Validity and limitations of the annulene-within-an-annulene (AWA) model for macrocyclic π-systems

Abstract

Recently some macrocyclic π-systems have been proposed as possible candidates for the annulene-within-an-annulene (AWA) model. We found that bond resonance energies (BREs) for spokes that link outer and inner annulenic subsystems can be used as an excellent measure for deciding whether or not a given macrocycle conforms to the AWA picture. In general, macrocycles with small BRE(spoke) values, such as [10,5]coronene and [14,7]coronene, can be described well in terms of the AWA model.

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Introduction

Hückel's 4n + 2 rule of aromaticity had long been the only meaningful rule in aromatic chemistry.^1,2 It, however, is applicable only to monocyclic π-systems, such as annulenes and heteroannulenes.² Therefore, some attempts have been made to extend this rule to polycyclic π-systems. Among them are the proposals of the perimeter model for polycyclic π-systems^3,4 and the annulene-within-an-annulene (AWA) model for macrocyclic π-systems.^5,6 According to the perimeter model,^3,4 naphthalene and pyrene must be aromatic with (4n + 2)-membered perimeters. However, there has been no theoretical justification for this model. The bridged annulene model for porphyrins^7,8 may also belong to such attempts. We recently showed that this model can be used rather safely to justify the macrocyclic aromaticity of porphyrins.^9,10 Global aromaticity of porphyrins, however, cannot be explained with the bridged annulene model. Validity and limitations of the AWA model is a subject to be examined in this paper.

Kekulene (1 in Fig. 1) is a polycyclic aromatic hydrocarbon (PAH) formed by an [18]annulene (the hub) joined through 12 bonds (spokes¹¹) to an external [30]annulene (the rim).⁶ Hereafter, CC bonds that form spokes, rim and hub in macrocyclic π-systems will be referred to simply as ‘spoke’, ‘rim’ and ‘hub’, respectively. If the two annulenes were totally uncoupled, the molecular stability of 1 would reflect straightforwardly the aromaticity of individual annulenes. This is the essence of the AWA model. However, if not, many circuits that extend over the hub and rim will play a dominant role in aromatic stabilization^12,13 and invalidate the AWA picture. Some candidates for the AWA model, such as [10,5]- (2), [14,7]- (3), [8,5]- (4) and [12,5]coronenes (5), have recently been proposed (see Fig. 2 and 3 for their molecular structures).^14–18 Coronene (6) and corannulene (7) have likewise been considered as candidates for the AWA bonding model (Fig. 4).^5,19,20 We will examine the applicability of the AWA model to these macrocycles by calculating some graph-theoretical quantities, such as topological resonance energy (TRE),^21,22 bond resonance energy (BRE)^23,24 and superaromatic stabilization energy (SSE).^25,26

Fig. 1 BREs in units of |β| for nonidentical π-bonds in kekulene (1).

Fig. 2 BREs in units of |β| for nonidentical π-bonds in [10,5]coronene (2) and [14,7]coronene (3).

Fig. 3 BREs in units of |β| for nonidentical π-bonds in [8,5]coronene (4) and [12,5]coronene (5).

Fig. 4 BREs in units of |β| for nonidentical π-bonds in coronene (6) and corannulene (7).

Computational procedures

The main tools to tackle the validity of the AWA model is TRE,^21,22 BRE^23,24 and SSE,^25,26 all defined within the framework of simple Hückel molecular orbital theory. In brief, TRE is the extra stabilization energy arising from all possible circuits in the π-system.^21,22 When one wants to evaluate TRE for a given cyclic π-system, a characteristic polynomial is constructed for the polyene reference by removing the contribution of all circuits from the coefficients of the Hückel characteristic polynomial for the actual π-system.^21,22 Here, circuits stand for all possible closed cycles that can be chosen from a cyclic π-system. BRE for a given π-bond is the extra stabilization energy arising from all circuits that pass through the bond.^23,24 SSE is the extra stabilization energy arising from all macrocyclic circuits in a macrocyclic π-system.^25,26 As shown in Fig. 5, circuits in macrocycles can be classified into two groups: local and macrocyclic circuits.¹² Macrocyclic circuits are circuits that enclose a central cavity, whereas local circuits do not enclose it. For examples of local and macrocyclic circuits in kekulene (1), see ref. 12.

Fig. 5 Examples of local (2a–e) and macrocyclic (2f–j) circuits in [10,5]coronene (2).

Bond resonance energy

First, we outline the method for calculating BRE for a π-bond in a cyclic π-system.^23,24 Nonidentical rings and π-bonds in kekulene (1) are labeled as in Fig. 1. A hypothetical reference π-system for evaluating the BRE for any π-bond in 1 (e.g., a C_aC_b bond in Fig. 1) is given simply by modifying a pair of resonance integrals for the π-bond in the manner:^23,24

β_a,b = iβ₀ and β_b,a = −iβ₀ (1)

where β₀ is the standard resonance integral for a CC π-bond, and i is the square root of −1. This procedure removes contributions of all circuits that pass through the C_aC_b bond from the coefficients of the characteristic polynomial. As shown in Fig. 1, BRE for this bond is 0.1964 |β|. Note that BRE for the C_eC_f bond is exactly the same as that for C_fC_g bond, because they belong to the same arc of the same ring.

Superaromatic stabilization energy

All macrocyclic circuits in kekulene (1) pass through either the C_aC_b bond or the C_dC_e bond in Fig. 1. Such two π-bonds must belong to the same ring. A hypothetical reference π-system for calculating SSE can then be constructed by modifying the resonance integrals for these two π-bonds in the following manner:^25,26

β_a,b = β_d,e = iβ₀ and β_b,a = β_e,d = −iβ₀ (2)

Circuits that pass through both the C_aC_b and C_eC_f bonds are not removed from the resulting reference π-system; as can be seen from Fig. 1, these circuits are local ones and so do not contribute to the macrocyclic aromaticity of the π-system.²⁶ The resulting SSE for 1 is 0.0035 |β|.

All molecules studied are taken to be planar with all CC bonds being the same in length. These molecules are assumed to be in a closed-shell singlet state. We used Hückel parameters for heteroatoms reported by Van-Catledge²⁷ to depict π-systems of octathia[8,5]coronene (8) and decathia[10,5]coronene (9) in Fig. 6. TREs, BREs and SSEs calculated for macrocycles 1–9 are summarized in Table 1.

Fig. 6 BREs in units of |β| for nonidentical π-bonds in octathia[8,5]coronene (8) and decathia[12,5]coronene (9).

Table 1 TREs, SSEs and BREs for macrocycles studied

Species	TRE/|β|	SSE/|β|	BRE(rim)/|β|	BRE(hub)/|β|	BRE(spoke)/|β|
a Ring A.b Ring B.
Kekulene (1)	1.5688	0.0035	0.1955^a	0.1964^a	0.2210
			0.1079^b	0.1084^b
[10,5]Coronene (2)	−0.0663	0.0801	−0.0569	0.0786	0.0156
[14,7]Coronene (3)	−0.0882	0.0197	0.0171	−0.0277	0.0050
[8,5]Coronene (4)	−0.5741	−0.4652	−0.1342	−0.4362	−0.1073
[12,5]Coronene (5)	−0.4667	−0.2836	−0.0780	−0.2987	−0.0631
Coronene (6)	0.9474	0.0922	0.1701	0.2188	0.2459
Corannulene (7)	0.7352	0.0123	0.1668	0.1752	0.2429
Octathia[8,5]coronene (8)	1.0344	−0.0027	0.1246	0.1266	0.2330
Decathia[10,5]coronene (9)	1.2967	0.0003	0.1267	0.1269	0.2339

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Results and discussion

Naphthalene and azulene

For the purpose of reviewing the physical meanings of TRE, BRE and SSE, we first examine the aromatic character of naphthalene (10) and azulene (11) in Fig. 7. Dewar resonance energies of 10 and 11 are 0.869 eV (20.0 kcal mol⁻¹) and 0.169 eV (3.9 kcal mol⁻¹), respectively,²⁸ whereas Hess–Schaad resonance energies are 0.55 |β| (0.3888 |β|) for 10 and 0.23 |β| (0.1511 |β|) for 11,²⁹ where values in parentheses are the corresponding TREs.^21,22 These aromatic stabilization energies indicate that 10 is significantly more aromatic than its nonbenzenoid isomer 11. NICS(1) values for each benzene ring of 10 and the five- and seven-membered rings of 11 are −10.71, −17.77 and −7.82 ppm, respectively,³⁰ at the RB3LYP/6-311+G** level of theory.³¹ However, we cannot assess the relative degrees of local aromaticity in 10 and 11 reliably from these NICS(1) values.³²

Fig. 7 BREs in units of |β| for nonidentical π-bonds in naphthalene (10) and azulene (11).

In 2011 we found that the sequential line plot of TRE against the number of π-electrons (N_π) for any PAH is similar with the same number of major extrema to that for benzene.³³ Here, N_π varies from 0 to twice the number of conjugated atoms. These plots always exhibit three major maxima and two major minima. Thus, global aromaticity of neutral and charged PAH molecules strongly reflects that of constituent benzene rings. In general, the N_π dependence of TRE for any polycyclic π-system formed by fusion of two or more rings of the same size reflects that for a monocyclic species of the same ring size.³³ The line plots of TRE against N_π for naphthalene (10) and azulene (11) are presented in Fig. 8. The TRE vs. N_π plot for 10, the simplest PAH, really resembles that for benzene with three major maxima and two major minima (Fig. 8a). In contrast, the TRE vs. N_π plot for 11 (Fig. 8b) is somewhat complicated, because it consists of two rings of different sizes. We cannot decide immediately which circuit is the main origin of aromaticity in this nonbenzenoid hydrocarbon.

Fig. 8 Sequential line plots of TRE and BRE(cross-link) against the number of π-electrons (N_π) for naphthalene (10) and azulene (11). Neutral species are denoted by asterisks.

BRE is useful for examining the geometric distribution of aromaticity in a polycyclic π-system.^23,24 As can be seen from Fig. 7, a π-bond shared by two rings (i.e., a cross-link) in naphthalene (10) exhibit a large positive BRE of 0.2668 |β|, indicating that both six-membered rings contribute significantly to aromaticity, whereas BRE for the cross-link in azulene (11) is negligibly small, suggesting that neither five- nor seven-membered ring contributes appreciably to aromaticity.³² Thus, the main origin of aromaticity in 10 must be two six-site circuits, whereas that in 11 it must be a ten-site circuit. These aspects of BREs will further be confirmed by calculating circuit resonance energies (CREs) for all circuits.^34,35 Here, CRE for a given circuit is an aromatic stabilization energy estimated from the magnetic response of the circuit.³⁶ Fig. 8 shows that the plot of BRE(cross-link) against N_π for 10 is similar in appearance to that of TRE against N_π for the same hydrocarbon. Such a similarity of the plots also supports the view that the main origin of aromaticity in 10 is six-site circuits. Note that no π-current flows through the cross-link in 10, because π-currents induced in two six-site circuits cancel out there.³² This never means that this bond does not contribute to aromaticity; it contributes much to global aromaticity as part of the two six-site circuits.

The extended Hückel rule proposed by Hosoya et al. states that the global aromaticity of a neutral polycyclic conjugated hydrocarbon can be attributed qualitatively to individual circuits in it.³⁷ They then noted that relatively small conjugation circuits contribute predominantly to aromaticity. The importance of small conjugation circuits itself had been pointed out by Herndon and Ellzey³⁸ and Randic.³⁹ Three circuits in 10 are (4n + 2)-site conjugation ones, which necessarily contribute significantly to the large negative NICS(1) values in 10. Large negative NICS(1) values for two rings in 11 must be associated mainly with the peripheral ten-site circuit, which is the only conjugation circuit in 11. It then follows that the Platt perimeter model³ happens to apply to 11; the perimeter cycle indeed is the main origin of global aromaticity.

Kekulene and the AWA model

Analogous graph-theoretical reasoning is applicable to kekulene (1). This hydrocarbon has been represented either as a D_6h Clar structure containing six sextet rings or an [18]annulene-within-a-[30]annulene π-system.⁶ Current–density analysis revealed that, when placed in a perpendicular magnetic field, 1 has induced counter-rotating ring currents with a diamagnetic rim and a paramagnetic hub, in obvious contradiction of the prediction based on the [18]annulene-within-a-[30]annulene model.^19,20 The AWA model would give diamagnetic rim and hub ring currents.¹³ An apparently paramagnetic current induced along the inner periphery is associated with currents induced in many local circuits.¹² The proposal that circularly conjugated kekulene possesses superaromaticity, i.e., enhanced stabilization energy due to macrocyclic conjugation,⁴⁰ led to controversy resolved finally by the calculated current–density map and the negligibly small SSE of 0.0035 |β| (Table 1).^12,13,41

Kekulene (1) consists of 12 benzene rings, so that the plot of TRE against N_π for 1 (Fig. 9a) is similar to that for benzene (Fig. 10b); the central 18-site conjugation circuit in 1 has no appreciable effect on the gross feature of the plot. Although all six-site circuits contribute much to global aromaticity, all circuits, including the six-site circuits, must be responsible for the details of the plot. Likewise, the plot of BRE(spoke) against N_π for 1 (Fig. 9b) is similar not only to that for benzene (Fig. 10b) but also to the plot of TRE against N_π for 1 (Fig. 9a). BRE(spoke) is as large as 0.2210 |β| in the neutral species (Fig. 1). This large BRE(spoke) value is never compatible with the AWA model, indicating that the two annulenoid substructures are strongly coupled. Local circuits that pass through spokes are all (4n + 2)-site conjugation circuits and must contribute significantly to global aromaticity.^37–39 Thus, the BRE concept is very useful for validating or invalidating the AWA model.

Fig. 9 Plots of TRE and BRE(spoke) against N_π for kekulene (1). Neutral species are denoted by asterisks.

Fig. 10 Plots of TRE against N_π for cyclopentadienyl, benzene and cycloheptatrienyl.

[10,5]Coronene and the AWA model

[10,5]Coronene (2) comprises ten fused pentagons around a central decagon (Fig. 2).^14,15,17 This yet elusive molecule has four Kekulé structures with all spokes always being formal single bonds. Thus, inner and outer cycles are formally decoupled, suggesting that an AWA picture of electronic structure might be applicable to 2. However, one should note that the linking of two subsystems by formal single bonds is not always a sufficient condition for decoupling the subsystems. For example, formal single bonds in pentalene (12) and dibenzo[cd,gh]pentalene (13) exhibit large positive BREs (Fig. 11), because these bonds tend to relax the antiaromaticity of 4n-site circuits. Monaco et al. calculated the π-current density in 2 and predicted that 2 must be the first macrocycle that sustains a diatropic-hub/paratropic-rim pattern of induced currents.¹⁵ In this connection, we have seen that kekulene (1) sustain a paratropic-hub/diatropic-rim pattern of induced currents.^12,13 Is it true that the AWA model is applicable to 2?

Fig. 11 BREs in units of |β| for nonidentical π-bonds in pentalene (12) and dibenzo[cd,gh]pentalene (13). Those for formal single bonds are shown in blue.

TRE for [10,5]coronene (2) is slightly antiaromatic with a TRE of −0.0663 |β| (Table 1). Since the lowest unoccupied molecular orbital (LUMO) is nonbonding, the molecular dianion is aromatized to some extent. The line plot of TRE against N_π for 2 is shown in Fig. 12a. This plot is similar in shape to that for cyclopentadienyl (Fig. 10a), both with two major maxima and two major minima. These extrema indicate that the main origin of global aromaticity or antiaromaticity in the neutral and charged species must be individual cyclopentadienyl rings. Likewise, the plot of BRE(spoke) against N_π (Fig. 12b) is more or less similar not only to that for cyclopentadienyl (Fig. 10a) but also to the plot of TRE against N_π (Fig. 12a).

Fig. 12 Plots of TRE and BRE(spoke) against N_π for [10,5]coronene (2). Neutral species are denoted by asterisks.

It is in contrast to that for kekulene (1) that BRE(spoke) for 2 is only 0.0156 |β| in the neutral state (Table 1), which amounts to only 7% of that for kekulene (1). This small BRE(spoke) value is compatible with the AWA model, in that local circuits and macrocyclic circuits that extend over the hub and rim cycles as a whole contribute only slightly to the hub current. Note that all circuits but 2f and h in Fig. 5 extend over the hub and rim cycles. BRE(rim) and BRE(hub) for 2 are negative and positive, respectively, in sign (Fig. 2). This fact indicates that local circuits in 2 are not the determinant of the sign of BRE(rim) and/or that of BRE(hub). Remember that BRE for a given π-bond is approximately equal to the sum of CREs for all circuits that pass through the bond.^34–36 If local circuits were much more important than macrocyclic ones in determining the degree of global aromaticity, BRE(rim) and BRE(hub) would be of a similar magnitude with the same sign; BRE(spoke) would of course be much larger. As can be seen from Table 1, this is the case for kekulene (1) but not for 2. As BRE(spoke) for 2 is very small, the rim and hub cycles (i.e., circuits 2h and f in Fig. 5) are presumed to sustain paramagnetic and diamagnetic ring currents, respectively, in harmony with Hückel's 4n + 2 rule.^42,43 The sign of CRE determines not only the aromaticity of the circuit but also the sense of the π-current induced in the circuit.³⁶ As already stated, the paramagnetic current induced along the hub cycle in 1 does not conform to the AWA model.

[14,7]Coronene and the AWA model

[14,7]Coronene (3 in Fig. 2) was theoretically designed by Dickens and Mallion.¹⁸ They called it 7-coronene. We prefer the former name, because it specifies not only the size of constituent rings but also the number of rings. This molecule comprises 14 circularly fused heptagons with all spokes being formal single bonds. The neutral molecule is slightly antiaromatic with a TRE of −0.0882 |β| (Table 1). Since the highest occupied molecular orbital (HOMO) is nonbonding, the molecular dication must be more aromatic with fewer π-electrons on each heptagonal ring. Dickens and Mallion predicted this hydrocarbon to conform to the AWA model,¹⁸ based on their Hückel–London π-current calculations.^44,45 They found that the current around the 42-membered outer perimeter flows in the diamagnetic direction and that the current around the 28-membered central ring flows in the paramagnetic direction.¹⁸

The line plots of TRE and BRE(spoke) against N_π for 3 are shown in Fig. 13. Both plots are similar in shape to that for cycloheptatrienyl (Fig. 10c). It then follows that the main origin of aromaticity/antiaromaticity in neutral and charged [14,7]coronene (3) must be individual cycloheptarienyl rings. BRE(rim) and BRE(hub) are 0.0171 and −0.0277 |β|, respectively, in the neutral state. Since BRE(spoke) for the neutral species is as small as 0.0050 |β|, 3 really conforms to the AWA model. Local circuits and macrocyclic ones that extend over the hub and rim cycles as a whole then contribute little to the hub current. Thus, in line with Dickens and Mallion's π-current calculations,¹⁸ the outer rim and inner hub cycles are predicted to sustain diamagnetic and paramagnetic ring currents, respectively.

Fig. 13 Plots of TRE and BRE(spoke) against N_π for [14,7]coronene (3). Neutral species are denoted by asterisks.

However, geometry optimization recently reported by Monaco and Zanasi⁴⁶ revealed that 5, a very large antiaromatic π-system, is non-planar with bond length alternation, which was then predicted to bring about the mixing and localization of rim and hub π-currents. The π-circulation pattern will depend on the direction of the external magnetic field, but, as pointed out by Monaco and Zanasi,⁴⁶ it must be very sensitive to the geometry change of the π-system. Therefore, the present discussion is limited to the ideal planar structure. In our view, the above AWA picture of 3 still does not lose its importance as a model macrocyclic system. Relative magnitudes of BREs for different π-bonds must be much less sensitive to the non-planarity than the π-circulation pattern is.

[4n,5]Coronenes

In a sense, the validity of the AWA model is a matter of degree. Monaco et al. designed the [4n,5]coronene molecules, such as [8,5]coronene (4 in Fig. 3) and [12,5]coronene (5 in the same figure), with two 4n-membered cycles.¹⁶ [8,5]Coronene has three nonbonding orbitals with two π-electrons and is necessarily antiaromatic with a TRE of −0.5741 |β| (Table 1). The line plots of TRE and BRE(spoke) against N_π for 4, shown in Fig. 14, are similar in shape to that of TRE for cyclopentadienyl (Fig. 10a) with two major maxima and two major minima. Monaco et al. presumed that rim and hub cycles are not strongly coupled and then predicted the occurrence of conrotating paratropic ring currents leading to overall ring-current paramagnetism.¹⁶ In fact, BRE(spoke) for 4 exhibits a fairly large negative value of −0.1073 |β|. We presume that the strong antiaromatic current induced in the hub cycle overwhelmed the counter-rotating currents induced in the local circuits and macrocyclic ones that extend over rim and hub cycles. One should note that both 4n- and (4n + 2)-site conjugation circuits often bear much stronger antiaromatic and aromatic currents, respectively, when they coexist in the same π-system.^47,48 Both BRE(rim) and BRE(hub) are negative in sign, suggesting that the rim and hub cycles with large areas sustain fairly strong paramagnetic currents.

Fig. 14 Plots of TRE and BRE(spoke) against N_π for [8,5]coronene (4). Neutral species are denoted by asterisks.

[12,5]Coronene (5) likewise has three nonbonding orbitals with two π-electrons and is necessarily antiaromatic with a TRE of −0.4668 |β| (Table 1). The line plots of TRE and BRE(spoke) against N_π for 5, shown in Fig. 15, are similar in shape to that of TRE for cyclopentadienyl (Fig. 10a), all with two major maxima and two major minima. BRE(spoke) for 5 exhibits a fairly large negative value, although it is smaller than that for [8,5]coronene (4). We again presume that the strong antiaromatic current induced along the hub cycle overwhelmed the couter-rotating currents induced along the local circuits and macrocyclic ones that extend over rim and hub cycles.

Fig. 15 Plots of TRE and BRE(spoke) against N_π for [12,5]coronene (5). Neutral species are denoted by asterisks.

Octathia[8,5]coronene and decathia[10,5]coronene

Octathia[8,5]coronene (C₁₆S₈, 8 in Fig. 6) was prepared in 2006 by Chernichenko et al.⁴⁹ Decathia[10,5]coronene (C₂₀S₁₀, 9 in the same figure) was theoretically studied by Napolion et al.⁵⁰ as a higher homologue of 8. Although these carbon sulfides are never candidates for the AWA model, we chose these molecules here, because the hypothetical octacation of 8 and decacation of 9 are iso-π-electronic with 4 and 2, respectively. Both π-systems likewise are formed by fused five-membered rings located around the central polygon. Therefore, the plots of TRE and BRE(spoke) against N_π for 8 (Fig. 16) and 9 (Fig. 17) are in close resemblance to those for 4 (Fig. 14) and 2 (Fig. 12), respectively, and to the TRE vs. N_π plot for cyclopentadienyl (Fig. 10a). Unlike those in 4 and 2, five-membered rings in 8 and 9 are aromatic thiophene rings, so that BRE(spoke) values are very large as in the case of kekulene (1). It is obvious that the AWA model does not apply to these molecules. Since local circuits in 8 and 9 contribute much to their respective TREs, BRE(rim) and BRE(hub) are of similar magnitude with the same sign. Like kekulene (1), 8 and 9 must sustain counter-rotating ring currents with a diamagnetic rim and a paramagnetic hub. In fact, the NICS(1)_zz value calculated at the center of 6 is 15.8 ppm at the GIAO/HF/6-311+G(d,p)//B3LYP/6-31G(d,p) level of theory.⁵¹

Fig. 16 Plots of TRE and BRE(spoke) against N_π for octathia[8,5]coronene (8). Neutral species are denoted by asterisks.

Fig. 17 Plots of TRE and BRE(spoke) against N_π for decathia[12,5]-coronene (9). Neutral species are denoted by asterisks.

Coronene and corannulene

It has been firmly established that coronene (6) and corannulene (7), presented in Fig. 4, do not conform to the AWA model.^19,20 π-Currents induced along the central six-membered rings in 6 and 7 are in the paramagnetic direction,^19,20 thereby violating the AWA model. According to our way of reasoning, 6 and 7 are not compatible with the AWA model, simply because their BRE(spoke) values are as large as those for kekulene (1) (see Table 1).

Concluding remarks

We have examined the aromatic character of recently proposed candidates for the AWA model in graph-theoretical terms. These candidates are macrocyclic π-systems that comprise fused polygons of the same size around a central polygon. If positive or negative BRE for each spoke is very small and if circuits chosen along the rim and the hub are conjugation circuits, the AWA model will possibly hold for the macrocycle. At least two macrocycles, [10,5]coronene (2) and [14,7]coronene (3), fulfill these conditions. In contrast, kekulene (1), octathia[8,5]coronene (8) and decathia[10,5]coronene (9) are incompatible with the AWA picture, because constituent rings are highly aromatic and because BRE(spoke) necessarily is very large.

There may be another type of AWA pictures. Even if BRE(spoke) for a macrocycle is not very small, there is some possibility that the AWA model is apparently applicable to it. In the case of [8,5]coronene (4) and [12,5]coronene (5), BRE(spoke) is not very small, suggesting that the rim and hub cycles are moderately coupled. However, owing to the strong paramagnetic current induced along the 4n-membered hub cycle, both the hub and rim cycles bear paramagnetic currents. It follows that the senses of hub and rim currents apparently conforms to Hückel's naive 4n + 2 rule. Such a situation must be highly probable if a 4n-site conjugation circuit can be chosen from the hub cycle. The present BRE-based approach may also be applicable to the analysis of three contra-rotating currents in altan-molecules, such as altan-corannulene and altan-coronene.^52,53 Note that any cyclic π-system with outgoing C–H bonds can be converted into the corresponding altan-molecule by substituting the C–H bonds with C–C bonds to alternating carbon atoms of an annulene.⁵²

Acknowledgements

We thank Prof. G. Monaco (University of Salerno), Prof. R. B. Mallion (University of Kent) and Prof. T. K. Dickens (University of Cambridge) for critically reading the early manuscript of this paper.

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