p -Electron ring-currents and bond-currents in [10,5]-Coronene and related structures conforming to the ‘Annulene-Within-an-Annulene’ model

A series of hypothetical conjugated structures is defined; the series is called the p -Coronenes and the first four members of it are shown to respect the ‘Annulene-Within-an-Annulene’ (AWA) model when tested by means of Hu¨ckel–London–Pople–McWeeny (HLPM) p -electron ring-current and bond-current calculations. The first member of this series, 5-Coronene, is also a member of the regular [ r , s ]-Coronene series, where it is known as [10,5]-Coronene. It is shown that, as p is varied (with p always odd, and with p > 3) through the values 5, 7, 9, 11, etc. , the resulting structures alternate between a ‘[4 n + 2]-Annulene-Within-a-[4 m ]-Annulene’ (if ( p (cid:2) 1) is divisible by 4) and a ‘[4 n ]-Annulene-Within-a-[4 m + 2]-Annulene’ (if ( p (cid:2) 1) is not divisible by 4). It is therefore claimed that the p -Coronenes constitute an ideal series for testing the AWA model. It is also remarked that each member of the p -Coronene series has only four Kekule´ structures, and that the ‘spokes’ or ‘transverse’ bonds connecting the central [ p ( p (cid:2) 3)]-membered ring to the outer [ p ( p (cid:2) 1)]-membered periphery always have a Pauling bond-order of zero, ensuring that the outer and inner rings are ‘decoupled’; such bonds also bear zero bond-current, by symmetry. It is argued that the former property of these transverse bonds, rather than the latter, determines that the p -Coronenes obey the AWA rule—which is in fact an exception, rather than a ‘rule’ per se . The paper concludes by explicitly stating our philosophy that a conceptually simple model depending on no subjective (or any other) parameters whatsoever can give intuitive chemical insight for certain systems equal to that available from far-more complex methods such as ab initio calculations—what Coulson once famously called ‘primitive patterns of understanding’.

We draw attention here to the fact that [10,5]-Coronene (I) is merely the first of a homologous series of hypothetical conjugated structures that may be formed by joining a number, 2p, of p-sided regular polygons in such a manner that they join back onto themselves.The first four members of the series that we are here calling the 'p-Coronenes' are shown in Fig. 1: they are (I)-referred to, when considered as a member of this series, as '5-Coronene'-and 7-Coronene (II), 9-Coronene (III), and 11-Coronene (IV).We further demonstrate that these p-Coronenes form an ideal series for testing the AWA model 4-9 and, in view of the esteem in which the original London approach 13 is still held, 6,10,19,20 we investigate the magnetic properties of (I)-(IV) by applying the HLPM formalism 12-18 to calculate p-electron ring-currents and bond-currents in these structures.These more-simplistic calculations agree with the conclusions of Monaco et al. 2 -based, in the main, on moresophisticated methods of calculation-that the bond currents in [10,5]-Coronene (I) are in accord with the qualitative predictions of the AWA model 4-9 and we extend this conclusion to the other structures, ((II)-(IV)), of the p-Coronene series that are here investigated. 21,22e p-Coronenes By considering, in the general case, the internal-angle sum in the central ring, we have verified that, on purely geometrical grounds-though this is not taking into account, of course, any considerations about potential molecular strain, if these structures were actually extant molecules-such p-Coronenes tessellate, forming a structure that has (a) An outer perimeter of length p( p À 1); (b) A central ring of length p( p À 3)-which, however, is a regular [ p( p À 3)]-gon only in the case where p = 5 (that is to say, only in the case of [10,5]-Coronene). 1 Furthermore, we have proved (inductively) that the following rules apply:  (ii) the central ring will be of length [4n] (with n = 1 4 p( p À 3)).( p = 7 (structure (II)) and p = 11 (structure (IV)) are like this.)It will be seen, therefore, that this p-Coronene series has, in the present context, the very pertinent property that, as p is varied (with p always odd, and with p > 3) through the values 5, 7, 9, 11, etc., the resulting structures alternate between a '[4n + 2]-Annulene-Within-a-[4m]-Annulene' (if ( p À 1) is divisible by 4) and a '[4n]-Annulene-Within-a-[4m + 2]-Annulene' (if ( p À 1) is not divisible by 4).
In addition, there is one other singular feature of this series, which is highly relevant for our purposes.Monaco et al. 2 observed that [10,5]-Coronene (I) has only four Kekule ´structures.These comprise the two ways in which the bonds in the outer perimeter may alternate, single and double, and, for each of these two ways for the outer ring, there are two ways in which the perimeter of the central ring may likewise be depicted with alternating single-and double-bonds.What Bochvar et al. 1 have called the 'transverse' bonds and Balaban et al. 7 have dubbed the 'spokes' bonds (that is, those bonds that symmetrically connect the outer perimeter to the inner ring) are formally 'single' in any Kekule ´structure that can be devised for the system as a whole.Another way of succinctly indicating this property is to note that these spokes 7 /transverse 1 bonds have zero 7 Pauling bond-order. 23It can be shown by induction that this is indeed the case for this entire series of p-Coronenes.In other words, the outer and inner rings of the structures in this series are always what Fowler et al. 2,[5][6][7] have previously described as 'decoupled'.(As an example, the four Kekule ´structures for 7-Coronene (structure (II)) are explicitly illustrated in Fig. 2

Calculations
HLPM 12-15 p-electron ring-currents 24 were calculated as described in ref. 16-18, by application of the equation where J i is the ring-current intensity in the ith ring of the conjugated system and J benzene is the corresponding ringcurrent intensity calculated, by the same method, for benzene (which has a numerical value, in these units, of (1/9)). 15It is immediately seen, therefore, that the HLPM ring-currents-being, as they are, expressed as a ratio ((J i /J benzene ))-are entirely dimensionless quantities, devoid of any units; the same is true of the bond currents that are derivable from these ring currents.The calculation is based on an arbitrary spanning-tree 15,25,26 15 defined as (respectively) the self-and mutual imaginary bond-bond polarisabilities of circuit-completing bonds m and n (likewise in the absence of a magnetic field).It is important for the philosophy of our approach to emphasise that all these quantities may be calculated solely from a knowledge of the molecular graph 25,27 of the conjugated system in question, and the eigenvalues and eigenvectors of its vertex adjacency-matrix, 25,28 which are latent in the structure and which are pre-determined as soon as the carbon-atom adjacencies are specified. 16It is also vital to note that P (m) , b% p (m)(m) and b% p (m)(n) are calculable solely from knowledge of that molecular graph, 25,27,28 without recourse to any parameters.Details of their explicit definition are to be found in the appendix of ref. 25 (which also corrects some typographical errors in McWeeny's original paper 15 ).S (m) is the signed (algebraical) area of the mth circuit (i.e., the enclosed area formed when the mth circuit-completing bond, only, is inserted into the spanning tree on which the calculation is being based 15,16,25,29 ).These areas are to be counted positive if the arrow on the mth circuit-completing bond (which bears an arbitrarily assigned direction) points in the anticlockwise sense around the circuit that it completes (the mth circuit), and are to be counted negative if that arrow points in the clockwise sense around the circuit that the mth circuit-completing bond completes. 16,29The quantities C i (m) are likewise purely topological in nature and take on the values 0, +1 or À1 according to whether (respectively) (a) the ith ring does not lie within the mth circuit, (b) the ith ring lies within the circuit completed by the mth circuitcompleting bond and that circuit-completing bond is directed in the anti-clockwise sense around the circuit (the mth one) that it completes, (c) the ith ring lies within the mth circuit but the mth circuit-completing bond points in the clockwise direction around the mth circuit.For detailed examples and more explanation of how the method is applied in practice the reader is referred to ref. 15,  16, 25, 29 and 34.It is important to specify the assumptions that have been made about ring areas.Strictly, our 'topological' approach [16][17][18] prescribes that the areas of all rings should be taken as the areas of regular polygons of the appropriate number of sides (of unit length).Whilst there is no difficulty in implementing this policy for the p-sided polygons on the peripheries of structures (I)-(IV), this assumption is not realistic for the (irregular 1 ) polygon that forms the central ring in structures (II)-(IV); this is because the areas when the central polygons are considered as if they were regular polygons are all much larger than the actual geometrical areas that the central rings are calculated to have if the tessellated polygons are to be arranged as they are in Fig. 1.Therefore, in the calculations presented in Table 1, (a) that on [10,5]-Coronene ((I)) is genuinely a 'topological' HLPM result (because the geometrical requirements determine that the central ring in (I) is, in any case, a regular polygon 1,2 ), but (b) the others (for structures (II)-(IV)) are not purely topological.Our computations on (II)-(IV) should therefore properly be described as being HLPM calculations [12][13][14][15][16] with the peripheral polygons being taken as regular polygons but with the central polygon having the actual geometrical area that the arrangement of the peripheral polygons (Fig. 1) requires.These centralring areas were calculated by elementary trigonometry.The values for these geometrical areas of those central rings are as follows (expressed as a ratio to the ring area of a standard benzene-hexagon), with the quantity in brackets being the area of a regular polygon of the corresponding number of sides (of unit length): It can be seen that these central-ring areas grow very rapidly, as the series is progressed.
Calculations were initially carried out in single precision and were, in the first instance, based on un-branched spanningtrees 25 -in order to be able to take the usual advantage of McWeeny's original unitary-transformation, 15 which requires the ring-current calculation to be founded on a spanning tree that represents a semi-Hamiltonian path 25 through the molecular graph under study.This policy, however, gave rise to considerable, and unacceptable, instability in the ring-current intensities calculated for the ten symmetrically-equivalent fivemembered rings in [10,5]-Coronene (I).We therefore sought to correct this by repeating the calculations, but this time basing them on branched spanning-trees, for which circuit areas 15,16 are usually smaller; in such cases, the more-general unitarytransformation proposed by Gayoso and Boucekkine 26 had to be invoked in order to be able to effect the ring-current calculations.Still, however, there was instability, with the independently calculated ring-current in each of the symmetrically equivalent five-membered rings not converging to the same values when rounded to three places of decimals.It was only when we combined (a) using double-precision in the computational manipulations and (b) basing the calculations on a branched spanning-tree that we finally obtained values of ring-current intensities in all ten of the symmetrically equivalent peripheral pentagonal rings that were identical when corrected to three decimal-places, the accuracy to which, in the past, 8,9,[16][17][18] we have routinely quoted our calculated HLPM ring-currents and bond-currents.Because round-off errors could also potentially arise from the very large circuit-areas that are encountered in the case of structures (II)-(IV)-even when a branched spanning-tree 25,26,29 is used-we judge that it is appropriate in those cases to limit quotation of bond-and ring-currents to only two places of decimals; for consistency, therefore, the corresponding data for structure (I) ([10,5]-Coronene) are likewise quoted to only two decimal-places, even though, in this case, we would have been confident in expressing our results to three decimal-places.(Furthermore, these calculations on [10,5]-Coronene, reported in the first row of Tables 1 and 2, have since been independently verified by Professor P. W. Fowler. 30)

Results and discussion
The results are summarised in Table 1.It should be emphasised that the set of ring currents and the set of bond currents for each structure are consistent with each other and are connected by a common compliance with the microscopic analogy of Kirchhoff's first law ('Conservation of Currents at a Junction') for macroscopic electrical-networks. 25,29,31,32 As is implied from eqn (1), ring currents are expressed as a ratio to the corresponding value calculated, by the same method, for benzene and the same goes for the bond currents that are derived from these ring currents.All currents are, accordingly, dimensionless quantities, as was already noted in the section entitled 'Calculations'.We discuss the results with reference to the specific example of [10,5]-Coronene ((I)).It is seen from Table 1 that the ringcurrent intensities in the peripheral five-membered rings are À1.57(that is, paramagnetic) and the ring-current intensity calculated for the central decagonal ring is also paramagnetic, at À0.16.Because the outer periphery is formed by bonds in the five-membered rings that are unshared with any other ring, the direction of current flow around the periphery is seen to be paramagnetic (that is, clockwise), with an intensity of 1.57-in qualitative accord with the AWA model, 4-9 as this outer periphery is of length 20; (4m, with m = 5).
The situation with the central ten-membered ring is as follows: the bonds in that ring form a part of the peripheral pentagonal rings and, consequently, those rings would provide a current of intensity of 1.57 in what is, from the point of view of the central ring, the anti-clockwise (diamagnetic) direction.But these bonds are also a part of the central, decagonal ring around which there is a ring-current of À0.16.This would therefore make a contribution of 0.16 in the clockwise (paramagnetic) direction around the central ten-membered ring.The overall effect of this electronic competition between the outer rings and the inner ring, therefore, is that those bonds in the central, decagonal ring (each of which is shared with a peripheral pentagonal ring) have a net current of (1.57À 0.16) = 1.41 in the anti-clockwise (diamagnetic) direction, so far as the central ring is concerned.Thus, despite the paramagnetic ringcurrent calculated for the decagonal ring, the actual p-electron flow around the central ten-membered ring-a [4n + 2]-ring, with n = 2-is in the anti-clockwise (diamagnetic) direction.This is entirely in accord with the qualitative predictions of the AWA model. 4-9Finally, we note that the bond current in the spokes bonds 7 /transverse bonds 1 in (I) are all zero, by symmetry-as they are in (II)-(IV) and, indeed, in all subsequent p-Coronenes.
A similar analysis to the above may be given for the peripheral and central rings of structures (II)-(IV) and the results are summarised in the last three rows of Table 1.In every case, the 'Annulene-Within-an-Annulene' model 4-9 is seen to be respected.

Conclusions
Monaco et al. 2 showed that [10,5]-Coronene (I) conforms to the qualitative AWA rule 4-9 when tested by means of calculated ab initio ipso-centric p-electron current-density maps, 5-7,10 'pseudo-p' calculations, 11 and simple Hu ¨ckel 12 considerations.Our present investigation, in which the AWA model 4-9 was tested by means of the more rudimentary but intuitive HLPM approach, 12-15 has come to the same conclusion.Structures (II)-(IV) also comply with the AWA rule, 4-9 though we emphasise again that our calculations in the case of these three structures are not purely topological, for we incorporated actual, calculated, geometrical ring-areas for the central rings, which (as discussed in ref. 1) are not regular polygons in the case of (II)-(IV).In view of the very large ring-areas involved in these calculations, we did have sufficient curiosity to investigate the influence of the various assumptions that we have made about ring areas by repeating all these calculations using purely 'topological' ring-areas (those given in brackets in the section labelled 'Calculations')-that is, assuming that the central rings in (II)-(IV) have the area of a Yes regular polygon with the same number of sides; the results are given in Table 2.][8][9][10]34 Following Fowler et al., 2,[5][6][7]35 we suggest that the answer may possibly lie in the fact that, in most 'superring' 3 structures, 5-10 the spokes bonds 7 /transverse 1 bonds connecting the outer perimeters to the inner rings are 'double' bonds in some Kekule ´structures that represent the systems as a whole, and 'single' in others. Inother words, there is what Fowler et al. 2,[5][6][7] call 'coupling' between the outer perimeter and the inner ring and, hence, the bonds in question generally have non-zero 7 Pauling bond-orders.23 We have already pointed out, however, in the section headed 'Calculations', that, in the p-Coronene series, the inner ring and the outer perimeter are always 'decoupled' in this sense because there are only ever four Kekule ´structures-all illustrated, in the case of 7-Coronene ((II)), in Fig. 2-and in none of these four Kekule ´structures are the spokes 7 /transverse 1 bonds represented as anything other than 'single' bonds.The central ring and the outer peripheryconnected, as they are, by transverse 1 bonds that all have zero 7 Pauling bond-order 23 -are thus always 'decoupled' 2,[5][6][7] in the p-Coronene series.We therefore concur that the reason that the members of the p-Coronene series that we have examined respect the AWA rule, 4-9 whereas most super-ring 3 structures do not, 5-10,34 is that, in the p-Coronene series, the 'decoupling' just described is always extant, whereas, in most 'super-ring' 3 systems, it is not.5-10, [34][35][36] The AWA-rule seems generally (and often indiscriminately) to be invoked by physical organic chemists apparently unaware of the relevance in this context of the intricacies of Graph Theory, Kekule ´structures, and Molecular Orbital Theory-such as have been gone into here. Inshort, it should perhaps be emphasised more strongly than is usually the case that the original AWA model 4 does in fact represent an exception, rather than a general rule.2,[5][6][7][8][9] In other words, we submit that the so-called 'AWA rule' is, in truth, a misnomer.
In conclusion, the Editor has invited us explicitly to spell out in detail why we favour using the HLPM 12-15 'topological' formalism [16][17][18] to complement ostensibly more justifiable ab initio approaches, such as ref.5-7, 10, 11, 21 and 24, and many others.Our philosophy here (and elsewhere 8,9,[16][17][18] ) is that a conceptually simple model that depends on no subjective (or any other) parameters can give intuitive chemical insight for certain systems equal to that available from far-more complex methods such as ab initio calculations-what Coulson once famously called 'primitive patterns of understanding'. 38ur whole aim here has been to demonstrate how we can tie down when the AWA concept 2,4-9 is and is not expected to work on the basis of a model that needs absolutely no subjective-or, indeed, any other-parameters whatsoever, and that requires no more, as a starting-point, than a mere knowledge of the carbon-carbon s-bond connectivity of the conjugated system under study (i.e., its 'molecular graph' 27 ) and the areas of its constituent rings.In the case of the one structure ([10,5]-Coronene) of the general series defined and presented here that has been treated by a more-sophisticated method, 2 we have shown that qualitative agreement regarding that structure's compliance with the AWA model can be achieved between the simple and intuitive HLPM 12-15 'topological' formalism [16][17][18] and an ostensibly more-refined calculation, effected by means of an ab initio approach. 2Furthermore, another advantage of the HLPM 'topological' method [12][13][14][15][16][17][18] is that, although it is crude, it is capable of yielding quantitative ring-current and bond-current intensities (quoted in Tables 1 and 2 to two decimal-places) rather than pictorial current-density diagrams. 2s argued, once this molecular graph and these ring areas are established, the results and conclusions of an HLPM calculation depend on no parameters whatsoever.When investigating ring currents in polycyclic conjugated systems, it should be borne in mind that using more-sophisticated models, even though they may be labelled 'ab initio', does involve a choice of numerical values for parameters.To see this one need only glance, for example, at the section headed 'Computational Details' (page 848) of ref. 21b or that labelled 'Ab Initio Calculations' (page 7448) of the paper by Monaco et al. 2 (which was the starting point for the present investigation).It is clear that such methods-though, of course, properly classified as 'ab initio'do nevertheless need extensive 'parameterisation'.
We therefore offer the HLPM approach for consideration as a vehicle that offers the Chemist an intuitive, parameterindependent appreciation-entirely complementary to, and not in any way intending to be competing with, ab initio calculationsof what happens to conjugated structures, such as those considered here, when they are in the presence of an external magnetic-field.is named 'Coronene-10,5'.The original authors 1a pointed out that this structure could be '. ..considered as two rings symmetrically connected by ten transverse bonds: an inner ten-membered ring for which Hu ¨ckel's criterion is satisfied, and an outer 20-membered ring not satisfying Hu ¨ckel's criterion.'According to Bochvar et al., 1a a structure is denoted as 'Coronene-n,m' if n is the number of vertices in what those authors call 1a the 'framed polygon' (that is, the central ring) and m is the number of vertices in what they term (in the English translation 1a from the original Russian) the 'framing polygon'-by which appears to be meant, not the grand polygon serving as the outer perimeter (as might initially be thought from the above expression), but the number of vertices in each polygon that forms a part of the periphery.Thus, in the case of Coronene-10,5, a ten-membered central-ring (hence the '10') is tessellated with surrounding pentagons (hence the '5').Furthermore, Bochvar et al.Following Fowler et al., 2,[5][6][7]35 we suggest that it is the former ((a)), and not the latter ((b)), which is responsible for the fact that (I)-(IV) obey the AWA rule. 4- As a counterexample to any suggestion otherwise we cite the case of Coronene ('regular' [6,6]-Coronene 1 ) itself (structure 1 of ref. 7-9): the spokes 7 /transverse 1 bonds in that conjugated system likewise bear zero bond-current by symmetry, but those bonds are 'double' in some of the 20 Kekule ´structures, 7 and 'single' in others; they thus have a non-zero Pauling bond-order 23 (of 0.407) 7 and so the 18-membered perimeter and the six-membered inner-ring in Coronene are 'coupled', in the sense that Fowler et al. 2,[5][6][7] introduced that term-and in the sense in which we are also using the term, here.As is well known (e.g.ref.
7-9 and 37), Coronene itself violates the AWA rule.On the other hand, the fact that the spokes bonds 7 /transverse bonds 1 have zero Pauling bondorder 23 in a given conjugated system does not seem inevitably to guarantee compliance with the AWA rule.A counterexample to any suggestion otherwise is to be found amongst the non-alternant isomers of Coronene that labelled 16, 17, and 18 in ref. 7 and 9; each of these has only four Kekule śtructures 7 and 16-18 thus possess transverse 1 bonds of zero Pauling bond-order 23 that connect the six-membered central-ring to the 18-membered perimeter in each of 16, 17 and 18.Both Balaban et al., 7 by means of their ab initio ipsocentric calculations, 5,6 and we, 9 by means of our HLPM 'topological' approach, [16][17][18] agree that 16 and 17 conform to the AWA, but that 18 does not.(Application of Randic ´'s 'conjugated-circuit' formalism, 36b however, gives rise to the conclusion 9,36c,d (a) If ( p À 1) is divisible by 4, then (i) the periphery will be of the '[4m]' type (with m = 1 4 p( p À 1)), and (ii) the central ring will be of the '[4n + 2]' type (with n = 1 4 {( p À 1)( p À 2) À 4}).( p = 5 (structure (I)) and p = 9 (structure (III)) are examples of this.)(b) If ( p À 1) is NOT divisible by 4, then exactly the opposite is true: (i) the periphery will be of length [4m + 2] (with m = 1 4 ( p + 1)( p À 2)) and
1a explicitly state that they are dealing only with what they call 'regular Coronenes-n,m', defined as being those that exhibit a regular polygon their centres.(What Bochvar et al. 1 called 'Coronene-n,m' we are, from here on, calling 'Coronene-r,s', in order to avoid potential confusion with our use of the symbols 'n' and 'm' to have other meanings in this paper; on the terminology of Monaco et al. 2 , the Coronene-r,s is called '[r,s]-Coronene'-the nomenclature that we adopt here.)It is easy to verify the assertion of Bochvar et al. 1a that, for such regular Coronenesr,s, the following simple relationholds: s ¼ 4r r À 2 .These authors pointed out 1a that only four regular [r,s]-Coronenes are actually realisable in practice-namely [3,12]-, [4,8]-, [6,6]-and [10,5]-Coronenes.([6,6]-Coronene does, of course, represent the carbon-atom connectivity of the extant benzenoid hydrocarbon Coronene itself.)In this paper, we are dealing only with entities in which a number, 2p, of regular polygons, each with p sides, are joined together as in Fig. 1 and for which (a) p is odd and (b) p > 3. It is straightforward to show that the only value of p that simultaneously satisfies these two conditions, as well as the above relation requiring, in this case, that p ¼ 4p p À 3 ð Þ p p À 3 ð ÞÀ2 , is in fact p = 5.Hence, [10,5]-Coronene or Coronene-10,5 is the only member of the series that we are studying here in which the central ring is a regular polygon.A final point should be noted: in regular [r,s]-Coronenes, each surrounding polygon on the periphery shares just one edge with the central ring; in the extension to the more-general series considered here, in which 2p regular polygons (p odd and p > 3), each with p sides, are joined in a closed form as in Fig. 1, each such regular polygon on the periphery shares (1/2)(p À 3) edges with the polygon-in general, a non-regular one-that constitutes the central ring.In order to avoid potential confusion between these two generally different series, therefore, we shall refer to our series defined here-the first four members of which are illustrated in Fig. 1-as the 'p-Coronenes'.What is called '[10,5]-Coronene' in the [r,s]-Coronene series 1,2 is thus coincident with what would be called '5-Coronene' in the p-Coronene series.That, however, is the only structure that these two series have in common.
that the entire trio of 16, 17, and 18 obeys the AWA rule); (b) M. Randic ´, Chem.Phys.Lett., 2010, 500, 123-127; (c) M. Randic ´, M. Novic ´, M. Vrac ˇko, D. Vukic ˇevic ánd D. Plavs ˇic ´, Int.J.Quantum Chem., 2012, 112, 972-985; (d) M. Randic ´, D. Vukic ˇevic ´, A. T. Balaban and D. Plavs ˇic ´, J. Comput.Chem., 2012, 33, 1111-1122.37 (a) I. J. Maddox and R. McWeeny, J. Chem.Phys., 1962, 36, 2353-2354; (b) N. Jonathan, S. Gordon and B. P. Dailey, J. Chem.Phys., 1962, 36, 2443-2448.38 (a) C. A. Coulson, Rev. Mod.Phys., 1960, 32, 170-177; (b) Commenting on the fact that the simple HLPM approach 16-18 shows that the ring-current intensity in the central ring of Peropyrene (structure (I) of ref. 18) is fully six times 38c that in the ostensibly similar central ring of Perylene (structure (II) in ref. 18), the late Professor Coulson remarked (in a letter to one of us [RBM] that is now amongst the Coulson Papers in the Bodleian Library, University of Oxford 38d ): 'I think your conclusions 38c about these ''ring currents'' are very exciting and clear cut. ... This seems to me to be one of those rare cases where a simple theory predicts a very profound difference in two kinds of situations and where even a rough measurement should be enough to show that the theory is either right or wrong.'That is the sort of philosophy that we are trying to invoke by means of this paper; (c) R. B. Mallion, J. Med.Chem., 1971, 14, 824-826; (d) C. A. Coulson, Letter to R. B. Mallion, December 1st, 1970, The Papers of Charles Alfred Coulson FRS 1910-1974 (Collection Reference CSAC 60.4.78),MS.Coulson, Box 175, Item No. 46, Department of Special Collections and Western Manuscripts, Bodleian Library, University of Oxford.We are very grateful to Mr Michael Hughes (Senor Archivist, Department of Special Collections and Western Manuscripts) and Mr Colin Harris (Superintendent, Special Collections Reading Rooms) for their kind help in locating the reference to this letter in the Coulson Papers at the Bodleian Library.
and the summations in the above equation are over what McWeeny, in his classic paper, 15 referred to as 'circuit-completing bonds'.b is the standard Hu ¨ckel resonance-integral, 12 P (m) is the ordinary 'fieldfree' Coulson bond-order, 12 and the quantities b% p (m)(m) and b% p (m)(n) are what McWeeny

Table 1 p
-Electron ring-currents and bond-currents in structures (I)-(IV) based on calculated geometrical areas for the central rings and areas of the appropriate regular-polygons for the surrounding, peripheral rings Yes Paper PCCP Open Access Article.Published on 10 April 2013.Downloaded on 8/10/2024 7:49:34 PM.This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Online This journal is c the Owner Societies 2013 Phys.Chem.Chem.Phys., 2013, 15, 8245--8253 8249

Table 2 '
Topological' p-electron ring-currents and bond-currents in structures (I)-(IV), where the areas of all polygons-including, even, the central one-are considered to be those appropriate to regular polygons with the corresponding number of sides