Ï-Electron Ring-Currents and Bond-Currents in Some Conjugated

Article pubs.acs.org/JPCA

π‑Electron Ring-Currents and Bond-Currents in Some Conjugated Altan-Structures Timothy K. Dickens*,† and Roger B. Mallion‡ †

University Chemical Laboratory, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, England, United Kingdom School of Physical Sciences, University of Kent, Canterbury, CT2 7NH, England, United Kingdom

‡

ABSTRACT: Ring-currents and bond-currents calculated using the Hückel− London−Pople−McWeeny (HLPM) method are reported for a series of altanstructures recently explored by Monaco et al. and to which these authors have applied the ab initio ipso-centric formalism in order to calculate the structures’ πelectron current-density maps. Two aspects of the conclusions of their ab initio study are confirmed by means of the much more simplistic HLPM formalism adopted here: namely (a) that the inner core in these altan-structures generally displays similar current patterns to those previously reported for the parent fragments, and (b) that the unexpected diamagnetic circulation in the outer [4n]perimeter of altan-kekulene reported by Monaco et al. is not an artifact of their method but appears to be confirmed. Attention is drawn to the conceptual advantages of discussing bond currents rather than just (equivalent) ring currents.

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denote “inner” and “outer” (respectively), altan-naphthalene (3) is such that 3-i has the carbon−carbon connectivity of naphthalene, and 3-o possesses the molecular graph of [16]annulene. Monaco and Zanasi have observed8 that all altanmolecules with an even outer-annulene share the characteristic that the additional (“spokes”7) bonds formed by this process of “altanization” are formally fixed as single bonds8,5 in all Kekulé structures that may be devised for the conjugated system as a whole. Monaco and Zanasi and co-workers have invariably invoked3,5,6 the ab initio ipso-centric approach9 in effecting their calculations of bond-current maps for these structures, basing their computations on optimized geometries obtained via the Gaussian MO program.10 The present authors, on the other hand, have championed the Hückel−London−Pople− McWeeny (HLPM) formalism11−18 (reviewed in exhaustive detail in ref 18), based on idealized geometries, because its predictions do not depend on any assumed parameters: ringcurrent maps (and, henceby Kirchhoff’s Conservation Law19bond-current maps) for a given conjugated system are implicitly predetermined in this approach as soon as the following have been specified:11−18 (i) the carbon-atom connectivity manifested in the molecular graph4 of the conjugated system in question, and (ii) the areas of its constituent rings. Much recent work has been devoted to comparing how numerical predictions of the sophisticated ab initio ipso-centric formalism9 compare with those of the much more rudimentary

INTRODUCTION In an attempt to investigate the magnetic propertiesand, by extension, the aromaticity1,2of extant and hypothetical conjugated hydrocarbons, Monaco and Zanasi have recently explored3 an intriguing series of unsaturated systems, which they have named altan-structures. The first to be proposed3 were altan-corannulene and altan-coronene (whose molecular graphs4 are labeled 1 and 2, respectively, in Figure 1), and Monaco and Zanasi later considered anionic and cationic derivatives of the former.5 The same authors, with Memoli,6 have recently introduced (Figure 1) altan-naphthalene (3), altan-pyrene (4), altan-pyracylene (5), and altan-kekulene (6). Also illustrated in Figure 1, though not studied by Monaco, Memoli, and Zanasi in ref 6, is altan-azulene (7), which will also eventually be considered in the present article. The process of what might be called “altanization” is described by Monaco and Zanasi3 in the following way: they think of it as a metamorphosis that arises when an unsaturated hydrocarbon is notionally placed “inside” a [4n]-annulene, and bonds are then altered in such a way that an outgoing C−H bond from the inner structure and an internal C−H bond from the annulene are replaced by a C−C bond; at the end of the process, such C−C bonds join certain carbon atoms of the inner structure to alternating carbon-atoms of the surrounding Annulene, and only to those carbon atoms. For example, in altan-naphthalene (the species labeled 3 in Figure 1), the inner structure has the carbon-atom connectivity of naphthalene and the outer structure, to which the inner one is joined as just described, has the carbon−carbon connectivity of [16]annulene; the two moieties are thereby joined by connecting “spokes” bonds,7 as shown in structure 3 of Figure 1. In the nomenclature of Monaco et al.,6 which uses “i” and “o” to © 2014 American Chemical Society

Received: March 14, 2014 Revised: April 22, 2014 Published: April 22, 2014 3688

dx.doi.org/10.1021/jp502585f | J. Phys. Chem. A 2014, 118, 3688−3697

The Journal of Physical Chemistry A

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Figure 1. Molecular graphs4 of altan-structures 1−7, showing their carbon−carbon connectivities.

HLPM approach. 11−18 The two methods qualitatively agree20−22 in the case of [10,5]-coronene (structure (IV) of ref 21), but not when they are both applied to 7-coronene (structure (II) of ref 22), which, like [10,5]-coronene (there22 called “5-coronene”), is a member of the series that we have recently named22 the “p-coronenes”.22,23 In addition, the present authors have recently demonstrated24 remarkably close qualitativeand even, to some extent, semiquantitativeagreement between the two methods when they are used to predict bond-current patterns in the first two altan-structures that were proposed,3 altan-corannulene (1) and altan-coronene (2). In the present study, we continue this comparison between these two very different methods of computation by using the HLPM formalism11−18 to calculate the ring-current and bondcurrent maps associated with the four newly considered6 altanstructures (3−6 of Figure 1). In addition, because Pople13 tested his version of the HLPM formalism on azulene (7-i), whereas McWeeny14 illustrated his own formulation of the theory by appeal to the alternant isomer of azulenenamely, naphthalene (3-i)we also consider altan-azulene (7) (Figure 1) in this study, even though it is not one of the structures treated by Monaco and Zanasi in ref 6.

area of the central 18-membered ring was taken (as on previous occasions28−30,18) to be seven benzene-hexagon units, which is the area of the “missing” hexagons in the center of 6, a structure that, apart from its peripheral rings, is otherwise ideally envisaged as being completely tessellated with regular hexagons of carbon atoms. The symmetries assumed were D2h (for altannaphthalene (3), altan-azulene (7), altan-pyrene (4) and altanpyracylene (5)) and D6h (for altan-kekulene (6)). It should also be emphasized that, in the course of calculating ring-current and bond-current intensities by the HLPM method,11−18 the simplest assumption of a planar carbon− carbon framework has to be made. In this context, the following points may be borne in mind: (a) The following remark of Monaco, Memoli, and Zanasi:6 “...although the altan molecules studied here are not planar, previous experience indicates that the ring currents can still be ascribed to a good extent to a subset of orbitals that bear most similarity with the πorbitals.” (b) The following observations, by Haigh and one of the present authors,31 made when they were attempting to adapt the McWeeny version14 of the HLPM11−18 approach in order to make it applicable to nonplanar structures such as the helicenes: they claimed31 that their approach was viable when molecular overcrowding is such that “...the skeletal distortion about any bond is comparatively mild...” and they concluded that “...because of the way in which the strains from overcrowding are spread over many degrees of freedom, this condition does appear to be satisfied in the case of the helicenes, even though the overall non-planarity between wellseparated parts of such molecules may in fact be very large.” (c) The previous acknowledgments32−34 that ring−current calculations often do appear to be surprisingly insensitive to what assumptions are made about ring areas.

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CALCULATIONS Ring-current intensities in structures 3−7 were calculated using the standard HLPM regimen,11−18 the details of which are described in ref 18. Equation 14 of ref 18 was applied and, with the exception of the calculations on structure 6, the simplest (“topological”25−27) version of this method11−18 was adopted (please see eqs 14 and 16 of ref 18). In the latter, it is assumed that regular polygons of uniform side-length form idealized planar structures,17 with the area of a five-membered ring taken13 to be {(5 cot π/5)/(6 cot π/6)} times the area of a standard benzene-hexagon and the area of a seven-membered ring similarly assumed13 to be bigger than that of the standard benzene-hexagon by the factor {(7 cot π/7)/(6 cot π/6)}. Exceptionally, in the sole case (here) of altan-kekulene (6), the 3689



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Figure 2. Maps for the topological ring-currents (in black) and the associated topological bond-currents (in red) for altan-naphthalene (3) (left) and naphthalene (3-i) (right). The topological ring-currents and bond-currents are dimensionless quantities. Positive (diamagnetic) ring-currents are considered to circulate counterclockwise around their respective rings while negative (paramagnetic) ring-currents flow in the clockwise sense around those rings. The various bond-currents flow in the direction indicated by the arrow pointing along each bond.

Figure 3. Maps for the topological ring-currents (in black) and the associated topological bond-currents (in red) for altan-pyrene (4) (left) and pyrene (4-i) (right). For the conventions on displaying topological ring-currents and bond-currents, please see the caption to Figure 2.

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Ring-current intensities were expressed as a (dimensionless) ratio to the ring-current intensity calculated, by the same method, for benzene; continued application of Kirchhoff’s Law of Conservation of Current at a Junction19 yielded the bond currents, also expressed as a ratio to the unique carbon−carbon bond-current in benzene, itself likewise calculated by the HLPM method.11−18 Maps for ring-current intensities for altans 3−7, and for the bond currents that are consistent with them, are depicted in Figures 2−6, together with analogous currentmaps for 3-i−7-i, the corresponding “inner cores” of structures 3−7.

RESULTS AND DISCUSSION

(i). Altan-Naphthalene (3) and Naphthalene (3-i). Monaco, Memoli and Zanasi6 state that, on the basis of ab initio ipso-centric calculations,35 naphthalene (3-i) has “... a single diatropic loop.35” and they report that, from their computations on 3,6 they “...nicely get a paratropic annulenic loop enclosing the same patterns previously reported35 for the parent...” It can be seen from Figure 2 that the bond-current intensities in altan-naphthalene (3) and in naphthalene (3-i) calculated by the HLPM approach11−18 are in qualitative agreement with the description of Monaco, Memoli, and Zanasi,6 just quoted. Naphthalene itself (3-i) carries a diamagnetic current of size 3690



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Figure 4. Maps for the topological ring-currents (in black) and the associated topological bond-currents (in red) for altan-pyracylene (5) (left) and pyracylene (5-i) (right). For the conventions on displaying topological ring-currents and bond-currents, please see the caption to Figure 2.

(ii). Altan-Pyrene (4) and Pyrene (4-i). In considering the ab initio ipso-centric calculation39 on pyrene (4-i), Monaco, Memoli and Zanasi observed6 that 4-i “...presents a diatropic flow around the perimeter and some current bifurcation in the middle portion”. They further remarked6 that the corresponding altan (4) carries an avowedly paramagnetic circulation6 around its [20]-membered perimeter, which has the same molecular graph as [20]-annulene (4-o). These features are likewise reproduced in the HLPM calculations presented in Figure 3. In both the outer perimeter of the parent structure (4-i) and the corresponding bonds of the altan (4) itself, a strong diamagnetic (that is, counterclockwise) circulation is predicted. We note also that the calculated current-flows in the “...bifurcation in the middle portion...”,6 which were observed in the ab initio calculations on 4-i,39,6 are semiquantitatively reproduced in 4 by the HLPM computations. What is more, Monaco, Memoli, and Zanasi’s prediction6 of a substantial paramagnetic (clockwise) flow around the [20]-membered perimeter of the altan (4) is also verified by the HLPM calculation, with paramagnetic bondcurrents in the periphery varying between ca. 68% and 118% of the magnitude of the diamagnetic current in benzene. Other points worthy of note in Figure 3 are (a) the diamagnetic flows around the “north” and “south” rings of pyrene (4-i) and around the corresponding six-membered rings in 4; (b) the paramagnetic circulations around the “west” and “east” fivemembered rings of 4; and (c) the paramagnetic direction of flow around the pentalenic moieties in the “north” and “south” of 4. We mention in passing that the HLPM ring-currents in pyrene were previously reported in refs 36 and 32. Those depicted in Figure 3 are recomputed values, employing a branched spanning-tree and Double Precision in the calculation. (Please see ref 18 for details of these aspects of the computations.) The values in Figure 3 agree with those of ref 32 to the three places of decimals quoted, and they agree with the corresponding ring-currents quoted in ref 36 to within two digits in the third decimal-place. (iii). Altan-Pyracylene (5) and Pyracylene (5-i). The ab initio ipso-centric calculation40,41 on pyracylene (5-i) is

1.09 around its perimeter, while the corresponding bonds in altan-naphthalene (3) bear a diamagnetic current of between ca. 72% and 94% of the benzene value; furthermore, a large paramagnetic flow, which has an average size of more than 2.5 times the (diamagnetic) benzene-value, is predicted to run around the outer [16]-annulenic perimeter of 3. It is important to observe that each and every ring-current in 3 is paramagnetic and yet there is a diamagnetic flow around the bonds of the central naphthalenic core. This results from the algebraic competition that is evident in the bonds that the two central six-membered rings of altan-naphthalene (3) share with the sixmembered rings and the two types of symmetrically distinct five-membered rings forming its perimeter. The clockwisecirculating (paramagnetic) ring-currents (ca. −1.7) associated with the two six-membered rings of the naphthalenic core are overwhelmed by stronger paramagnetic currents in the outer rings (which range from ca. −2.4 to ca. −2.8, on the scale where the diamagnetic ring-current in benzene is +1.0); as a result of this swamping of the paramagnetic ring-currents in the naphthalenic core by paramagnetic ring-currents of larger size associated with the three types of symmetrically distinct rings on the periphery, the net direction of flow around the inner naphthalenic core is in fact counterclockwise (that is, diamagnetic). It may be observed in passing that naphthalene (3-i) was treated by means of the HLPM approach in McWeeny’s elegant and classic paper14 of 1958. As a checkand to be consistent with the other calculations being presented here and in recent papers of our own18,24we have repeated McWeeny’s HLPM calculation, basing it on a branched spanning-tree and using Double Precision in the computations. (Please see the account in ref 18 for these aspects of the HLPM calculations being reported here.) These new ring-current computations on naphthalene agree, to the three decimal-places quoted, with the original ones of McWeeny,14 as well as with those of Jonathan, Gordon, and Dailey36 (who used Pople’s method13) and of Haigh and Mallion,37 (who applied McWeeny’s14 formalism). They are also consistent with the Hückel11− London12 calculation of the diamagnetic susceptibility of naphthalene, reported even longer ago.38 3691



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Figure 5. Maps for the HLPM ring-currents (in black) and the associated HLPM bond-currents (in red) for altan-kekulene (6) (left) and kekulene (6-i) (right). For the conventions on displaying HLPM ring-currents and bond-currents, please see the caption to Figure 2.

described as “topological”.25 It was based on a continuous spanning-tree18 and was computationally carried out in what would now be called Single Precision.18 In order to obtain the ring currents and bond currents presented in Figure 4, we have recomputed these quantities making appeal to a branched spanning-tree 18 and using Double Precision18 for the calculation. The ring-current intensities for 5-i presented in Figure 4 are not precisely those of Coulson and Mallion25 but they do agree with the older values25 to within four digits in the third decimal-place. One of the present authors1,43 and collaborators25,44−47 have investigated making resonance integrals iteratively self-consistent with respect to the corresponding Coulson bondorders11 and (simultaneously) making Coulomb integrals iteratively self-consistent with calculated charge-densities11 on the carbon atoms; pyracylene (5-i) was the paradigm for this study.1,25,43,45−47 Incorporating this refinement is particularly worthwhile when predominantly paramagnetic systems are being treated, where the HOMO−LUMO gap might be especially small,43 even more so if the ring-currents so calculated are to be used, for example, for the prediction of 1 H NMR chemical-shifts.48 However, as in a recent companion paper,24 this has not been done here because such a refinement would ensure that the calculationseven once ring-areas had been establishedwould no longer be properly described as being “purely topological”.15−18 (iv). Altan-Kekulene (6) and Kekulene (6-i). Altankekulene (6) is the first and only one of the structures newly examined here by the HLPM approach11−18 which, like altancorannulene (1 in Figure 1) and altan-coronene (2), may formally be considered24,6in an intuitive extension of the “annulene-within-an-annulene” (AWA) model49−51topologically to be a three-layered “annulene-within-an-annulene-

described by Monaco, Memoli, and Zanasi as indicating “...a diatropic circulation on the naphthalenic moiety and a paratropic circulation on the two pentagons...” and these patterns are reproduced in the corresponding parts of 5 (see, for example, the middle section of Figure 3 of ref 6). The corresponding altan (5) supports a paramagnetic flow in its [16]-membered perimeter.6 The HLPM calculations presented in our Figure 4 reproduce (a) the paramagnetic flow in the [16]-membered perimeter of 5 reported by Monaco et al.6 and (b) the paramagnetic circulation in the five-membered rings of 5-i and the corresponding rings in 5; however, the diamagnetic flow around the naphthalenic moiety reported after applying the ipso-centric9 method in ref 6 is not reproduced by the HLPM calculations, where such a flow is interrupted by a change of direction in some bonds. However, it can be seen from Figure 4 that, within the HLPM calculations, the pattern of directions of flow in the peripheral bonds of 5i and in its central bifurcations are very similar on passing from 5i to the corresponding bonds of 5. It should be noted that the predicted paramagnetic circulation in the [16]-membered perimeter of 5 is substantial, ranging from nearly 90% of the size of the diamagnetic current in benzene to nearly 170% of it (in the six-membered rings on the periphery and the two types of symmetrically distinct fivemembered rings that contribute bonds to the outer perimeter of 5). Pyracylene (5-i) was synthesized by Trost et al.42 more than 40 years ago because it was considered potentially to be what was thought of as a “perturbed annulene”.42 Soon afterward, Coulson and one of us (R.B.M.)25 studied the π-electron magnetic-properties of pyracylene in the context of what the present authors now call the HLPM “topological” approach.11−18 That calculation was the first HLPM one to be 3692



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data were plotted as a regression line) of more than 0.8that (reproduced in Figure 6) between bond-currents calculated by

within-another-annulene”. By contrast, as Monaco, Memoli, and Zanasi have observed,6 in the altans 3−5 of Figure 1, the “inner parents”6 are formed by an “ethylene-within-anannulene”. (This latter remark applies also to altan-azulene (structure 7 in Figure1), not considered by Monaco et al. in ref 6, but dealt with later in this paper.) A point of major interest, noted by Monaco et al.,6 is that, unlike the other three altan-structures (3, 4, and 5) examined in ref 6 by the ipso-centric method9 and studied here by means of the HLPM approach,11−18and also unlike 7, discussed lateraltan-kekulene (6) supports a diamagnetic circulation in its [4n]- ([36]-membered) periphery. Monaco et al.6 remarked that this is a “...minor failure...” from a “...design point of view...”, which they intend to investigate further.6 The magnetic nature of the inner/middle/outer cycles of altan-kekulene (6) was declared by Monaco et al.6 to be (respectively) paramagnetic/diamagnetic/diamagnetic. The important point that we wish to emphasize here is that the much more rudimentary HLPM calculations11−18 predict the aforementioned two features of the π-electron magneticproperties of 6 as clearly as do those based on the much more sophisticated ab initio ipso-centric approach.9 This is illustrated by the HLPM ring-current and bond-current maps depicted in Figure 5. The pattern (for the innermost, middle and outer cycles, respectively) of paramagnetic/diamagnetic/diamagnetic currents, noted by Monaco, Memoli, and Zanasi,6 is clearly seen there, emphasizing the diamagnetic circulation in the outer periphery, also encountered by Monaco et al.6 in the course of their ab initio ipso-centric calculations. According to the computations by the HLPM approach that are presented in Figure 5, the diamagnetic currents in the peripheral bonds are ca. 33% and 57% of the benzene value in the two types of peripheral bonds which  on the symmetry that we have assumed here (D6h)  are symmetrically distinct. If altanKekulene (6) were considered to be an [18]-annulene-within-a[30]-annulene-within-a-[36]-annulene respecting an intuitive extension of the AWA model,49−51 the expected magnetic behavior of the three layers  innermost, middle and outer, respectively  would be diamagnetic/diamagnetic/paramagnetic. In fact, paramagnetic/diamagnetic/diamagnetic is what is predicted, both by the ab initio ipso-centric method9 used in ref 6, and the HLPM method,11−18 applied here. Therefore, as was the case with altan-corannulene (1) and altan-coronene (2),6,24 both methods9,11−18 are in agreement that altan-kekulene (6) does not conform to the intuitive extension of the AWA rule envisaged above. In order to investigate the apparent discrepancies6 discrepancies which, as just noted, are fully confirmed by our topological HLPM calculations and hence would seem not to be an artifact of either methodMonaco, Memoli, and Zanasi6 (in addition to devising the traditional semiquantitative currentdensity maps) presented actual numerical values of individual calculated bond-current susceptibilities in altan-kekulene (6), and displayed them in Figure 5 of ref 6. Potentially, these are extremely useful for our purposes because, by dividing each of these by 12.8 nanoamperes per tesla (the appropriate value, on the same approximations and assumptions,6 for benzene), individual bond-current susceptibilities relative to benzene may be obtained which are then directly comparable with the HLPM bond-currents in our Figure 5 (above). We attempted some “comparator diagrams”, such as were found especially illuminating in our previous study24 of 1 and 2 (of Figure 1), but only one showed a correlation coefficient (when the same

Figure 6. “Comparator” diagram, based on the data presented in Table 1, for those bonds (labeled according to the scheme adopted in part (4) of Figure 2 of ref 6) in altan-kekulene (6) that it has in common with kekulene (6-i). The comparison is between the predictions of ab initio ipso-centric calculations6 (blue line) and those of topological HLPM calculations11−18 (brown line). (Dimensionless) bond-currents are displayed along the vertical axis, and symbols denoting the individual bonds6 are ranged along the horizontal axis.

the HLPM approach11−18 (brown line) and by the ab initio ipso-centric method9,24 (blue line) for those bonds (labeled according to the scheme adopted in part 4 of Figure 2 of ref 6) in altan-kekulene that it has in common with kekulene. The corresponding numerical data are presented in Table 1. It is, however, enlightening to compare the HLPM ringcurrent and bond-current intensities in the current maps for altan-kekulene (6) and kekulene (6-i) displayed consecutively in Figure 5. Immediately striking is the similarity of the ring currents in the two types (on our assumed symmetry of D6h) of six-membered rings on the outer cycle of 6-i and the middle cycle of 6: these are, respectively, 0.998 (compared with 1.000) and 1.359 (compared with 1.303). The HLPM-calculated ringcurrent intensities in the large innermost cycles (i.e., the central rings) of 6 and 6-i (of area seven benzene-hexagon units18,28−30) are also much smaller than the (likewise diamagnetic) ring-current intensities in all the other rings of these two structures: 0.190 and 0.265, respectively. Accordingly, despite their large ring-area, they will make only relatively modest contributions to the overall “London” π-electron susceptibility12 perpendicular to the (assumed) molecular plane (cf. the discussion in ref 13). Altan-Azulene (7) and Azulene (7-i). Although these structures were not considered by Monaco, Memoli, and Zanasi in ref 6, we include azulene (7-i) and altan-azulene (7) in this study, as a companion to naphthalene (3-i) and altannaphthalene (3). Azulene and naphthalene appropriately form a pair as the former is the nonalternant isomer of the latter: furthermore, there were two (equivalent) classical formulations of the HLPM approaches independently published in Volume 1 of Molecular Physics (1958): that due to Pople,13 used azulene as an illustration of the method, while McWeeny applied his formalism to naphthalene, as an exemplar. It seems appropriate, therefore, that the “altanized” version of both of these fundamentally important conjugated structures should be examined together. Accordingly, we present the HLPM ringcurrent and bond-current maps for altan-azulene (7) and azulene (7-i) in Figure 7, for ready comparison with those of altan-naphthalene (3) and naphthalene (3-i), depicted in Figure 2. 3693



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Table 1. Comparison between Bond Currents Evaluated by the Ipso-Centric9 and HLPM11−18 Approaches for Those Bonds in Altan-Kekulene (6) That It Shares with Kekulene (6-i) bond in altan-kekulene and corresponding bond in kekulene (bond-symbolsa are ranged along the horizontal axis)

magnitude of the ipso-centric “bond-current”b expressed as a ratio to the intensity of the corresponding current in benzene, calculated by the same method (plotted along the vertical axis)

magnitude of the HLPM bond-currentc expressed as a ratio to the ring-current intensity in benzene, calculated by the same method (plotted along the vertical axis)

m1 m2 m3 n1 n2 n3 n4 n5 s1 s2

0.94 0.95 0.61 0.95 0.91 0.50 0.52 0.54 0.38 0.40

1.04 1.04 0.72 0.73 0.73 0.43 0.67 0.43 0.30 0.30

Bond labelings are as depicted in part 4 of Figure 2 in ref 6. bThese values are those of Figure 5 in ref 6, after division by 12.8 nAT−1, the appropriate value for benzene on the version of the ipso-centric method used in ref 6. cTranscribed from Figure 5 in the present paper.

a

Figure 7. Maps for the topological ring-currents (in black) and the associated topological bond-currents (in red) for altan-azulene (7) (left) and azulene (7-i) (right). For the conventions on displaying topological ring-currents and bond-currents, please see the caption to Figure 2.

clockwise-circulating (paramagnetic) ring-currents (ca. 70% of the size of the diamagnetic current in benzene) associated with the five- and seven-membered rings of the azulenic core are overwhelmed by stronger paramagnetic currents in the outer rings (which range from ca. −1.2 to ca. −1.9). The paramagnetic ring-currents associated with the eight rings on the periphery of 7 thus prevail over the smaller paramagnetic ring-currents in the five-membered and seven-membered rings of the azulenic moiety, and this results in the net direction of flow around the inner azulenic-core’s being counterclockwise (that is, diamagnetic).

Comparison of Figure 7 and Figure 2 reveals a number similarities between the pairs naphthalene (3-i)/altan-naphthalene (3) on the one hand, and azulene (7-i)/altan-azulene (7), on the other: (a) while naphthalene (3-i) and azulene (7-i) both bear strong diamagnetic ring-currents in their two rings, every one of the ten individual rings of each of altannaphthalene (3) and altan-azulene (7) supports substantial paramagnetic ring-currents; (b) as is consistent with (a), above, both altan-naphthalene (3) and altan-azulene (7) manifest strong paramagnetic circulations around their outer [4n]- ([16]-membered)peripheries; (c) despite statement (a), above, the net circulation around the perimeter of the azulenic inner core of 7 (Figure 6) is (as in azulene (7-i) itself) diamagnetic, as was also seen to be the case around the naphthalenic core of 3 (Figure 2). Observation (c) is again rationalized by observing that there is an algebraic competition in the bonds that the central azulenic core of altan-azulene (7) shares with the six-membered rings and the three symmetrically distinct types of fivemembered rings that contribute to its perimeter. The

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CONCLUSIONS

The three major conclusions from Monaco, Memoli, and Zanasi’s study6 of altan-structures by the ab initio ipso-centric formalism are here confirmed by the much less sophisticated HLPM approach: (a) “The current density patterns should be considered as the sum of the current density patterns of the parent molecules...”6 and the annulenic loop in 3−7 encloses “... the same patterns previously reported for the parent fragments.”6 This is also consistent with what was found 3694



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in our earlier study,24 on altan-corannulene (1) and altan-coronene (2). (b) The unexpected6 diamagnetic circulation in the outer perimeter of altan-kekulene (6) reported by Monaco et al.6 appears not to be an artifact of their method but is indeed confirmed by our HLPM calculations. (c) As a corollary of (b), above, we note that the one structure newly considered here that might potentially be a candidate for an intuitive extension of the AWA rule49−51 to three-layered systems, namely, altankekulene (6), does not in fact conform to that rule. To be compliant with the AWA model the current patterns in the innermost, middle and outer cycles, respectively, of 6, would have had to have been diamagnetic/ diamagnetic/paramagnetic, whereasas has been observed by Monaco et al. in ref 6, and by ourselves, herethe predicted pattern is actually paramagnetic/ diamagnetic/diamagnetic. This is analogous to the situation that was encountered for altan-corannulene (1) and altan-coronene (2) in refs 6 and 24. We conclude with three observations: (i) This analysis has once again emphasized the value of considering bond currents in addition to (entirely equivalent) ring currents when studying the π-electron magnetic properties of conjugated systems. Although bond currents were discussed by some of the early pioneers in the field,52,53 and, from time to time, have been resurrected by others (e.g., refs 54−57, and 15), the fashion for many decades was to emphasize ring currents over bond currents.2,48 In the present study, and in other contemporary work (e.g., refs 6 and 24),58 an examination of the bond currents, rather than just the ring currents, has, for example, enabled rationalization of net diamagnetic circulations around the bonds of the interior rings of systems (such as altan-naphthalene (3) and altanazulene (7)) in which every ring bears a paramagnetic ringcurrent. (ii) It was remarked more than a decade ago39−41 that the Hückel−London approach11−18 frequently well reproduces the trends in the magnetic properties of conjugated systems that are predicted by ab initio calculations. For example, Steiner et al.39 have observed that, in the case of pyrene (4-i) and related molecules, “... it is impressive...” that predictions based on the Hückel−London formalism “... exhibit all the main features of the ab initio maps in most cases, even if they exaggerate the relative strength of the paratropic currents...” (This sensitivity of paramagnetic ring-currents, especially, to the assumptions/ wave functions on which their calculation is based, has been stressed by one of us (R.B.M.), over many years.25,43−47) In addition, Fowler et al.,40,41 in the context of their ab initio calculations on pyracylene (5-i), commented40 that “... the ab initio current density map is in qualitative accord with the early semi-empirical calculations of Coulson and Mallion25 that predicted paramagnetic ring-currents in the pentagons but not the hexagons, in broad agreement with the claim of Trost et al.42 based on the 1H-NMR spectrum that this molecule should support a paramagnetic circulation.” (iii) Furthermore, from the present, and recent,24 work on the altan-structures,3,5,6 the fact that a simple-minded theory (the HLPM11−18 formalism) has yielded the same qualitative and, on occasions, even semiquantitativeconclusions as a more sophisticated ab initio one (the ipso-centric method9), once again brings to mind Coulson’s celebrated remark62 about

the conceptual value, even in what we might consider to be this “ab-initio age”, of simple theories that bring about intuitive and “... primitive patterns of understanding.”

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +44 1223 763 811. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are very grateful to Professor Riccardo Zanasi and Professor Gulielmo Monaco of the University of Salerno for some extremely helpful correspondence about the calculation of bond-current intensities by means of the ab initio ipso-centric approach.

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REFERENCES

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(41) Fowler, P. W.; Steiner, E.; Cadioli, B.; Zanasi, R. DistributedGauge Calculations of Current Density Maps, Magnetizabilities, and Shieldings for a Series of Neutral and Dianionic Fused Tetracycles: Pyracylene (C14H8), Acepleiadiene (C16H10), and Dipleiadiene (C18H12). J. Phys. Chem. A 1998, 102, 7297−7302. (42) Trost, B. M.; Bright, G. M.; Frihart, C.; Brittelli, D. Perturbed [12]-Annulenes. The Synthesis of Pyracylenes. J. Am. Chem. Soc. 1971, 93, 737−745. (43) Mallion, R. B. Ring-Current Effects in C60. Nature 1987, 325, 760−761. (44) Coulson, C. A.; Gomes, J. A. N. F.; Mallion, R. B. Ring Magnetic Susceptibilities in Conjugated Hydrocarbons. Mol. Phys. 1975, 30, 713−732. (45) Gomes, J. A. N. F.; Mallion, R. B. Calculated Magnetic Properties of Some Isomers of Pyracylene. J. Org. Chem. 1981, 46, 719−727. (46) de Castro, B. R. M.; Gomes, J. A. N. F.; Mallion, R. B. πElectron “Ring Currents” in Dipleiadiene. J. Mol. Struct. (THEOCHEM) 1992, 260, 123−132. (47) de Castro, B. R. M.; Gomes, J. A. N. F.; Mallion, R. B. Calculated “London” π-Electron Magnetic-Properties of Some Carbon-Atom Fragments Extant on the Surface of Icosahedral C60 (Buckminsterfullerene). Croat. Chem. Acta 1993, 66, 49−58. (48) Haigh, C. W.; Mallion, R. B. Ring Current Theories in Nuclear Magnetic Resonance. In Prog. Nucl. Magn. Reson. Spectrosc.; Emsley, J. W., Feeney, J., Sutcliffe, L. H., Eds; Pergamon Press: Oxford, England, United Kingdom, 1979/1980, Vol. 13, pp 303−344. (49) Barth, W. E.; Lawton, R. G. Dibenzo[ghi,mno]fluoranthene. J. Am. Chem. Soc. 1966, 88, 380−381. (50) Barth, W. E.; Lawton, R. G. The Synthesis of Corannulene. J. Am. Chem. Soc. 1971, 93, 1730−1745. (51) Benshafrut, R.; Shabtai, E.; Rabinowitz, M.; Scott, L. T. πConjugated Anions: From Carbon-Rich Anions to Charged Carbon Allotropes. Eur. J. Org. Chem. 2000, 1091−1106. (52) Longuet-Higgins, H. C.; Salem, L. The Alternation in Bond Length in Long Conjugated Chain Molecules. III The Cyclic Polyenes C18H18, C24H24, C30H30. Proc. Royal Soc. (London) 1960, A257, 445− 456. (53) Salem, L. Molecular Orbital Theory of Conjugated Systems; W. A. Benjamin, Inc.: New York, 1966; Chapter 4. (54) Amos, A. T.; Roberts, H. G. Ff. The Calculation of Chemical Shifts in Conjugated Molecules. I. A Current Density Approach. Mol. Phys. 1971, 20, 1073−1080. (55) Haddon, R. C. The Application of the Biot-Savart Law to the Ring-Current Analysis of Proton Chemical Shifts  I. A Comprehensive Investigation of the Annulenes. Tetrahedron 1972, 28, 3613−3633. (56) Haddon, R. C. The Application of the Biot-Savart Law to the Ring-Current Analysis of Proton Chemical Shifts  II. An Approach to Aromatic Character in the Annulenes. Tetrahedron 1972, 28, 3613− 3633. (57) Haigh, C. W.; Mallion, R. B. Rationalisation of Relative “RingCurrent” Sizes in Polycyclic, Conjugated Hydrocarbons. Croat. Chem. Acta 1989, 62, 1−26. (58) As claimed elsewhere,23,24 bond currents calculated by the ab initio ipso-centric method obey Kirchhoff’s Law of Conservation of Current at Junctions19,15,57 only in the limit of an infinite basis-set. Those bond currents obtained by the HLPM formalism do, by contrast, (automatically) comply rigorously with the requirements of the Kirchhoff Conversation Law.52,53,15,57 There are, however, likewise similar problems about current conservation at junctions even in the case of certain other semiempirical methods  such as the uncoupled Hartree−Fock approach54  in which (unlike in the case of the HMO conventions11) matrix elements between non-neighboring centers are, in general, nonzero. This point is considered in some detail in the long footnote on page 1420 of ref 15. In this context, it is conceptually helpful59,19 to think of a ring current, in a microscopic, conjugated hydrocarbon, as being the analogy of the familiar loop current60,61 in a macroscopic electrical network, while the bond-current is the 3696



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microscopic analogy of the current in a wire that constitutes one of the arms of the macroscopic electrical network.59−61 Thinking in this way provides an exact 1−1 analogy between the loop current in a macroscopic network and the ring current in a (microscopic) conjugated hydrocarbon, as well as between the current in a wire/ arm of a macroscopic electrical-network and a bond current in the corresponding microscopic, unsaturated system. (59) Gomes, J. A. N. F.; Mallion, R. B. The Concept of Ring Currents. In Concepts in Chemistry; Rouvray, D. H., Ed.; Research Studies Press Limited: Taunton, Somerset, England, United Kingdom, 1997, and John Wiley & Sons, Inc.: New York, 1997; Chapter 7, pp 205−253. (60) Cundy, H. M. Advanced Mathematics Book 3 [Metric]; Cambridge University Press: London & New York, 1970; p 91. (61) Smart, D. Linear Algebra & Geometry, S. M. P. Further Mathematics Series, Cambridge University Press: Cambridge, England, United Kingdom, 1988; pp 307−308. (62) Coulson, C. A. Present State of Molecular Structure Calculations. Rev. Mod. Phys. 1960, 32, 170−177.

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Ï-Electron Ring-Currents and Bond-Currents in Some Conjugated

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Ï-Electron Ring-Currents and Bond-Currents in Some Conjugated