J. Phys. Chem. A 2011, 115, 351–356
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Topological Ring Currents in the “Empty” Ring of Benzo-Annelated Perylenes Timothy K. Dickens*,† and Roger B. Mallion‡ UniVersity Chemical Laboratory, UniVersity of Cambridge, Lensfield Road, Cambridge, CB2 1EW, England, United Kingdom, and Ingram Building, School of Physical Sciences, UniVersity of Kent, Canterbury, CT2 7NH, England, United Kingdom ReceiVed: October 7, 2010
Cyclic conjugation in benzo-annelated perylenes is examined by means of the topological π-electron ring currents calculated for each of their constituent rings, in a study that is an exact analogy of a recent investigation by Gutman et al. based on energy-effect values for the corresponding rings in each of these structures. “Classical” approaches, such as Kekule´ structures, Clar “sextet” formulas, and circuits of conjugation, predict that the central ring in perylene is “empty” and thus contributes negligibly to cyclic conjugation. However, conclusions from the present calculations of topological ring currents agree remarkably with those arising from the earlier study involving energy-effect values in that, contrary to what would be predicted from the classical approaches, rings annelated in an angular fashion relative to the central ring of these perylene structures materially increase the extent of that ring’s involvement in cyclic conjugation. It is suggested that such close quantitative agreement between the predictions of these two superficially very different indices (energy effect and topological ring current) might be due to the fact that, ultimately, both depend, albeit in ostensibly quite different ways, only on an adjacency matrix that contains information about the carbon-carbon connectivity of the conjugated system in question. Introduction Forty years ago, one of us (R.B.M.) observed1 that the π-electron ring-current intensity calculated by means of the Hu¨ckel2-London3-Pople4-McWeeny5 formalism (hereafter HLPM) for the central ring (C) of the condensed benzenoid hydrocarbon peropyrene (I) (Figure 1) is some 6 times that calculated, by the same method, for the superficially analogous central ring (C) of perylene (II) (Figure 1). Almost twenty years later, Haigh and one of the present authors (R.B.M.) rationalized6 this striking difference by noting that the perylene molecule (II) may formally be considered to be constructed by joining together two conjugated systems; in the case of perylene they are both naphthalene units, each of which is possessed of at least one Kekule´ structure; in the course of this process, the two naphthalene fragments are thought of as being connected together by means of two further bonds (those indicated by the arrows in structure (II) of Figure 1, bonds which, for further emphasis, are depicted in that figure by bold lines), thereby creating a “new” ringsthe ring labeled “C” in perylene (structure (II) of Figure 1). The bonds shown in bold and indicated by the arrows never appear other than as single bonds in any Kekule´ structure that may be devised for the conjugated system (II) as a whole. Peropyrene (I), by contrast, may be considered to be formed in an analogous way but, this time, by joining together, by means of the two bonds drawn in bold and indicated by the arrows in structure (I) of Figure 1, two moieties that are radicals and thus, by their very nature, do not, individually, possess any Kekule´ structures. Consequently, the bonds shown in bold and indicated by the arrows in structure (I) appear as single bonds in some of the Kekule´ structures that may be devised for the peropyrene (I) molecule as a whole, * Corresponding author. E-Mail:
[email protected]. † University of Cambridge. ‡ University of Kent.
Figure 1. Central ring, C, in peropyrene (I) (left) and in perylene (II) (right), and two bonds of special interest in those rings, indicated by the arrows and drawn in bold.
and as double bonds in others. On what Gutman et al.7 call this “classical” criterion of Kekule´ structures, therefore, the central ring (C) in peropyrene (I) appears to participate fully in the conjugation between the upper and the lower parts of the molecule (I) whereas the central ring (C) in perylene (II) does not. Hence the frequent description of this latter ring as “empty”.7 Haigh and Mallion6 reconciled the above considerations about Kekule´ structures with predictions based on perturbation theory effected at the HMO2 level and, in this way, they rationalized why the ring-current intensity1 in the central ring, C, of peropyrene (I) is 1.446,1,8 when expressed as a ratio to the ring-current intensity calculated, by the same (HLPM) method,2-5,8 for the unique ring in benzene, while that in the central ring (C) of perylene (II) is less than one-sixth of this value, at only9,10,8 0.239. Recent Developments Much more recently, Gutman et al.7 have considered the “emptiness”, or otherwise, of ring C in perylene (II), and in every one of its two mono-, nine di-, eight tri-, and seven tetrabenzo derivatives. These authors likewise emphasized the
10.1021/jp1096103 2011 American Chemical Society Published on Web 12/20/2010
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Dickens and Mallion
“essentially single”7 nature of the bonds marked by the arrows and indicated in bold in structure (II) of Figure 1 because (a) (as already mentioned) they are single in all nine Kekule´ structures possessed by (II), (b) they are single in all four Clar aromatic-sextet formulas,11 and (c) they are not contained in any of the circuits of conjugation12 that may be devised for perylene (II). Gutman et al.7 concluded that these three classical criteria7 all concur that the bonds in perylene (II) that are indicated by the arrows and depicted as bold in Figure 1, as well as the ring (C) to which they belong, “... do not participate in cyclic conjugation”. Such classical criteria would thus declare rings such as ring C in perylene (II) to be strictly empty. As an alternative to these classical approaches, Gutman et al.7 studied the benzo-annelated perylenes by means of Gutman’s “energy-effect” index.13 This is an entirely topological quantity that may be calculated for each indiVidual ring of a polycyclic conjugated system. When expressed in units of the HMO2 carbon-carbon resonance-integral, β, the energy-effect index for a cycle Z is defined13 as
ef(G,Z) )
2 π
dx ∫0∞ ln φ(G,ix) +φ(G,ix) 2φ(G - Z,ix)
(1)
where G is the network representing the carbon-atom connectivity of the conjugated system under study, φ(G,x) is its characteristic polynomial,13 (G - Z) is the subnetwork obtained by deleting the cycle Z from G, and i ) (-1)1/2. In ref 7, Gutman et al. applied eq 1 to calculate the energy-effect values for each of the six-membered cycles ()rings) of perylene (II) and its 26 benzo-annelated derivatives. They asserted that the greater the energy effect for a particular ring, the greater the extent of cyclic conjugation in that ring.7 The results of Gutman et al.7 showed that, on the energyeffect criterion, the central ring (C) in perylene (II) is by no means strictly empty and that, furthermore, the magnitude of the energy effect calculated for that ring C varies by up to ca. 140%, according to the way in which the basic perylene (II) moiety has further benzenoid rings annelated to it. In the present paper, we undertake a study entirely parallel to that of Gutman et al.7 in which topological π-electron ring currents8 replace energy-effect values13 as a measure of what is interpreted as the extent of each ring’s participation in the overall conjugation that is considered to be extant in a given condensed, benzenoid hydrocarbon. The qualitative, even quantitative, agreement found in the course of this investigation between the trends predicted by these ostensibly independent criteria of conjugation is indeed remarkable, as will now be seen. Nomenclature Adopted for Benzo-Annelated Perylenes We adopt in its entirety the nomenclature for labeling the rings of perylene (II) and its benzo-annelated derivatives that was introduced by Gutman et al. in ref 7 and which is illustrated in Figure 2. The central ring in perylene (II), which, until now, we have labeled as “ring C”, is, hereafter, labeled “ring 5”, the other four rings in perylene (II) being denoted 1, 2, 3, and 4, as shown in the upper part of Figure 2. From the point of view of ring 5, annelation by further benzenoid rings may be effected either in an angular fashion (in one or more of the positions labeled a1, a2, a3, and a4 in the top part of Figure 2) or in a linear manner (in one or more of the positions b1, b2, b3, and b4). Any benzenoid rings annelated to rings 1, 2, 3, 4 of perylene (II) are labeled 1′, 2′, 3′, 4′, respectively, as indicated in Figure
Figure 2. Labeling of the rings of perylene and its benzo-annelated derivatives. Relative to the central ring (5), annelation may be either angular (that is, in one or more of the positions a1, a2, a3, and a4) or linear (in one or more of the positions b1, b2, b3, and b4). Any benzenoid rings annelated to rings 1, 2, 3, and 4 of perylene are labeled 1′, 2′, 3′, and 4′, respectively. The notation should be self-explanatory from the three examples shown in the lower part of the figure.
2. The notation should be self-explanatory from the three examples shown in the lower part of Figure 2. Topological π-Electron Ring Currents The concept of topological π-electron ring currents was precisely defined by one of us (R.B.M.) only very recently,8 though Coulson and Mallion14 informally used the term “topological” in the context of ring currents many years ago. To start with,8 the idea was proposed rigorously only for the condensed, benzenoid hydrocarbons but its formal definition has recently been extended to include the case of conjugated systems containing rings of any size.15 Topological ring currents8 for the condensed, benzenoid hydrocarbons are defined as being π-electron ring-current intensities that (a) are calculated by the HLPM formulation,8,2-5 with all Coulomb integrals assumed to be the same as that for a carbon atom in benzene, and with all resonance integrals set equal to the standard value appropriate for a carbon-carbon bond in benzene; (b) are calculated on the assumption that the areas of all benzenoid rings are the same as that of a standard benzenehexagon; and (c) are expressed as a ratio to the ring-current intensity calculated, by the same method, for benzene. For the purposes of documentation, we record here the expression that we used16,17 to calculate the topological ring current, Ji/Jbenzene, in each ring (i) of perylene (II) and its 26 benzo-annelated derivatives:
Ji Jbenzene
) 9{
i + ∑ [P(µ) + βπ¯ (µ)(µ)]S(µ)C(µ) (µ)
i S(ν)]} ∑ ∑ βπ¯ (µ)(ν)[S(µ)C(ν)i + C(µ)
(2)
(µ
