Topological Ring-Currents and Bond-Currents in ... - ACS Publications

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Topological Ring-Currents and Bond-Currents in Some Nonalternant Isomers of Coronene 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: The H€uckel
London
Pople
McWeeny (HLPM) (topological) approach for calculating π-electron ring-currents and bond-currents in conjugated systems is applied to coronene and the 17 nonalternant isomers of it for which Balaban, Bean, and Fowler recently published ab initio ipso-centric π-electron current-density maps. In a parallel investigation to that conducted by Balaban et al., the key findings of their quantum-mechanical study are confirmed by this topological one: (a) all seven rings in all of the 18 structures studied are found to bear substantial diamagnetic ring-currents, greater than the benzene value; (b) all 18 conjugated systems have a pronounced, overall London diamagnetic-susceptibility, perpendicular to their respective molecular-planes; (c) a substantial diamagnetic current runs around the periphery of each of these structures; and (d) there is general agreement between the two studies as to which structures, of those treated, respect the annulene-within-an-annulene model for conjugated systems of this sort. The same ideas are also tested against ring-current and bond-current intensities recently calculated by Randi
c et al. using purely graph-theoretical methods that are based on circuits of conjugation. Considerable qualitative agreement is found, and when there is disagreement, arguments are offered as to why the HLPM topological ring-currents and bond-currents are to be preferred.

’ INTRODUCTION The ab initio, ipso-centric current-density maps for coronene and its 17 nonalternant isomers, published by Balaban, Bean, and Fowler1 in 2010, have reawakened interest2
5 in the π-electron ring-currents6
9 and bond-currents10
12 that, classically, are considered to be extant in such structures when they are subjected to an external magnetic-field, at right angles to the molecular plane. Randi
c2 initially selected a particular one of these isomers for special study, that called [567567] on the nomenclature devised by Balaban, Bean, and Fowler,1 and this same structure was adopted by us and Gomes5 when comparing and contrasting various old13,14 and recent2,15 formalisms of the π-electron ring-current effect that are based on the concept of circuits of conjugation.13,16 Subsequently, Randi
c, with coworkers, has extended3,4 application of his own conjugatedcircuits approach2 to include all 18 of the isomers studied by Balaban et al.1 We have previously argued that the H€uckel
London
Pople
McWeeny (HLPM)17
23 approach to ring-current calculation is the most appealing of the methods that may be regarded as topological because (a) it requires knowledge only of the carbon
carbon connectivity of the conjugated system under investigation and the areas of its constituent rings; (b) it depends on no subjective (or any other) parameters; (c) it is founded on sound quantum mechanics and physical principles;5,17
23 r 2011 American Chemical Society

(d) it is less labor-intensive to apply to large conjugated systems than recently revived methods2
4,13
15 that depend on the concept of circuits of conjugation.13,16 The idea of topological π-electron ring-currents was alluded to in general terms many years ago,20 but it has only recently been rigorously and formally defined, first for benzenoid hydrocarbons,21 and then it was generalized so that the concept of topological π-electron ring-currents can be applied to conjugated systems containing rings of any size.22 We ourselves have recently invoked topological ring-currents when investigating23 the so-called24 empty rings in perylenes. We point out that the London model has rightly been described as venerable,25 while Aihara et al.26 have recently cited Fowler et al.27 when claiming that “...it is striking that the original H€uckel
London model is capable of reproducing the main features of the ab initio ring-current maps of polycyclic aromatic hydrocarbons. Quantitative problems about aromaticity and ring current diamagnetism can therefore be solved with H€uckel
London theory.” Accordingly, in this article, we report topological π-electron ring-currents and the concomitant bond-currents5,10
12,2
4,15 implicitly associated with them, for coronene and 17 isomers of it, structures 1
18 in Figure 1, for which ipso-centric π-electron Received: June 24, 2011 Revised: October 10, 2011 Published: October 11, 2011 13877

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The Journal of Physical Chemistry A

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Figure 1. Topological ring-currents (in black) and the associated bond-currents (in red) in structures 1
18 calculated (as specified in paragraphs a
c of the section entitled Calculations) by the HLPM topological approach,17
23 which invokes eqs 1 and 2. As explained in the text, the topological ringcurrents and bond-currents are dimensionless quantities; accordingly, all are appropriately depicted here as pure numbers, devoid of any units. The ring currents, all diamagnetic, are considered to circulate counterclockwise around the rings that are their respective domains, while the several bond currents flow in the direction indicated by the arrow depicted along each bond. The structures are labeled 1
18, but each also has a name in the form of a sixfigure code (given in column 2 of Table 1) devised by application of a systematic nomenclature that was introduced and defined by Balaban, Bean, and Fowler in ref 1.

current-density maps11,27,28 were computed by Balaban, Bean, and Fowler.1 We compare the trends within and the predictions of these maps with those deduced from our own HLPM topological ring-currents and bond-currents, and we also compare and contrast the corresponding quantities arising from the very recent calculations of Randi
c et al.,2
4 which rely on the idea of circuits of conjugation.13,16 In doing so, we pay particular attention to the following: (a) the size and the sign of the topological ring-current intensities in each of the 95 distinct and symmetrically nonequivalent rings in the 18 structures (1
18) that have been studied here; (b) the overall London π-electron diamagnetic-susceptibility6
12 computed, for each of these structures 1
18, from our topological ring-current intensities displayed in Figure 1; (c) the direction of circulation of bond currents around the peripheries of each of the 18 isomers in question; (d) the algebraical average of the bond-currents in the central rings; (e) the direction of the bond currents in each of the central rings of structures 1
18 and the identification of which structures, among the 18 studied, display bond-current behavior that is in accord with what is known1,29 as the annulene-within-an-annulene (AWA) model; (f) the average of the magnitudes of the bond currents in what Balaban, Bean, and Fowler1 call the spokes bonds connecting the central ring of each structure to its outer, peripheral, rings and whether these average magnitudes

are related in any way to Pauling bond-orders30 and/or to Coulson bond-orders.31 The quantities referred to in a
f, above, will all be of relevance when, as we intend here, an investigation parallel to the one conducted by Balaban et al.1 is undertaken, but based on the HLPM topological18
22 bond-currents and ring-currents, instead of the π-electron current-density maps that were adopted by Balaban et al.1

’ CALCULATIONS As described elsewhere,21
23 topological ring-currents (Ji/ JBenzene) in the ith ring of a polycyclic, conjugated system are defined as being π-electron ring-current intensities calculated according to the following specifications: (a) They are evaluated by application of the HLPM formulation,17
19,21,22 via eq 1.




Ji Jbenzene þ

¼ 9f

∑ ½PðμÞ þ βπ̅ ðμÞðμÞ S ðμÞCðμÞ i

ðμÞ

∑ðμ > > > > > > B C> > > B C > > >
C> 6B Jbenzene Jbenzene C> > > > B all peripheral rings, C> > > > @ A> > > > > : ; i ¼ B, C, D, E, F, and G

∑

where JA/JBenzene is the ring-current, relative to the benzene value, calculated for the central ring, regarded as being labeled A; the summation runs over all six of the peripheral rings (considered as being labeled B, C, D, E, F, and G, as in Figure 11 of ref 5) of each of structures 1
18. e A central six-membered ring in which every bond-current is in the same, clockwise (i.e., paramagnetic), direction around that ring. This occurs whenever the magnitude of the (diamagnetic) ring-current in the central ring (A) is less than the diamagnetic ring-current in every one of the surrounding peripheral rings (called B
G). Such structures (indicated in bold, in this Table) violate the annulene-within-an-annulene (AWA) model.1,29 f A central six-membered ring in which the bond-currents in some bonds are in the clockwise (paramagnetic) sense around that ring, and other bond-currents in the central ring point in the counterclockwise (diamagnetic) direction; however, paramagnetic is dominant, and so the algebraical average is negative (paramagnetic). g A central six-membered ring in which every bond-current is in the same, counterclockwise (i.e., diamagnetic), direction around that ring. This occurs whenever the magnitude of the (diamagnetic) ring-current in the central ring (A) is greater than the diamagnetic ring-current in every one of the surrounding peripheral rings (called B
G). Such structures (indicated in italic, in this Table) are in accord with the annulene-within-an-annulene (AWA) model.1,29 h A central six-membered ring in which the bond-currents in some bonds are in the counterclockwise (diamagnetic) sense around that ring, and other bond-currents in the central ring point in the clockwise (paramagnetic) direction (or, in the case of Randi
c's results,3 in the right-hand column, for structures 10 and 13, with zero bond-current); however, diamagnetic is dominant, and so the algebraical average is positive (diamagnetic). i These are computed from the raw (integral) bond-current data displayed in Tables 6, 8, and 10 in ref 3, after they have been normalized35 by division by the quantity K(K
1), where K is the number of Kekul
e-structures in the structure in question; K(K
1) is in fact the total number40 of sets of conjugation circuits.13,16 (Some of these K-values are listed in ref 3; the remaining ones were kindly communicated by Professor Randi
c.37 They are as follows: Structure 1: K = 20. Structures 2, 6, 9, 10, 14, and 15: K = 8. Structure 3: K = 12. Structure 4: K = 10. Structures 5, 8, 11, and 12: K = 7. Structures 7 and 13: K = 9. Structures 16, 17, and 18: K = 4.)

ring-current in benzene, while the central rings in three of these structures (7, 16, and 17, the ones that Balaban et al.1 noted as obeying the annulene-within-an-annulene rule1,29) bear diamagnetic ring-current intensities of more than twice the size of the ring-current in benzene.

Furthermore, it is immediately evident from the third column of Table 1 that the overall London diamagneticsusceptibilities of the hepta-cyclic structures 1
18, perpendicular to their (assumed) molecular planes, are all approximately 12 times the benzene value (mean, 11.6; standard deviation, 0.85). 13880

dx.doi.org/10.1021/jp205964t |J. Phys. Chem. A 2011, 115, 13877–13884

The Journal of Physical Chemistry A These susceptibilities (calculated via eq 3 from the topological ringcurrents computed by means of eq 1 and the ring areas obtained from eq 2), together with the substantial sizes of the diamagnetic ring-current intensities in the 95 nonequivalent rings in 1
18, referred to above, are thus in close accord with the concluding statement of the paper by Balaban, Bean, and Fowler1 that “...all 18 isomers have an equal claim to the label 'aromatic' (on the ring-current criterion41,42) in spite of their likely differences in stability and reactivity.” Another of the observations of Balaban et al.1 upheld, equally unequivocally, by the data presented in Figure 1, is the assertion that “...all isomers 1
18 support strong diatropic [i.e., diamagnetic] perimeter currents” which, Balaban et al.1 point out, “...corresponds to a prediction of (magnetic) aromaticity for all these species.” 41,42 All three of these findings, the large individual diamagnetic ring-current intensities, the substantial overall London diamagnetic-susceptibilities, and the strong peripheral diamagnetic bond-currents, are consistent with the fact that structures 1
18 are all formally related to coronene (1) by means of a gradual transition from that structure. During the course of that transition, a pair of peripheral hexagons in coronene is replaced, in a stepwise fashion, by a pentagon and a heptagon, in various positions around the central hexagon. More drastic changes correspond to replacing nonadjacent pairs of hexagonal rings, the process of which may lead to there being adjacent pentagons and/or heptagons which, Balaban et al.1 claim, “...will be associated with lower aromaticity.” 41,42 Finally, it should be borne in mind that, as a result of these various metamorphoses from coronene (1), some of the structures 1
18, if they exist at all, are very likely to be geometrically nonplanar.43 Of particular interest to Balaban, Bean, and Fowler1 was what they termed the annulene-within-an-annulene (AWA) model.29 This idea emphasizes that the outer peripheries of 1
18 (which they call1 the rim) are 18 (= [4k + 2], with k = 4) carbon
carbon bonds in length, while the inner hexagon (what they call1 the hub) is also a cycle of length [4k + 2] (but, this time, with k = 1). Therefore, by virtue of the [4k]- and [4k + 2]-rules for paramagnetic and diamagnetic ring-currents, respectively, in the annulenes,41 if the rim and hub circuits functioned independently, one would expect the direction of circulation of each of the six bond-currents in the central ring also to be in the counterclockwise (diamagnetic) sense, thereby bringing about what Balaban et al.1 call con-rotating diatropic rim-and-hub currents. These authors found this behavior in their current-density maps for structures 7, 16, and 17. Such a pattern is also reflected in our own calculations (Table 1, fourth column) with the diamagnetic circulation around the inner hexagon being predicted to be especially strong in the case of 7 and 16 and less pronounced in 17. We also find AWA behavior, though more mildly, and with varying degrees of intensity, in 5, 6, 8, and 9. The data of Randi
c et al.2
4 (the right-hand column of Table 1) lead to the prediction of con-rotating diamagnetic rim-and-hub currents in all the isomers 1
18 except for 1, 3, 10, and 13. It is clear from footnote g of Table 1 that the annulene-within-an-annulene (AWA)1,29 model is validated whenever the magnitude of the (diamagnetic) ring-current in the central ring (A) is greater than the diamagnetic ring-current in every one of the surrounding peripheral rings (called B
G; see, for example, Figure 13 of ref 5). Average algebraical bond-currents in the central ring of the structures that respect the AWA model are indicated in italic (and flagged with footnote g) in Table 1. Exactly the opposite occurs whenever the magnitude of the (diamagnetic) ring-current in the central ring (A) is less than the

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diamagnetic ring-current in every one of the surrounding peripheral rings (called B
G). Such structures (for which the average algebraical bond-currents in the central ring are indicated in bold and are flagged with footnote e in Table 1) violate the annulene-within-an-annulene (AWA) model.1,29 When this happens, the direction of the bond-currents in all six of the bonds in the central ring is in the clockwise (paramagnetic) sense around that ring. Balaban et al.1 describe this behavior as counterrotating. When what Balaban et al. call the coupling between the rim and hub cycles is what they describe as strong, they claim1 that a simple H€uckel
London treatment17
19 predicts that the aforementioned counter-rotating phenomenon will be evident; (note, however, that in our topological approach,22,23 there is no scope for varying the HMO β-parameters). The three methods of calculation considered here, the Balaban, Bean, and Fowler current-density maps,1 our HLPM toplogical bond- and ringcurrents being reported here (Table 1, fourth column), and the (normalized3,4,37) bond- and ring-currents of Randi
c et al.2
4 (Table 1, right-hand column), all agree that coronene (1) is like this, and that, consequently, the AWA model29 is, thereby, violated in 1 (as, indeed, had already been predicted in ref 25). The HLPM topological method predicts such a violation also for 2, 3, and 18, whereas the only other structure for which the method of Randi
c et al.2
4 predicts counter-rotating behavior is 3. There are two other possibilities for the bond-currents in the central ring: (a) the bond-currents in some bonds are in the clockwise (paramagnetic) sense around that ring, and other bondcurrents in the central ring point in the counterclockwise (diamagnetic) direction; however, paramagnetic is dominant, and so the algebraical average is negative (paramagnetic); this situation is indicated by footnote f in Table 1. (b) the bond-currents in some bonds are in the counterclockwise (diamagnetic) sense around that ring, and other bond-currents in the central ring point in the clockwise (paramagnetic) direction (or, in the case of Randi
c’s results for structures 10 and 13, have zero bond-current); however, diamagnetic is dominant, and so the algebraical average is positive (diamagnetic). This situation is indicated by footnote h in Table 1. The above considerations show that discussion of the magnetic character of the central rings is a complex matter. These rings are entirely surrounded by other rings, and it should be noted that, just because the calculated ring-current intensity in the central ring may be positive (such as, for example, it is in coronene (1) and indeed as it is in each of 1
18), this does not necessarily mean that the currents actually flowing in the bonds of the central ring all circulate in the counterclockwise direction around that ring.45 Indeed, in coronene (1), the opposite is the case, for each and every bond in the central ring, as, in fact, the current-density maps of Balaban et al.1 and Randi
c’s graphtheoretical currents readily concur.3 This is because the actual (net) current flowing in any one bond of the central rings in these structures is the algebraical resultant of diamagnetic currents in the central ring and in the relevant peripheral ring that shares a bond with it. Each of these diamagnetic currents is competing for the direction of net flow in a given bond in the central ring, and the current in such a bond will thus be the difference between two quantities which, frequently, are of comparable size. The consequence of this finely balanced sensitivity of bond-currents in 13881

dx.doi.org/10.1021/jp205964t |J. Phys. Chem. A 2011, 115, 13877–13884

The Journal of Physical Chemistry A the central ring to the ring-current intensities calculated for that ring and the peripheral rings is that the quantitative HLPM currents and the semiquantitative ispo-centric current-density maps1 will, on the face of it, be as likely to disagree as to agree on the individual directions of bond-currents in the central rings of 1
18. Thus, the two agree on directions quite well for 1, 5, 7, 8, 9, 16, and 17; they agree partially in the case of 3, 4, 6, 10, 13, and 15; and they totally disagree in the cases of 2, 11, 12, 14, and 18. For the reasons outlined above, this is to be expected. It may be observed in passing that the best agreement in this respect (structure 16) and the worst agreement (structure 18) both occur among the structures (16
18) whose spokes bonds1 have a zero Pauling bond-order.1,30 Finally, other bonds in 1
18 that were of interest to Balaban, Bean, and Fowler in their original investigation are those that connect the inner six-membered hub to the outer, 18-membered periphery of these structures. Balaban et al.1 called these six bonds (pursuing the wheel analogy) the spokes bonds, and they tried to correlate the bond-current activity in these bonds with their Pauling bond-orders,30 with, however, only limited success. As an alternative approach, therefore, we attempted to confront the average of the moduli of the bond-current in the spokes bonds with the total Coulson bond-order31 for each of the species 1
18, but we found correlation little better than that observed by Balaban et al.1 when they tried to relate Pauling bond-orders in the spokes bonds to the ab initio current-densities in the vicinity of these bonds. This is probably because (i) the bond orders (of both types) in the spokes bonds show very little variation from one structure to another, and (ii) in general, the bond-currents in the spokes bonds are, in any case, very small (see Figure 1) being, in effect, (as we have already seen to be the case with the bonds in the central rings) just the algebraical difference between two quantities (the topological diamagnetic ring-current intensities in adjacent rings on the periphery) that, as can be seen from Figure 1, are themselves frequently not very different in size.

’ CONCLUSIONS We should properly preface any conclusions by stating that the ab initio π-electron current-density maps of Balaban, Bean, and Fowler1 show the current-densities 1a0 above the molecular plane after they have been projected into the molecular plane; as such, the projected current-densities do not necessarily respect Kirchhoff’s Law34
36 of Conservation of Currents at a junction because they are just one component of the real current.5 Consequently, although they are visually appealing, these currentdensity maps are not directly comparable with the bond-currents (which are regarded strictly as classical line-currents10
12,5) that are being considered elsewhere in this article. Nevertheless, given that the current-density maps themselves are in any case only qualitative, while we claim that our HLPM topological ring-currents17
19 are (within the strict limitations of their definition22,23) quantitative, we permit ourselves to draw the following main conclusions from this investigation: (a) All seven rings in all the structures 1
18 bear substantial diamagnetic ring-currents, all greater (and some considerably greater) than the ring-current intensity calculated by the same method for benzene. (b) The overall London diamagnetic-susceptibility, perpendicular to the (assumed) molecular planes of these 18

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heptacyclic systems, is consistently between nine and 14 times the benzene value. (c) A substantial diamagnetic ring-current runs around the periphery of each of the structures 1
18. (d) Structures 7 and 16, and, to lesser extents, 17, 5, 6, 8, and 9, respect the annulene-within-an-annulene (AWA) model. The observations a
c are in accord with the deductions of Balaban et al.1 that “...all 18 isomers have an equal claim to the label 'aromatic' (on the ring-current criterion41) in spite of their likely differences in stability and reactivity” and that “...all isomers 1
18 support strong diatropic [i.e., diamagnetic] perimeter currents” which, Balaban et al.1 point out, “...corresponds to a prediction of (magnetic) aromaticity for all these species,”41 and which is also generally in accord with the conclusions of Randi
c et al. on the aromaticity of these species.42 Furthermore, conclusion d partially agrees with Balaban et al.1 that 7, 16, and 17 are the most likely candidates among 1
18 for being structures that are in accord with the AWA model.29 The calculations of Randi
c et al.2
4 (Table 1, right-hand column) also support conclusions a and c, above, while, in connection with conclusion d, the calculations of Randi
c et al.2
4 indicate that all of structures 1
18 should respect the AWA model, to greater or lesser extents, with the exception of 1, 3, 10, and 13. Where the Randi
c2
4 graph-theoretical ring-currents and bond-currents disagree with the corresponding topological HLPM quantities,21,22 we are inclined to put more faith in the latter. This is because (a) the HLPM topological approach does properly take into account the areas5 of the structures’ constituent rings, and (b) although, in practice, the HLPM bond-currents and ringcurrents are, as we have emphasized, likewise effectively graph-theoretical in nature, in that they depend only on the carbon
carbon connectivity of the conjugated system in question and the areas of its constituent rings, the formulations used to calculate them are founded on the long-established methods of London, Pople, and McWeeny,17,18 which have their roots firmly fixed in conventional quantum mechanics rather than being free-standing graph-theoretical algorithms.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT We are very grateful to Professor Milan Randi
c for kindly allowing us sight of preprints and proofs of several manuscripts (refs 3, 4, and 37) before their actual publication, and we also appreciate friendly correspondence with Professor Randi
c on the topics covered in this article. ’ REFERENCES (1) Balaban, A. T.; Bean, D. E.; Fowler, P. W. Acta Chim. Slov. 2010, 57, 507–512. (2) Randi
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(27) (a) Fowler, P. W.; Steiner, E.; Acocella, A.; Jenneskens, L. W.; Havenith, R. W. A. J. Chem. Soc. Perkin Trans. 2 2001, 1058–1065. (b) Fias, S.; Fowler, P. W.; Delgado, J. L.; Hahn, U.; Bultink, P. Chem.—Eur. J. 2008, 14, 3093–3099. (28) (a) Steiner, E.; Fowler, P. W.; Jenneskens, L. W. Angew. Chem., Int. Ed. 2001, 40, 362–366. (b) Fowler, P. W.; Steiner, E. J. Phys. Chem. A 2001, 105, 9553–9562. (c) Fowler, P. W.; Steiner, E. Chem. Phys. Lett. 2002, 364, 259–266. (29) Benshafrut, R.; Shabtai, E.; Rabinovitz, M.; Scott, L. T. Eur. J. Org. Chem. 2000, 6, 1091–1106. (30) Pauling, L.; Brockway, L. O.; Beach, J. Y. J. Am. Chem. Soc. 1935, 57, 2705–2709. (31) Coulson, C. A. Proc. R. Soc. London 1939, 169, 413–428. (32) (a) Mallion, R. B. Nuclear Magnetic Resonance: A Theoretical and Experimental Study of the Spectra of Condensed, Benzenoid Hydrocarbons. Ph.D. Thesis, University of Wales, U.K., 1969; Chapter 7, pp 160
177, Appendix J and Appendix F. (b) Haigh, C. W.; Mallion, R. B.; Armour, E. A. G. Mol. Phys. 1970, 18, 751–766. (c) Haigh, C. W.; Mallion, R. B. Mol. Phys. 1970, 18, 767–772. (d) Haigh, C. W.; Mallion, R. B. Mol. Phys. 1971, 22, 955–970. (e) Haigh, C. W.; Mallion, R. B. J. Chem. Phys. 1982, 76, 4063–4066. (33) The data for structure 13 (only) were presented in ref 5. (34) (a) Kirchhoff, G. Anal. Phys. Chem. 1845, 64, 497–514. (b) Kirchhoff, G. Anal. Phys. Chem. 1847, 72, 497–508. (35) (a) Cundy, H. M., Ed. S.M.P. Advanced Mathematics Book 3 [Metric]; Cambridge University Press: London, 1970; p 911. (b) Smart, D. Linear Algebra & Geometry, S. M. P. Further Mathematics Series; Cambridge University Press: Cambridge, U.K., 1988; pp 307
308. (36) (a) Historical note: In Kirchhoff’s first paper34a on the subject (published in 1845 from the physikalischen Seminars of the Albertina University, K€onigsberg, when he was a 21-year-old undergraduate (see refs 36b and 36c), Kirchhoff labeled his Law of Current Conservation as Law 1, and his other Law (about the fall of potential between two points in a network evaluated along any path that connects them) as Law 2. Two years later, in a second pioneering paper,34b he reversed this labeling, calling the same Laws II and I, respectively. Nowadays, Kirchhoff’s initial labeling-scheme34a is almost universally adopted, and consequently, the Law of Conservation of Current at a junction is now generally regarded as Kirchhoff’s First Law.36b,36c (b) Mallion, R. B. MATCH 2007, 58, 15–52. (c) Mallion, R. B. The Bulletin of the British Society for the History of Mathematics 2008, 23, 24–36. (37) Randi
c, M. Unpublished work, 2010. (38) Maddox, I. J.; McWeeny, R. J. Chem. Phys. 1962, 36, 2353–2354. (39) Jonathan, N.; Gordon, S.; Dailey, B. P. J. Chem. Phys. 1962, 36, 2443–2448. (40) Gutman, I.; Randi
c, M. Chem. Phys. 1979, 41, 265–270. (41) (a) The suggestion that ring-currents could be a material symptom in the diagnosis of aromaticity (strictly, though, only in the context of monocyclic conjugated systems) was first put forward nearly 50 years ago41b
f and has since been much debated, not least by one of the present authors (e.g., refs 41g and 9); here, however, we are deliberately avoiding any possible polemic about whether aromaticity is connected with ring-currents. (b) Elvidge, J. A.; Jackman, L. M. J. Chem. Soc. 1961, 859–866. (c) Elvidge, J. A. Chem. Commun. 1965, 160–181. (d) Abraham, R. J.; Sheppard, R. C.; Thomas, W. A.; Turner, S. Chem. Commun. 1965, 43–44. (e) Davies, D. W. Chem. Commun. 1965, pp 258
258; (f) Abraham, R. J.; Thomas, W. A. J. Chem. Soc. B 1966, 127–131. (g) Mallion, R. B. Pure Appl. Chem. 1980, 52, 1541–1548. (42) Note that Randi
c et al.3,4 have recently introduced several gradations of aromaticity based on a combination of criteria that involves ring-currents and conjugation circuits.13,16 Randi
c’s scheme ranges its categorizations on a seven-point scale and classifies a structure as aromatic if none of its ring-currents is clockwise (paramagnetic), and describes it as fully aromatic if that condition applies and all its circuits of conjugation are of size (4k + 2). On this basis, 1
4, 8, 10
12, and 16
18 are fully aromatic, whilst 5
7, 9, and 13
15 are only aromatic. 13883

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(43) It should be noted that some of the structures 1
18 may be nonplanar. Strictly, the unmodified HLPM method should not be applied to these molecules (though this has been done from time to time, in the past, as discussed in refs 21 and 22). Because the topological ring-current is here effectively being regarded as a mathematical, essentially graph-theoretical, index, calculated without assumed knowledge of any parameters (apart from assumptions about ring areas), and is not necessarily being regarded as an assumed physical quantity per se,21,22 this will again be done here. It is not strictly legitimate to perform HLPM ring-current calculations on such conjugated systems and then to attribute to the results any physical significance because ringcurrent in the HLPM model is precisely defined as an exact quantity only for planar molecules. The topological ring-currents presented in Figure 1 for any such geometrically nonplanar molecules are thus, in effect, those for a hypothetical molecule having the same carbon
carbon connectivity as the actual molecule under consideration but envisaged as if it were (geometrically) planar.21,22 (44) (a) In ca. 1966, five groups44b
e,10b independently gave prominence to the observation that paramagnetic ring-currents are to be expected in the [4k]-annulenes, though this conclusion was actually implicit in the work of the French school in the early 1950s44f
i and, a decade later, in the paper of Wagniere and Gouterman.44j Refs 44k
44p are a selection of papers discussing this topic during the succeeding decade. Nowadays, the parametric/diamagnetic nature of the [4k]-/ [4k + 2]-annulenes is part of what is commonly accepted canon, in this field. (b) Dorfman, Y. G. In Diamagnetism and the Chemical Bond; Poole, C. P., Translation Ed.; Elsevier: New York, 1965; pp 15
16. (c) LonguetHiggins, H. C. Aromaticity; Chemical Society Special Publication No. 21; The Chemical Society: London, 1967; p 109. (d) Pople, J. A.; Untch, K. G. J. Am. Chem. Soc. 1966, 88, 4811–4815. (e) Baer, F.; Kuhn, H.; Regel, W. Z. Naturforsch., A: Phys. Sci. 1967, 22, 103–112. (f) Mayot, M.; Berthier, G.; Pullman, B. J. Phys. Radium 1951, 12, 652–658.(g) Berthier, G.; Pullman, B.; Pullman, A. Le Diamagn
etisme des Compos
es Aromatiques. In Les Th
eories Electroniques de la Chimie Organique; Pullman, B., Pullman, A., Eds.; Masson: Paris, 1952; Chapter IX, pp 527
550. (h) Pullman, B.; Pullman, A.; Bergmann, E. D.; Berthier, G.; Fischer, E.; Hirshberg, Y.; Pontis, J. J. Chim. Phys. Phys.
Chim. Biol. 1952, 49, 24–28. (i) Mayot, M.; Berthier, G.; Pullman, B. J. Chim. Phys. Phys.
Chim. Biol. 1953, 50, 176–182. (j) Wagniere, G.; Gouterman, M. Mol. Phys. 1962, 5, 621–627. (k) Haddon, R. C.; Haddon, V. R.; Jackman, L. M. Top. Curr. Chem. 1971, 16, 103–220. (l) Sondheimer, F. Acc. Chem. Res. 1972, 5, 81–91. (m) Corradi, E.; Lazzeretti, P.; Taddei, F. Mol. Phys. 1973, 26, 41–47. (n) Benassi, R.; Lazzeretti, P.; Taddei, F. J. Phys. Chem. 1975, 98, 5451–5456. (o) Aihara, J.-I. J. Am. Chem. Soc. 1979, 101, 558–560. (p) Haddon, R. C. J. Am. Chem. Soc. 1979, 101, 1722–1728. (45) Dickens, T. K.; Mallion, R. B. Chem. Phys. Lett. 2011, DOI: 10.1021/cplett111169.

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