Comparisons of π-Electron Ring-Current and Bond-Current Patterns Calculated by Topological ( ‘ HLPM ’ ) and Ab Initio ( ‘ Ipso-Centric ’ ) Formalisms for Two Isomeric Conjugated Hydrocarbons , Corazulene and Cornaphthalene

The π-electron ring-currents and bond-currents associated with the isomeric structures corazulene (1) and cornaphthalene (2) are calculated by means of the rudimentary topological Hückel–London–Pople–McWeeny (HLPM) method (which is entirely equivalent to the recently named ‘graph-theoretical CD–HL’ approach).These currents are compared with analogous quantities computed by Lillington et al. by use of the more-sophisticated ipso–centric ab initio approach. The simple HLPM method is seen to exhibit a remarkable ability to reproduce complex patterns of current in large polycyclic hydrocarbons — the successful prediction of which, ostensibly, might naïvely be expected to be the preserve only of more sophisticated, and much less intuitive, ab initio calculations. This conclusion is entirely consistent with findings from contemporary work on other structures by the present, and other, authors.


INTRODUCTION
N THIS PAPER, we take the opportunity afforded by the availability of certain ab initio calculations [1] to assess how well predictions of π-electron ring-current and bondcurrent intensities calculated for conjugated systems by the rudimentary Hückel [2] -London [3] -Pople [4] -McWeeny [5] ('HLPM') formalism [6] mimic those of more-sophisticated approaches, notably the ab initio 'ipso-centric' [7] and 'pseudo-π' [8] methods.Ab initio procedures frequently depend first on the application of elaborate software, such as the Gaussian program, [9] in order to optimise starting geometries, and then on the choice of wave-function basis-set employed (see, for example, 'Computational Details' [p.848] of Ref 10, 'Ab Initio Calculations' [p.7448] of Ref. 11 and 'Method' [p.655] of Ref. 1.) Application of the HLPM formalism, [2][3][4][5][6] on the other hand, depends only on knowledge of how the carbon atoms forming the conjugated system in question are joined to each other in the σ-bonded network, and on the areas of its constituent rings. [6]Once the matter of ring areas has been decided, topological π-electron bond-currents and ring-currents in such systems are thus effectively latent in the molecular graph of the hydrocarbon under study and, accordinglyeven though their actual computation may still have to be effected -these currents are at least implicitly and immediately predetermined as soon as the structure's molecular graph [12] has been written down. [6]n isolated cases (for example, that of 7-coronene [13][14][15] ) there have been discrepancies; however, in the large majority of the structures examined so far -such as coronene and many of its 17 non-alternant isomers, [16,17] [10,5]-coronene, [11,13,14] and especially the novel series of 'altan' structures, [18][19][20][21][22][23][24] recently defined and introduced by
We here effect comparison between the topological HLPM and the ab initio philosophies by comparing predictions from the two approaches relating to a pair of isomeric structures: namely, the non-alternant [2] corazulene (1) and its alternant isomer cornaphthalene (2) (Figure 1).These were studied by Lillington et al. [1] by means of the ipso-centric method. [7](These structures were considered of interest because it has been suggested that 1 and 2 could act as building blocks for carbon nano-structures. [27]) Lillington et al. [1] have observed that if a 'circulene' (also known [6] as a 'super-ring' system) can be visualised as 'a wheel with spokes', [11,16] 1 and 2 can be thought of as a wheel with the addition of what they [1] call 'crossing chords'.Both 1 and 2 may thus be regarded as arrangements of [4]-membered central-rings inside a [28]-membered perimeter, '...with four spokes and four chords cross-linking the outer cycle.' [1]

CALCULATIONS
The computations were carried out by the HLPM method [2][3][4][5] as described in step-by-step detail in Ref. 6, the main crux of which is equation ( 14) of that reference.As is a requirement of the 'topological' version [6,28,29] of the HLPM approach, [2][3][4][5] adopted here, both 1 and 2 are assumed to be planar structures whose ring areas are those of regular polygons [4,29] of the appropriate number of sides, each of which is of uniform length (and equal to the carbon-carbon bond-length in benzene) -please see equation ( 16) of Ref. 6.In the present calculations, this implies adoption of the following ratios for the ring areas in 1 and 2: [4] if An is the area of a regular polygon of n sides, then, : : : 4cot : 5cot : 6cot : 7cot .4 5 6 7 (The only circumstances in recent work where a different assumption about ring areas needs to be made are in the case of structures with 'holes', such as kekulene [6,22,[30][31][32][33] and the p-coronenes. [13,14]) Lillington et al. [1] have pointed out that 1 is indeed planar (with C4h symmetry) and so also is 2, in a transitional form (with D4h symmetry); in reality, however, 2 is bowlshaped in its equilibrium conformation, with symmetry C4v.
In the present calculations we adopt C4h symmetry for corazulene (1) and D4h symmetry for cornaphthalene (2).The topological [6,28,29] HLPM ring-current and bond-current maps of 1 and 2, calculated as just described, are illustrated in Figures 2 and 3.As is conventional, [6] the ring currents and bond currents are presented as dimensionless numbers, expressed as a ratio to the corresponding quantities calculated, by the same method, for benzene.

(ii) Comparison of Ab Initio and HLPM Topological Currents in Corazulene (1)
Lillington et al. [1] reported a strong paramagnetic circulation around the central four-membered ring with a somewhat weaker diamagnetic current around the perimeter.Qualitatively, this pattern is confirmed by the topological HLPM calculations displayed in Figure 2. Quantitatively, the comparison is less impressive: Lillington et al. [1] find that what they call jmax, the largest value in their 'plotting plane' [1] of the current density per unit inducing field, is (when expressed as a ratio to the benzene value) about -0.91 for the central ring, whereas for the diamagnetic circulation on the perimeter it is ca.0.65.][40][41][42][43][44] We have, however, resisted applying this refinement here because, as we are testing a method that claims to depend on no empirical or subjective parameters whatsoever, we wish to maintain the description of the resulting π-electron currents as 'topological'. [6,28,29]illington et al. [1] further observe that the 'chord bonds' completing the four pentagonal rings (C) in 1 also support current that closes a ring-current loop in which the circulation is in the diamagnetic direction, '...but this is much smaller than the main perimeter current.' [1] This finding is also supported by the topological HLPM calculations presented in Figure 2: the 'chord bonds' carry a current of about 0.15, whereas the main perimeter current varies between approximately 0.65 and ca.0.79.Consistent with what Lillington et al. [1] called the 'circulene analogy', the current in the 'spokes' bonds was said [1] to be 'negligible', even though, with the symmetry that Lillington et al. assumed in Ref. 1, it was not forced to be zero by symmetry (as was the case in D5h planar corannulene and D6h coronene. [7,16,17,30]) In the course of carrying out the HLPM calculations displayed in Figure 2, however, C4h symmetry was assumed (as described under the heading 'Calculations') and, hence, in the case of our own computations being presented here, bond currents in the spokes bonds are indeed constrained to be zero by symmetry.

(iii) Comparison of Ab Initio and HLPM Topological Currents in Cornaphthalene (2)
Lillington et al. [1] observed that, among the induced currents that they calculated for the planar transition-state of structure 2, there is again a strong paramagnetic circulation around the central four-membered ring (A) (with a jmax value -relative to benzene -of -1.0) but with stronger local diamagnetic currents (jmax = 1.05) on the periphery of the four outer hexagons (C), and with weak currents linking them on the perimeter.Once again, the topological HLPM bond-currents for cornaphthalene (2) shown in Figure 3 qualitatively confirm this view.Quantitatively, the position is as follows: (a) the current in the bonds of the central fourmembered ring (A) is ca.-2.28 (again an overestimate compared with the ab initio values, an observation which is rationalised by the same explanation [39][40][41][42][43][44] as was offered in the previous section for a similar phenomenon encountered in corazulene (1)); (b) the peripheral bonds in the four outer hexagons (C) bear a diamagnetic bond-current of about 1.02 (very close to the ab initio estimate of 1.05); and (c) the currents in the bonds linking those outer hexagons to the rest of the structure are likewise weaker, at ca. 0.39.Lillington et al. [1] remark that when they relax their assumption of a planar (D4h) geometry for 2, assuming instead a 'bowl-shaped' structure with C4v symmetry, the features observed are reduced in intensity compared with the planar form, but these authors do retain in their conclusions the identification of (a) a paramagnetic π-electron circulation around the bonds of the four-membered central-ring (A), and (b) a diamagnetic circulation, of magnitude equal to that of the ring-current in benzene, in the peripheral bonds of the four outer hexagonal rings (C).These claims are again qualitatively -and, in the case of the four outer hexagonal rings, even quantitatively -consistent with the topological HLPM currents presented in Figure 3.

CONCLUSIONS
Comparison of predictions from the ipso-centric ab initio calculations of Ref. 1, and those from the topological [6,28,29] HLPM computations [2][3][4][5][6]28,29] displayed in Figures 2 and 3, confirm the following conclusions of Lillington et al.: [1] 2) are in such a direction that they complete diamagnetic current-loops around each of those hexagons (as seen in Figure 3). This i in agreement with the claim of Lillington et al. [1] that 2 exhibits a 'Clar-like' structure [45] with diamagnetic currents circulating around the four outer hexagonal rings (C).(d) Corazulene ( 1) has a single global diamagnetic circulation on its perimeter whilst cornaphthalene (2) has strong local diamagnetic circulations in separated parts of the perimeter.[1] Once again, therefore, the simple topological HLPM approach [2][3][4][5][6]28,29] -which depends on knowledge only of the molecular graph [12] of the conjugated system in question (in the form of a vertex-adjacency matrix describing it), [46] and the areas of its constituent rings -has been seen to demonstrate a possibly unexpected ability to reproduce complex patterns of current in large polycyclic hydrocarbons.According to common belief, such predictive success should ostensibly be the preserve only of sophisticated, and much less intuitive, [47,48] ab initio calculations.This conclusion is entirely consistent with our own previous findings [21,22] and with the view expressed in the recent, independent and simultaneous work of Gershoni-Poranne et al. on phenylenes, [34] that the relative currents calculated by the (equivalent of the) HLPM topological approach '...are remarkably similar to those extracted from the pseudo-π maps, [8] which themselves mirror the full ab initio maps [7,11,15,19,20,49] '…and that, in general, the HLPM topological method '...has a remarkable ability…to capture essential features of delocalised systems including patterns of current…' There is therefore now a growing body of evidence [17,21,22,32−37] that this is the case.[6,28,29] ring-currents (in black) and the associated topological [6,28,29] bond-currents (in red) for (alternant [2] ) cornaphthalene (2).For the conventions on displaying topological ring-currents and bond-currents please see the caption to Figure 2.
Figure 2. Maps for the topological[6,28,29] ring-currents (in black) and the associated topological[6,28,29] bond-currents (in red) for (non-alternant[2] ) corazulene(1).The topological ring-currents and bond-currents are dimensionless quantities.Positive (diamagnetic) ring-currents are considered to circulate anti-clockwise around their respective rings whilst 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.

Structures 1 and 2 Structures 1
and 2 both contain a central four-membered ring (labelled A, in Figure1) in which the topological HLPM paramagnetic ring-currents are similar (ca.-1.81 and -1.65, respectively).In 1, each edge of the central square (A) is shared with a seven-membered ring (B) (with diamagnetic ring-current ca.0.65) and in 2 each bond of the central fourmembered ring (A) is shared with a six-membered ring (B) bearing very nearly the same diamagnetic ring-current (0.63) as the corresponding (seven-membered) ring (B) in 1, just described.Each of the seven-membered rings (B) in 1 shares a bond with an outer five-membered ring (C) that bears a diamagnetic ring-current of ca.0.79, whilst the sixmembered rings (B) in 2 that share an edge with the central four-membered ring (A) also share a bond with an outer sixmembered ring (C) bearing a qualitatively similar, but quantitatively larger, diamagnetic ring-current (ca.1.02).The ring-current pattern for both 1 and 2 is, therefore, summarised as follows:(a) Central four-membered rings (A): paramagnetic (-1.81 and -1.65, in 1 and 2, respectively).(b) Rings (B) sharing a bond with the central square: diamagnetic (0.65 and 0.63).(c) Peripheral rings (C): diamagnetic (0.79 and 1.02).