Superaromatic Stabilization Energy as a Novel Local Aromaticity Index

Sep 12, 2013 - Superaromatic stabilization energy (SSE), previously proposed by us, can be used as a novel local aromaticity index for benzene rings i...
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Superaromatic Stabilization Energy as a Novel Local Aromaticity Index for Polycyclic Aromatic Hydrocarbons Jun-ichi Aihara,*,†,‡ Masakazu Makino,‡ and Kenkichi Sakamoto† †

Department of Chemistry, Faculty of Science, Shizuoka University, Oya, Shizuoka 422-8529, Japan Institute for Environmental Sciences, University of Shizuoka, Yada, Shizuoka 422-8526, Japan

‡

S Supporting Information *

ABSTRACT: Superaromatic stabilization energy (SSE), previously proposed by us, can be used as a novel local aromaticity index for benzene rings in polycyclic aromatic hydrocarbons (PAHs). SSE can be interpreted as the first local aromaticity index explicitly related to all relevant circuits in a polycyclic πsystem, an origin of local aromaticity, being free of local aromaticity arising from adjacent six-site circuits. Therefore, this quantity is best suited for characterizing the electronic structure of large pericondensed PAHs and graphene nanoflakes.

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INTRODUCTION Clar’s sextet formula (or a Clar structure) with a maximum number of disjoint aromatic sextets and a minimum number of localized double bonds has been used to characterize the chemistry of a polycyclic aromatic hydrocarbon (PAH).1,2 An aromatic sextet or a sextet ring is defined as six π-electrons localized in a single benzene ring separated from adjacent sextet rings by formal single bonds. Fully benzenoid hydrocarbons, i.e., PAHs in which all carbon atoms belong to sextet rings, are not only thermodynamically but also kinetically very stable.1,2 In general, a PAH molecule with a given number of aromatic sextets is more stable than the PAH isomers with fewer aromatic sextets. These empirical facts strongly support the view that each ring in a PAH molecule must have a different degree of aromatic character.1−5 As comprehensively reviewed by Bultinck,6 many indices have been proposed to estimate the degree of local aromaticity in individual benzene rings of PAHs. Some local aromaticity indices make different predictions on the relative degrees of local aromaticity in different benzene rings of a PAH molecule. In particular, it is not easy to estimate the local aromaticity in benzene rings surrounded or flanked by six other benzene rings. The central ring in coronene (1 in Figure 1) is one of such benzene rings. In this hydrocarbon, πbonds shared by the central and peripheral benzene rings are strengthened not only by the aromaticity of the central ring but also by that of the adjacent peripheral ring. Therefore, local aromaticity indices based on the bond lengths, bond strengths, and bond orders might possibly overestimate the local aromaticity of inner benzene rings. Two of the typical local aromaticity indices, the harmonic oscillator model of aromaticity (HOMA) index6−9 and the bond-order index of aromaticity (BOIA) value,6,10 may belong to this type of indices. HOMA and BOIA represent the degrees of CC bond© 2013 American Chemical Society

length and bond-order equalization, respectively, in a given benzene ring of a PAH molecule. Therefore, it is not surprising that Aihara11 and Giambiagi et al.12 discussed the local aromaticity of benzene rings in PAHs, leaving out the inner benzene ring of coronene. Since 1996, nucleus-independent chemical shift (NICS) values have been used as a convenient index for local aromaticity.13,14 According to our theory of ring-current diamagnetism,15,16 π-currents are induced almost independently in individual circuits. This explains why no π-current flows through each spoke bond in coronene (1).17 Here, a circuit stands for a closed cyclic path that can be chosen from a πsystem. The π-current map of 1 in Figure 1 reveals that the sum of circuit currents that pass through a π-bond shared by two peripheral rings completely cancels out there. Likewise, πbonds that form the central ring sustain weak π-currents,18,19 because most circuit currents again cancel out there.17 Such a πcirculation pattern is closely associated with a much diminished NICS(1) value at the center of the central ring.20,21 Thus, the NICS(1) values for inner benzene rings of large pericyclic PAHs often underestimate the degrees of local aromaticity. In 1995, we proposed a new energetic quantity, bond resonance energy (BRE), to estimate the contribution of each π-bond to the global aromaticity of a polycyclic π-system.22,23 BRE for a given π-bond represents the aromatic stabilization energy arising from all possible circuits that pass through the πbond. This aromaticity-based quantity has proven to be very useful for identifying the reactive sites in polycyclic π-systems. BREs for fullerene π-bonds shared by two pentagonal rings are Received: July 13, 2013 Revised: August 6, 2013 Published: September 12, 2013 10477

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Figure 1. Coronene (1) and benzo[ghi]perylene (2). Hückel−London π-current intensities in units of the benzene value are shown in 1a and 2a and bond resonance energies (BREs) in units of |β| in 1b and 2b. In π-current maps 1a and 2a, diamagnetic currents flow counterclockwise.

kinetically very unstable with BREs < −0.100 |β0|.22,23 We noticed that BRE can also be used as an indicator of local aromaticity for PAHs.11 We observed an excellent correlation of the BRE for a peripheral π-bond of a PAH molecule with two typical local aromaticity criteria, HOMA6−9 and BOIA6,10 values for the benzene ring to which the π-bond belongs. However, BRE cannot be used to estimate the degree of local aromaticity in the inner benzene rings of coronene (1) and other pericyclic PAHs. As shown in Figure 1, π-bonds shared by two benzene rings exhibit much larger BREs, simply because these bonds are strengthened by the local aromaticity of the two benzene rings.24 We have long been studying the superaromaticity of macrocyclic π-systems, such as cycloarenes, coronoid hydrocarbons, and porphyrins.24−34 Here, superaromaticity stands for macrocyclic aromaticity, or aromaticity due to macrocyclic conjugation. In this paper, we report our finding that our previous definition of superaromatic stabilization energy (SSE)25 can be used as a physically sound local aromaticity index for all benzene rings in large pericondensed hydrocarbons. SSE originally represented the extra stabilization energy due to macrocyclic conjugation or to a set of macrocyclic circuits.25 As will be seen, SSE is theoretically identical with BRE when it is used as a local aromaticity index for peripheral benzene rings. We can now discuss consistently the relative local aromaticities of outer and inner benzene rings in large pericondensed PAHs and graphene nanoflakes.

Figure 2. Nonidentical rings and π-bonds in coronene (1c) and benzo[ghi]perylene (2c).

(e.g., a CaCb bond) in 2 is given simply by modifying a pair of resonance integrals for this π-bond in the manner:22,23 βa,b = iβ0

and

βb,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 deletes contributions of all circuits that pass through the CaCb bond from the coefficients in the characteristic polynomial. As displayed in Figure 3, 47 circuits can be chosen from the benzo[ghi]perylene π-system (2). BRE for the CaCb bond represents the extra stabilization energy arising from 21 of the 47 circuits listed in Table 1. All these circuits surround ring A, sharing the CaCb bond with the ring. Note that circuits such as b2, b6, b8, and b12−b15 are not found in the set of 21 circuits. These circuits share one or more π-bonds with ring A, but their contribution to BREs for the peripheral π-bonds of ring A is prohibited, because they surround ring B but not ring A. Thus, the BRE for a peripheral π-bond is free of the local aromaticity arising from nearby benzene rings. This is why BRE for the peripheral CaCb bond can be interpreted as a local aromaticity index for peripheral ring A to which the CaCb bond belongs. This method for calculating BRE will be referred to as method I. Method II for Calculating BRE. A hypothetical reference π-system lacking the local aromaticity of ring C in 2 can likewise be given by modifying a pair of resonance integrals for the CiCj bond in the following manner:

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COMPUTATIONAL PROCEDURES As will be described below, BRE and SSE were defined and calculated within the framework of simple Hückel molecular orbital (HMO) theory.22−25 We also calculated the intensities of π-electron currents induced in some typical PAHs, using our variant of the Hückel−London theory of ring currents.35−37 Method I for Calculating BRE. First, we briefly show how to calculate the BRE for a peripheral π-bond that belongs to ring B in benzo[ghi]perylene (2). Nonidentical rings and πbonds in 1 and 2 are labeled as in Figure 2. A hypothetical reference π-system for evaluating the BRE for a given π-bond 10478

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Figure 3. All possible circuits in benzo[ghi]perylene (2). Nonconjugation circuits are denoted by asterisks; others are conjugation circuits.

PAHs of D6h symmetry.38,39 Therefore, BRE for a peripheral πbond can be regarded as an SSE for the peripheral benzene ring to which the π-bond belongs and so the BRE for the CiCj bond in 2 can be referred to as an SSE for ring C. It indeed is an SSE arising from all circuits that surround the ring C. Because the main contributor to the BRE is the six-site circuit concerned, the SSE is supposed to strongly reflect the local aromaticity of the benzene ring. The SSE value for ring C, obtained using the above two methods, is consistently 0.0728 |β|; those for rings A, B, and D are 0.2082, 0.1438, and 0.1827 |β|, respectively.40 SSE as a Local Aromaticity Index. The standard method for calculating BRE (method I) cannot be used to estimate the local aromaticity of inner benzene rings flanked by six other benzene rings. We instead noticed that the method for calculating SSE (method II) can be used for this purpose. The reference π-system that is devoid of the local aromaticity in the central benzene ring (ring B) of coronene (1) can be constructed by modifying the resonance integrals for the CaCb and CcCd bonds in the following manner:25,32,34

Table 1. List of Circuits Associated with the SSE Values for Nonidentical Rings in Benzo[ghi]perylene (2) benzene ring

no. of circuits

A

21

B

24

C

32

D

25

a

circuitsa b1, b7, b11, b16, b20, b23, b26, b27, b29, b32, b34, b35, b36, b38, b40, b42, b43, b44, b45, b46, b47 b2, b7, b8, b12, b16, b17, b20, b21, b24, b28, b29, b30, b32, b33, b35, b37, b38, b39, b40, b41, b43, b44, b46, b47 b6, b11, b12, b13, b14, b15, b16, b17, b18, b19, b23, b24, b25, b26, b27, b28, b29, b30, b31, b32, b33, b34, b35, b36, b37, b38, b39, b42, b43, b44, b45, b46 b3, b8, b9, b13, b17, b18, b20, b21, b22, b23, b25, b29, b30, b31, b33, b36, b38, b39, b40, b41, b42, b44, b45, b46, b47

For the labels of circuits, see Figure 3.

βi,j = iβ0

and

βj,i = −iβ0

(2)

This procedure deletes the contributions of 32 circuits that pass through the CiCj bond from the coefficients in the original characteristic polynomial. All these circuits surround ring C. Note that these 32 circuits pass through either the CeCf bond or the CgCh bond. A π-system lacking the local aromaticity of ring C can then be constructed by modifying the resonance integrals for these two π-bonds in the following manner:25,32,34 βe,f = βg,h = iβ0

and

βf,e = βh,g = −iβ0

βa,b = βc,d = iβ0

and

βb,a = βd,c = −iβ0

(4)

As displayed in Figure 4, 94 circuits can be chosen from 1. Among them 64 circuits surround the central benzene ring (Table 2); all of the 64 circuits pass through either the CaCb bond or the CcCd bond. Therefore, eq 4 excludes the contributions of these circuits from the coefficients in the characteristic polynomial, thus defining the reference π-system for estimating the local aromaticity of ring B. The SSE defined in this manner does not contain any contributions from circuits that do not surround ring A, such as c1−c6 and c7−c13 in Figure 4. It is in this point that the SSE can be used safely as a local aromaticity index. If SSE values calculated for many PAH molecules are numerically reasonable and consistent with their chemistry, method II will really be useful for estimating the degrees of local aromaticity for all rings in large pericondensed PAHs. One may note that SSEs for the inner rings of 1 and the like ones may contain the contribution of large circuits that do not pass any π-bond of the ring concerned. For example, the hypothetical SSE reference π-system for the central ring in 1 does not contain the contribution of such a circuit (i.e., c88), which of course surrounds the central ring. This circuit shares no π-bond with ring B but surrounds the central ring at a

(3)

Both eqs 2 and 3 bring about exactly the same reference polynomial and necessarily the same aromatic stabilization energy due to ring C. The method based on eq 3 will be referred to as method II. As can be seen from Table 1, circuits that pass through both the CeCf and CgCh bonds (i.e., b3, b8, b9, b20, b21, b22, b40, b41, and b47 in Figure 3) are not deleted from the resulting reference polynomial; these circuits do not surround ring C and so do not contribute to the local aromaticity of this ring.25 At first, method II was proposed for calculating the energy of the hypothetical reference π-system for the purpose of determining the superaromatic stabilization energy (SSE) for macrocyclic π-systems.25,32,34 In the present case, every benzene ring in 2 is regarded as a cavity and all circuits surrounding the ring are viewed as macrocyclic circuits. The same viewpoint has been taken by Hajgáto et al. when they analyzed the NICS(1) values at the inner benzene rings of large 10479

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Figure 4. All possible circuits in coronene (1). Nonconjugation circuits are denoted by asterisks; others are conjugation circuits.

Table 2. List of Circuits Associated with the SSE Values for Rings A and B in Coronene (1) benzene ring

no. of circuits

A

47

B

64

a

circuitsa c6, c12, c13, c19, c24, c25, c29, c30, c31, c35, c37, c38, c44, c45, c46, c49, c51, c52, c54, c57, c58, c59, c62, c63, c64, c65, c68, c69, c70, c72, c74, c75, c77, c79, c80, c81, c83, c84, c85, c86, c87, c88, c90, c91, c92, c93, c94 c7, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c71, c72, c73, c74, c75, c76, c77, c78, c79, c80, c81, c82, c83, c84, c85, c86, c87, c88

For the labels of circuits, see Figure 4.

benzenoid hydrocarbons; all the carbon atoms belong to sextet rings with large SSEs. Thus, at least, SSEs for familiar PAHs seem to represent a physically meaningful measure of local aromaticity.11 Unfortunately, all the PAHs but coronene (1) in Figure 5 have no inner benzene rings. Therefore, what should be done next is to see whether or not the SSE concept is applicable on the same basis not only to peripheral rings but also to inner ones of coronene (1) and larger pericondensed PAHs. Coronene (1) is the smallest PAH molecule with an inner benzene ring. As shown in Figure 4, 94 circuits in all can be chosen from the coronene π-system, which are of course the source of global and local aromaticity. These circuits consist of 64 conjugation circuits (or conjugated circuits in Randić’s terminology4) and 30 nonconjugation ones. All of the

distance. However, this 18-site circuit is large in size and so the contribution to the SSE value must anyways be very small.

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RESULTS AND DISCUSSION SSEs for Familiar PAHs Including Coronene. We have seen that, when the SSE concept is applied to individual benzene rings, it can be interpreted as a local aromaticity index associated explicitly with all relevant circuits. SSEs calculated for all nonidentical rings in benzene and 27 familiar PAHs (1− 28 in Figure 5) are listed in the Supporting Information, Table S1. All benzene rings in these PAHs were found to be aromatic with positive SSEs. Relative degrees of aromaticity in individual benzene rings are also observable in Figure 5. Highly aromatic rings with large SSEs correspond to the locations of sextet rings in the Clar structures.23 PAHs 13, 25, and 27 are fully 10480

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Figure 5. Benzene and 27 familiar PAHs. Numerical data for PAHs outlined in black and pink are plotted in Figures 7 and 8. NICS(1) values for PAHs outlined in black and orange are employed in Figure 10. Indigo, blue, and gray filled circles indicate benzene rings with SSEs larger than 1.8, 1.4, and 1.2 |β|, respectively.

conjugation circuits in 1 are aromatic (4n + 2)-site ones. In general, small (4n + 2)-site conjugation circuits, particularly sixsite conjugation circuits (c1−c7), make a major contribution to SSEs.3,4,41,42 Rings A and B in 1 (Figure 2) are peripheral and inner benzene rings, respectively. As listed in Table 2, 47 and 64 circuits are the origins of SSEs for rings A and B, respectively. SSEs for rings A and B were calculated to be 0.1701 and 0.0882 |β|, respectively, with the ratio of SSE(B) to SSE(A) being 0.54. Hereafter, X(A) and X(B) represent the values of a given local aromaticity index X for rings A and B in 1, respectively. We previously reported that BRE for a given π-bond can be estimated roughly by summing up the circuit resonance energies (CREs) for all the circuits that pass though the πbond.16,28,40 Here, CRE is an aromatic stabilization energy of a given single circuit, felt by the external magnetic field; it is proportional to the intensity of the π-current induced in the circuit, dividend by the area of the circuit. SSE for a benzene ring flanked by six other benzene rings, if any, can likewise be estimated by summing up the CREs for circuits surrounding the benzene ring. SSEs estimated in this manner are 0.1462 and 0.0822 |β| for rings A and B in 1, respectively. The ratio of these two values is 0.60 and so is very close to the SSE(B)-to-SSE(A) ratio (0.54). Thus, the central ring in 1 is presumed to be considerably less aromatic than the peripheral benzene rings. This aspect of the coronene π-system is fully consistent with the Clar structures. As can be seen from Figure 6, two Clar structures, each with three aromatic sextets, can be written for 1, but an aromatic sextet cannot be placed on the inner ring, suggesting that ring B is much less aromatic than ring A. When, like coronene, more than one Clar structure can be written for a PAH molecule, we will superpose all possible Clar structures, in such a manner as shown in Figure 6, because it reflects the symmetry of the SSE pattern in the π-system more closely.21 Only one Clar structure can be written for benzo[ghi]perylene (2), in which three sextet rings exhibit relatively large SSEs. Comparison with HOMA and BOIA Values. We then compare the SSEs with the corresponding HOMA and BOIA

Figure 6. Clar structures for coronene (1d) and benzo[ghi]perylene (2d). Structure 1e is a superposed Clar structure for coronene.

values for familiar PAHs. HOMA and BOIA values employed are those calculated by Bultinck et al.6,10 at the B3LYP/631G*(6D,10F) level of theory.43 Figures 7 and 8 show plots of HOMA and BOIA values, respectively, against SSEs for benzene and 25 PAHs, where open circles represent peripheral benzene rings in PAHs other than coronene (1). For the PAHs adopted in these figures, see the caption of Figure 5. Correlations of SSEs with HOMA and BOIA values are very good; correlation coefficients are as large as 0.909 between SSE and HOMA values and 0.982 between SSE and BOIA values. This confirms our previous finding that, as long as peripheral benzene rings are concerned, SSE is as good a local aromaticity index as HOMA and BOIA.11 However, it is noteworthy that the HOMA value for the only inner ring (ring B in 1) is an outlier, deviating appreciably from the correlation in Figure 7. HOMA values for rings A and B in coronene (1) are 0.740 and 0.657, respectively,6,10 and so it follows that the ratio of HOMA(B) to HOMA(A) is 0.84. This ratio slightly increases to 0.89 when the MP2(fc)/6-31G* level of theory is employed.43,44 Thus, the HOMA(B)-to-HOMA(A) ratio is significantly larger than the SSE(B)-to-SSE(A) ratio (0.54). On the other hand, the BOIA values for rings A and B in 1 are apparently in harmony with the corresponding SSEs (Figure 8). To confirm this trend, however, it is necessary to examine the BOIA values for inner rings of some or many other PAHs. The ratio of BOIA(B) to BOIA(A) is not meaningful, 10481

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Figure 9. Correlation of BOIA with HOMA for all nonidentical rings in benzene and 25 PAHs. Filled circles denote benzene rings in coronene (1). Numerical values of HOMA and BOIA employed are given in the Supporting Information, Table S1; HOMA and BOIA values are those calculated at at the B3LYP/6-31G*(6D,10F) level of theory.6,10

Figure 7. Correlation of HOMA with SSE for all nonidentical rings in benzene and 25 PAHs. Filled circles denote benzene rings in coronene (1). Numerical values of SSE and HOMA employed are given in the Supporting Information, Table S1; HOMA values are those calculated at at the B3LYP/6-31G*(6D,10F) level of theory.6,10

Schleyer’s group14,45 and by us at the same B3LYP/6311+G(d,p) level of theory.43 Figure 10 shows a plot of the

Figure 8. Correlation of BOIA with SSE for all nonidentical rings in benzene and 25 PAHs. Filled circles denote benzene rings in coronene (1). Numerical values of SSE and BOIA employed are given in the Supporting Information, Table S1; BOIA values are those calculated at at the B3LYP/6-31G*(6D,10F) level of theory.6,10

Figure 10. Plot of the NICS(1) value against the BRE for all nonidentical benzene rings in benzene and 14 PAHs. Filled circles denote benzene rings in coronene (1). NICS(1) values employed are given in the Supporting Information, Table S2; NICS(1) values are those calculated at at the B3LYP/6-311+G(d,p) level of theory.14,45 Some NICS(1) values were calculated by us.

because the nonaromatic reference value for BOIA is not known. We then plotted the BOIA values6,10 against the HOMA values6,10 for benzene and 25 PAHs in Figure 9. Although the correlation coefficient between these two indices is as large as 0.963, ring B in 1 again deviates appreciably from the general correlation. This suggests that HOMA and BOIA must have somewhat different meanings when these concepts are applied to inner rings. It is highly probable that the HOMA value overestimates the local aromaticity of ring B, because all the CC bonds in such a ring are strengthened by the local aromaticity of six adjacent benzene rings. Comparison with the NICS(1) Value. We then examine how correlative BRE is with the NICS(1) value for familiar PAHs. NICS(1) values employed are those calculated by

NICS(1) value against the SSE for all rings in benzene and 19 PAHs; for the PAHs adopted in this figure, see the caption of Figure 5. One may see a gross correlation between NICS(1) values and the SSEs, although the correlation coefficient is as small as 0.797. The NICS(1) value seems to be saturated beyond SSE = 0.15 |β|. NICS(1) values for rings A and B in 1 are −11.99 and −4.60 ppm, respectively. Thus, the NICS(1) ratio of ring B to ring A is only 0.38. As has been pointed out, NICS(1) must underestimate the local aromaticity of inner benzene rings. NICS(0) is much less correlative with other local aromaticity indices, such as HOMA and BOIA.6 According to Bultinck et al.,6 correlation coefficients of 10482

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Figure 11. Thirteen large PAHs studied. Indigo, blue, and gray filled circles indicate benzene rings with SSEs larger than 1.8, 1.4, and 1.2 |β|, respectively.

Figure 12. Superposed Clar structures predicted from the SSEs.

PAHs are of moderate magnitude. As will be seen later, 41 is exceptional in some respects. Figure 12 shows Clar structures for 29−41, which were derived from their respective SSE patterns. Those for 30−34 and 39−41 are given in the superposed form, possessing more sextet rings than expected for their intuitively written Clar structures.21 The actual numbers of sextet rings in Clar structures are two for 32−34, four for 29−31, six for 36 and 38, seven for 35 and 37, ten for 39, and twelve for 40. Moran et al. reported the relationship found between Clar’s sextet formula and the NICS(1) values for all rings in 1, 35−37, and 40;21 the sites of sextet rings corresponded to the rings with large negative NICS(1) values. We observed an analogous relationship between the Clar structure and the parts of ring currents formally assigned to individual rings for 1, 29, 30, and 35−38.49 As for the way of formal division of π-currents into the rings, compare Figure 13 with Figure 1. Clar structures for 29−40, derived from the SSEs, are also exactly the same as those intuitively predicted from the molecular structures. PAHs 32−34 are higher members of the perylene/bisanthrene homologous series. Gutman et al. noted that benzene rings on both ends of the polyacene moieties in 33 and 34 exhibit the largest negative NICS(1) values, together with the largest HOMA values,50 supporting the Clar structures we derived for

NICS(0) values with HOMA and BOIA values are 0.51 and 0.59, respectively. SSEs for Large Pericondensed PAHs. We have seen that our SSE model represents well the degree of local aromaticity for both peripheral and inner benzene rings of relatively small PAHs. We then calculated the SSEs for all nonidentical benzene rings of 13 larger pericondensed PAHs in Figure 11, some of which have been employed in the modeling of graphene nanoflakes.46−48 These SSEs are graphically summarized in the Supporting Information, Figure S2. Many of large pericondensed PAHs necessarily have inner or internal benzene rings flanked by six other benzene rings. PAHs 29−31, 33−37, 39, and 40 indeed have one or more such benzene rings. Twelve benzene rings arranged along the inner periphery of 41 were dealt with as inner benzene rings, because they are adjacent to the 18-membered ring (i.e., internal cavity). All species were assumed to be in a closed-shell singlet electronic state.48 Benzene rings with relatively large SSEs in 29−41 can be distinguished from others in Figure 11. It again appears that all SSEs for 29−40 are numerically reasonable and are in congruence with Clar structures in the superposed form, in that the locations of sextet rings correspond to benzene rings with relatively large positive SSEs. SSEs for all outer and inner benzene rings in these large 10483

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sextets or six-site conjugation circuits cannot be chosen from them.1,2 One or more π-bonds that form empty rings are formal single bonds. Central-row benzene rings in the members of the perylene/bisanthrene series are all empty in this sense and have not been taken into account as a source of aromaticity.4,41,51 In a previous paper,52 we, however, found that even formal single bonds exhibit fairly large BREs if they are located in large πsystems. In line with this, some empty benzene rings in the higher members exhibit fairly large SSEs (e.g., SSEs > 0.10 |β| for inner central rings of 34). The same is true for hexagonal rings in semibenzenoid hydrocarbons, such as polyacene-2,3quinododimethides53 and p-polyphenyl-α,ω-quinododimethides,54 for which only one Kekulé structure can be written. These facts imply that even empty rings might be aromatized in large PAH π-systems. Local Aromaticity of Circumkekulene. Circumkekulene (41) is the smallest double-layered coronoid hydrocarbon of D6h symmetry.39 This PAH is exceptional in some respects. Interestingly, the Clar structure of 41 predicted from the SSEs (41a in Figure 12) is markedly different from those intuitively derived from the molecular structure (Figure 15). For reference, SSEs for all nonidentical rings of 41 are graphically shown in Figure 16, together with those for circumcircumcoronene (40). As seen from Figure 15, two pairs of Clar structures (four in all) can be written intuitively for 41, in which the maximum of nine aromatic sextets are located. These Clar structures imply that all benzene rings in 41 must contribute significantly to global aromaticity, because every benzene ring behaves as an sextet ring in one or two of the Clar structures. However, these Clar structures are not compatible with the ones derived from the SSEs. To get some clue to this rather complicated situation, we plotted in Figure 17 the NICS(1) values against the SSEs for all nonidentical benzene rings in 40 and 41. Capital letters in the plots denote nonidentical rings defined in Figure 18. We employed the NICS(1) values calculated by Hajgató et al.39 at the B3LYP/631G level of theory.43 The NICS(1) vs SSE plot for all nonidentical rings of 40 shows a reasonable linear correlation. This is a desirable side of the correlation between NICS(1) and local aromaticity; larger negative NICS(1) values correspond to larger SSEs. In contrast, the NICS(1) value for each ring in 41 is not highly correlative with the corresponding SSE. For example, ring C, located at the inner corner of the macrocycle, exhibits a large negative NICS(1) value of −16.4 ppm, although the SSE for this ring is relatively small (0.1110 |β|). This may be an undesirable side of NICS(1) as a local aromaticity index. We validated the accuracy of the NICS(1) values calculated for 41 by Hajgató et al.39 by recalculating them at the same level of theory. Fairly small SSE for ring C is supported by the optimized geometry of the molecule. As shown in Figure 19a, the HOMA value at the B3LYP/6-31G* level of theory43 is relatively small

Figure 13. Formal division of Hückel−London π-currents into constituent benzene rings. π-Current patterns 1f and 2e represent ring currents thus obtained for coronene (1) and benzo[ghi]coronene (2), respectively. Current intensities are given in units of the benzene value. All ring currents are diamagnetic, flowing counterclockwise.

33 and 34 (Figure 12). Hajgáto et al. calculated the NICS(1) values for all benzene rings in 37, 38, 40, and 41.39 It is clear that, for all but circumkekulene (41), reasonable Clar structures can be derived not only from the NICS(1) values but also from the SSEs. Two or more Clar structures can be written for 30−34, 36, and 39−41. However, one may note that superposed Clar structures derived from the SSEs (Figure 12) do not always cover all possible Clar structures. It is true that superposed Clar structures for 30−32, 39, and 40 cover all possible Clar structures. However, contributions from some Clar structures, such as those presented in Figure 14, are missing in the

Figure 14. Examples of apparently less important Clar structures.

superposed Clar structures for 33, 34, 36, and 39. These Clar structures are characterized by the shift of some peripheral sextet rings with the largest SSEs to the inner less aromatic rings. The same superposed Clar structure for 36 has been derived on the basis of NICS(1) values.21 As stated above, benzene rings on both ends of the polyacene moieties in 33 and 34 exhibit the largest HOMA values,50 supporting the reasonableness of our superposed Clar structures. It follows that there may be both important and less important Clar structures when two or more nonidentical Clar structures can be written. Clar structures, such as those presented in Figure 14, are less important ones. Clar called benzene rings such as central ones in perylene (17) and bisanthrene (32) empty rings, because aromatic

Figure 15. Possible Clar structures intuitively obtained for circumkekulene (41). 10484

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Figure 16. SSEs in units of |β| for all nonidentical rings in PAHs 40 and 41.

Figure 17. Plots of NICS(1) against SSE for nonidentical benzene rings in C96H24 (40) and C90H30 (41). NICS(1) values employed are those calculated at at the B3LYP/6-31G level of theory.39 For the notation of nonidentical rings, see Figure 18.

Figure 18. Notation of nonidentical benzene rings in 40 and 41.

on this benzene ring. The CaCb bond in 41 is the longest one in the π-system (1.451 Å). Further, the smallest Hückel π-bond order of 0.455 is assigned to this longest π-bond. Note that the longest CC bonds usually belong to nonsextet rings in a Clar structure. Therefore, it seems quite likely that, in marked contrast to most PAH molecules, the Clar structure for 41 contains fewer sextet rings than expected from the intuitively written Clar structure. As suggested in Figure 12, the number of sextet rings in 41 is six rather than nine. In harmony with this superposed Clar structure, the CbCc bond in 41 is as short as 1.393 Å (Figure 19b).

Figure 19. B3LYP/6-31G* lengths in Å (a) and Hückel π-bond orders (b) of nonidentical π-bonds in 41. In addition, the HOMA values, calculated at the B3LYP/6-31G* level of theory, are given in blue. All numerical values were obtained by us.

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Figure 20. SSEs in units of |β| for all nonidentical benzene rings in two larger double-layered coronoids, C126H42 (42) and C162H54 (43).

We further calculated SSEs for all nonidentical rings in two higher members of double-layered coronoids (42 and 43). They are graphically summarized in Figure 20. Like 41, benzene rings located at the outer and inner corners of the macrocycles 42 and 43 were found to exhibit relatively small SSEs. Despite that, the NICS(1) values for the benzene rings located at the inner corners are extremely large; they are −24.5 and −31.3 ppm for 42 and 43, respectively.39 This anomalous behavior may be a characteristic common to double-layered coronoids. Hajgáto et al. noted that strong diamagnetic πcurrents are induced along the macrocycles of these doublelayered coronoids.39 According to our Hückel−London ringcurrent calculation,37 macrocyclic π-circulation induced in 41 is 3.5 times as strong as the π-current induced in the benzene molecule.34 Note that Hückel−London ring currents reproduce at least qualitatively the ab initio current density patterns of PAHs.55,56 Large macrocyclic π-circulation in 41−43 may be responsible for the large negative NICS(1) values39 for benzene rings located at the inner corners of the macrocyclic π-systems. We divided the entire ring π-current induced in 41 formally into the individual rings and noticed that the π-current assigned to ring C becomes strongest (Figure 21). This π-circulation pattern is apparently consistent with the large negative NICS(1) value at ring C.

Figure 21. Hückel−London π-current map for 41 (a) and formal division of the π-current into the constituent benzene rings (b). In these π-current maps, diamagnetic currents flow counterclockwise. NICS(1) values, calculated at the B3LYP/6-31G level of theory, are given in blue. All numerical values were obtained by us.

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agree with those intuitively obtained by inspection of the molecular structure. This kind of difficulty may be a characteristic of some coronoid hydrocarbons. One should note that the SSE concept cannot be used to estimate the degree of local aromaticity in polycyclic nonbenzenoid hydrocarbons. For example, SSEs for five- and seven-membered rings of azulene are 0.1434 and 0.1290 |β|, respectively. However, our graph-theoretical analysis revealed that these SSEs arise mainly from the peripheral ten-site circuit but not from the five- or seven-site one;57 the BRE for the πbond shared by the five- and seven-membered rings is as small as 0.0031 |β|. In contrast, global and local aromaticity in PAHs (i.e., polycyclic benzenoid hydrocarbons) stems mainly from the six-site conjugation circuits.58 This must be why SSE can be used successfully as a local aromaticity index for PAHs.

CONCLUDING REMARKS We proposed a novel measure of local aromaticity to estimate the aromaticity of individual benzene rings in large pericondensed PAHs and graphene nanoflakes. This quantity has so far been used as a superaromatic stabilization energy (SSE). SSE is the first local aromaticity index associated explicitly with all relevant circuits in a polycyclic π-system, an origin of global and local aromaticity. It is defined without any approximations or parametrization; it presumably neither overestimates nor underestimates the local aromaticity of inner benzene rings in large pericondensed PAHs. Furthermore, this index is very easy to calculate even for benzene rings in graphene nanoflakes with more than 300 carbon atoms. We noticed that the Clar structures of circumkekulene (41) do not 10486

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(12) Giambiagi, M.; de Giambiagi, M. S.; Silva, C. D. dos S.; de Figueiredo, A. P. Multicenter Bond Indices as a Measure of Aromaticity. Phys. Chem. Chem. Phys. 2000, 2, 3381−3392. (13) Schleyer, P. v. R.; Maerker, C.; Dransfeld, A.; Jiao, H.; van Eikema Hommes, N. J. R. Nucleus-Independent Chemical Shifts: A Simple and Efficient Aromaticity Probe. J. Am. Chem. Soc. 1996, 118, 6317−6318. (14) Chen, Z.; Wannere, C. S.; Corminboeuf, C.; Puchta, R.; Schleyer, P. v. R. Nucleus-Independent Chemical Shifts (NICS) as an Aromaticity Criterion. Chem. Rev. 2005, 105, 3842−3888. (15) Aihara, J. Magnetotropism of Biphenylene and Related Hydrocarbons. A Circuit Current Analysis. J. Am. Chem. Soc. 1985, 107, 298−302. (16) Aihara, J. Circuit Resonance Energy: A Key Quantity That Links Energetic and Magnetic Criteria of Aromaticity. J. Am. Chem. Soc. 2006, 128, 2873−2879. (17) Aihara, J. The Origin of Counter-Rotating Rim and Hub Currents in Coronene. Chem. Phys. Lett. 2004, 393, 7−11. (18) Steiner, E.; Fowler, P. W.; Jenneskens, L. W. Counter-Rotating Ring Currents in Coronene and Corannulene. Angew. Chem., Int. Ed. 2001, 40, 362−366. (19) Acocella, A.; Havenith, R. W. A.; Steiner, E.; Fowler, P. W.; Jenneskens, L. W. A Simple Model for Counter-Rotating Ring Currents in [n]Circulenes. Chem. Phys. Lett. 2002, 363, 64−72. (20) Bühl, M. The Relation between Endohedral Chemical Shifts and Local Aromaticities in Fullerenes. Chem.Eur. J. 1998, 4, 734−739. (21) Moran, D.; Stahl, F.; Bettinger, H. F.; Schaefer, H. F., III; Schleyer, P. v. R. Towards Graphite: Magnetic Properties of Large Polybenzenoid Hydrocarbons. J. Am. Chem. Soc. 2003, 125, 6746− 6752. (22) Aihara, J. Bond Resonance Energy and Verification of the Isolated Pentagon Rule. J. Am. Chem. Soc. 1995, 117, 4130−4136. (23) Aihara, J. Bond Resonance Energies of Polycyclic Benzenoid and Nonbenzenoid Hydrocarbons. J. Chem. Soc., Perkin Trans. 2 1996, 2185−2195. (24) Aihara, J. Is Superaromaticity a Fact or an Artifact? The Kekulene Problem. J. Am. Chem. Soc. 1992, 114, 865−868. (25) Aihara, J. Non-superaromatic Reference Defined by Graph Theory for a Super-ring Molecule. J. Chem. Soc., Faraday Trans. 1995, 91, 237−239. (26) Aihara, J.; Tamaribuchi, T. Non-superaromatic Carbon Nanotube as a Quasi-One-Dimensional Model for Graphite. J. Math. Chem. 1996, 19, 231−239. (27) Aihara, J. Hückel-like Rule of Superaromaticity for Charged Paracyclophanes. Chem. Phys. Lett. 2003, 381, 147−153. (28) Aihara, J. A Simple Method for Estimating the Superaromatic Stabilization Energy of a Super-Ring Molecule. J. Phys. Chem. A 2008, 112, 4382−4385. (29) Aihara, J. Macrocyclic Conjugation Pathways in Porphyrins. J. Phys. Chem. A 2008, 112, 5305−5311. (30) Aihara, J.; Horibe, H. Macrocyclic Aromaticity in Hückel and Möbius Conformers of Porphyrinoids. Org. Biomol. Chem. 2009, 7, 1939−1943. (31) Dias, J. R.; Aihara, J. Antiaromatic Holes in Graphene and Related Graphite Defects. Mol. Phys. 2009, 107, 71−80. (32) Makino, M.; Aihara, J. Macrocyclic Aromaticity of Porphyrin Units in Fully Conjugated Oligoporphyrins. J. Phys. Chem. A 2012, 116, 8074−8084. (33) Aihara, J.; Nakagami, Y.; Sekine, R.; Makino, M. Validity and Limitations of the Bridged Annulene Model for Porphyrins. J. Phys. Chem. A 2012, 116, 11718−11730. (34) Aihara, J.; Makino, M.; Ishida, T.; Dias, J. R. Analytical Study of Superaromaticity in Cycloarenes and Related Coronoid Hydrocarbons. J. Phys. Chem. A 2013, 117, 4688−4697. (35) McWeeny, R. Ring Currents and Proton Magnetic Resonance in Aromatic Molecules. Mol. Phys. 1958, 1, 311−321. (36) Veillard, A. Le Déplacement Chimique en Résonance Magnétique Nucláire dans les Hétérocycles Aromatiques d’Intérêt Biochimique. J. Chim. Phys. Phys-Chim. Biol. 1962, 59, 1056−1066.

In 2005 Bultinck et al. proposed a new local aromaticity index, the so-called six-center index (SCI),6,10 which is a representative electron delocalization index for the constituent benzene rings of PAHs. SCI is based on diatomic delocalization indices between atoms, taking into account all possible resonance structures.6,10 Most of the index arises from πelectrons.6,10 They noted that the SCI value is moderately correlative with HOMA and BOIA values; the correlation coefficients are 0.82 with HOMA and 0.85 with BOIA values.6 Quite interestingly, Bultinck found that the SCI value is proportional to CRE, rather than to SSE.6,59 Therefore, this index must possibly represent the aromatic stabilization due not to a benzene ring but to a six-site circuit. For this reason this index was not discussed in this paper. The plots of the SCI values against the SSEs and CREs for familiar PAHs are given in the Supporting Information, Figures S3 and S4, respectively.

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ASSOCIATED CONTENT

S Supporting Information *

Numerical values of typical local aromaticity indices for benzene and 27 familiar PAHs, SSEs for all nonidentical rings in 13 larger PAHs, NICS(1) and SCI values and CREs for all nonidentical rings in familiar PAHs, and the SCI vs SSE and SCI vs CRE plots for benzene and 27 familiar PAHs. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*J. Aihara: e-mail, [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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