and experiment.49 All these cases (of this section) address occurrences along a single boundary which is translationally symmetric and well separated from other boundaries. For this case we can distill the underlying feature to its simplest form: weak pairing (or local unpairing) occurs with an absence of local balance between starred and unstarred sites of corresponding degree deficits (from the maximum degree of 3). To be more explicit, given a translationally symmetric boundary, let the number of degree d starred sites per unit cell of edge be denoted #d*, and, correspondingly, let #do the number of unstarred ones. Then the number of unpaired electrons per unit cell of edge is u = |(#2 * + 2#1 *) − (#2o + 2#1o)| /3 4333

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Thus, for the zig-zag edge, this predicts u = |(1 + 0) − (0 + 0)|/3 = 1/3 or one unpaired electron per three unit cells, just as already seen from Figure 13. For the arm-chair boundary, this gives u = |(1 + 0) − (1 + 0)|/3 = 0, in agreement with Figure 14. And for the hairy edge, u = |(0 + 2·1) − (0 + 0)|/3 = 2/3, in agreement with Figure 15. There is further validation43,52 in terms of simple tight-binding computations for about two dozen further edges.


∂P(x , y , z) = (∂end|T N − 1|start) +

(∂T) ·Tm − 1|start) + (end|T N − 1|∂start) (A.4)

Here the first term is 0, because of the constantive form of (end|, and the matrix ∂T, termed a connection matrix, typically ends up appearing even simpler than T itself. An eigen-resolution of T is illuminating. That is, if T is diagonalizable, with right and left biorthogonalized eigenvectors, as

CONCLUSION The utility of quantitative Clar-sextet treatments in various systems from molecules to polymers and onto grapheneic species has been illustrated. Starting with Clar,1 qualitative correlation has long been noted over a wide range of species (especially for nonextended benzenoid molecules), and quantitative correlation is here illustrated for the polyacene sequence, while further we find a consonance with observed (and computed) results for a few types of extended graphene ribbons as well as for what happens at the boundaries on semi-infinite graphene. In our previous article6 Clar theory was extended (in a quantitative format) to molecular radicals, arising from the overall imbalance of starred and unstarred sites, while here Clar theory is further extended to deal with local imbalances, as in fact occur with numerous nanostructures. Thence the idea of the utilization of Clar-sextet theory in a very wide context for (hexagon-rich) conjugated-carbon nanostructures seems promising. This offers the possibility of facile understandings26 (and qualitative predictions) which are further neatly quantifiable, as evidenced here (and elsewhere3,6). We hope to further address the quantitative estimates based on the Clar-theoretic ideas.

T|λ(r)) = λ|λ(r)), (λ(l)|T = λ(λ(l)| , (λ(l)|μ(r)) = δ(λ , μ) (A.5)

then P(x , y , z) =


∂P(x , y , z) =

μm − 1(μ(l)|start) +

∑ (end|λ(r))·λ N − 1·λ(l) λ



In the limit of long chains, the maximum eigenvalue Λ dominates, so that P(x , y , z) → (end|Λ(r))· (Λ(l)|start) ·ΛN − 1 ∂P(x , y , z) → (end|Λ(r))·(Λ(l)|start) ·(N − 1) · (Λ(l)|∂T|Λ(r)) ·ΛN − 2 E → (N − 1)(Λ(l)|(A1∂xT + A 2 ∂yT + A3∂zT)|Λ(r))/Λ (A.8)

For the total energy there are finite corrections (vanishing for the per-cell energy computed as E/N in the N → ∞ limit.). One needs to properly deal with these additional finite corrections if one wishes to get the singlet−triplet splitting (which we do not pursue here).

APPENDIX B: EXPECTATIONS FOR GLOBAL AND LOCAL DESCRIPTORS Clearly the expectations of eq 3 are given as


⟨s⟩G = ∂xPG(x , y , z)/PG(x , y , z)

As a consequence, one sees that T|i) gives the appropriately weighted sum of the substructures |j) following from |i). Moreover, the overall weight for an overall extended Clar structure is a product of such weights, so that for the N-acene N−1



⟨a⟩G = ∂yPG(x , y , z)/PG(x , y , z) ⟨l⟩G = ∂zPG(x , y , z)/PG(x , y , z).


⎛x⎞ ⎜0⎟ |start) = ⎜ 2 ⎟ ⎜z ⎟ ⎜ ⎟ ⎝1⎠


with the derivatives given via eq A.4. Asymptotically, they are given via



(end| = (1 1 1 0)

∑ ∑ (end|λ(r))·λ N − 1 − m·(λ(l)|∂T|μ(r))· λ,μ m=1


P(x , y , z) = (end|T


Moreover, various derivatives (say with respect to x, or y, or z, may be represented as

APPENDIX A: TRANSFER MATRIX FOR POLYACENEIC CLAR 3-NOMIAL Granted a subgraphical combinatoric computation on a repetitive polymeric structure, one can focus on how the (weighted) subgraphic features propagate from one unit of the polymer to the next. This is done in Figure 8, with the different associated contributions to the overall Clar-structure weight also indicated. We think of state vectors associated with each of these substructures: |i) for the ith substructure, i = 1,2,3,4. The propagation data of Figure 8 are then conveniently summarized in terms of a transfer matrix x xy ⎞ ⎟ 1 0 0⎟ ⎟ z2 z2 0 ⎟ ⎟ 0 1 1⎠

∑ (end|λ(r))·λ N − 1·(λ(l)|start) λ

⎛0 ⎜ ⎜y T=⎜ 2 ⎜⎜ z ⎝0

∑ (end|T N − 1 − m· m=1

⟨s⟩G → (N − 1)(Λ(l)|∂xT|Λ(r))/Λ ⟨a⟩G → (N − 1)(Λ(l)|∂yT|Λ(r))/Λ ⟨l⟩G → (N − 1)(Λ(l)|∂zT|Λ(r))/Λ (A.3)


Local invariants as needed for the (local) bond orders are accessible via a similar approach. These bond orders CBOG(e) can be easily computed by transfer-matrix technique. To achieve

To compute energies, one also needs derivatives of P(x,y,z). To this end, note that 4334 | J. Phys. Chem. A 2014, 118, 4325−4338

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where 1m is ⎧ = (end|T N − 1|∂ start); m = 0, ∀ X X0 ⎪ ⎪ ⎨ 1m = N−m−1 (∂XT)m Tm − 1|start); ∀ m( ⎪ = (end|T ⎪ ≠ 0), ∀ X , ⎩

Figure B.1. Schematic representation of the numbering scheme followed for assigning numbers at different positions of different bond lengths of the polyacene chain.

Using eq B.3 and B.4 in eq 6 one can easily estimate CBO of any bond em of the graph G, and in turn may be used to estimate other local invariants like Clar bond length and local aromaticity index. Similarly, using

this end, one needs to consider the transfer pattern of different substructures preceding and following the mth ring of the polyacene, which contains the particular bond em (= a, a′, b, b′, c, and c′ as in Figure B.1) keeping the bonding pattern unaltered for all other rings of C. The transfer pattern and the associated polynomials can be obtained using the connection matrices which are as follows: ⎛ 0 0 0 ⎜ 0 0 0 ⎜ ∂emTm = ⎜ 2 2 2 zδ z δm , N − 1 z δm , N − 1 ⎜⎜ m , N − 1 ⎝ 0 0 0 ⎛ 0 ⎞ ⎜ ⎟ ⎜ 0 ⎟ =⎜ 2 for (e = a , a′), zδ ⎟ ⎜⎜ a , a ′⎟⎟ ⎝ 0 ⎠ ⎛0 ⎜ 0 ∂emTm = ⎜ ⎜0 ⎜ ⎝0 ⎛0 ⎜ y ∂emT = ⎜ ⎜0 ⎜ ⎝0

0 0 0 0

0 0 0 1

CAIG(m) = 1m/PG(x , y , z) ⎛ ∂1 ⎞ CAI′G(m) = ⎜y m ⎟ /PG(x , y , z) ⎝ ∂y ⎠

0⎞ ⎟ 0⎟ and |∂e0start) 0⎟ ⎟⎟ 0⎠

in eqs 10 and 12b, one can calculate the Clar aromaticity CA.

*E-mail: [email protected]; phone: (409)740-4512. Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS D.B., A.P., and D.J.K. acknowledge the support from the Welch Foundation (via grant BD-0894), of Houston, Texas. AM thanks CSIR, India, for financial support. D.B. thanks the Managing Committee of Matikunda High School, Uttar Dinajpur, West Bengal, India, for sanctioning the leave for pursuing his post doctoral research in Texas A&M University at Galveston, TX, USA. A.P. also thanks the Governing Body of J. K. College, Purulia, India, for sanctioning the leave for pursuing postdoctoral research work in Texas A&M University at Galveston, TX.

0 0 0⎞ ⎛0⎞ ⎟ ⎜ ⎟ 1 0 0⎟ 0 and |∂e0start) = ⎜ ⎟ for (e = c , c′), ⎟ ⎜0⎟ 0 0 0 ⎜ ⎟ ⎟ ⎝0⎠ ⎠ 0 0 0


where m = 1, 2,..., N − 1, and N ≥ 2, and δm,N−1 = 1 (if m = N − 1) and 0 (if m ≠ N − 1). Also we designate polyacene fragment with m = 0 as “start” in all the above relations and the relations hereafter. The above connectivity matrices in eq B.3 is related to the expression for CBOG (em) in eq 6 by the following relations: G

= (end|T N − 1|∂emstart); ∀ e , ∀ m = 0


= (end|T N − m − 1(∂emT)m T m − 1|start) ; ∀ e , ∀ m( ≠ 0) G

∑ ηG(C , em)x sG(C)yaG(C)zlG(C) C


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0 x xy ⎞ ⎛x⎞ ⎟ ⎜0⎟ 0 0 0⎟ and |∂ X 0start) = ⎜ ⎟ ⎜⎜ 0 ⎟⎟ 0 0 0⎟ ⎟ ⎝0⎠ 0 0 0⎠

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Corresponding Author

⎛0⎞ 0⎞ ⎟ ⎜ ⎟ 0⎟ 0 and |∂estart) = ⎜ ⎟ for (e = b , b′, ), ⎜0⎟ 0⎟ ⎜ ⎟ ⎟ ⎝1 ⎠ 1⎠

⎛0 ⎜ 0 (∂XT)m = ⎜ ⎜0 ⎜ ⎝0


= 1m + 1m + 1; m = 1, 2, 3··· , ∀ am( ≠ a0 , aN − 1), a′0 = 1m; m = 0, 1, 2, 3···, ∀ e( ≠ am = 1,2,3..N − 2 , a′0 )

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4338 | J. Phys. Chem. A 2014, 118, 4325−4338

Clar theory extended for polyacenes and beyond.

An extension of Clar's classical sextet ideas is presented to allow resonance-based weak-pairing long bonds. As the prototypic illustration, the theor...
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