Validity and Limitations of the Bridged Annulene Model for Porphyrins

Article pubs.acs.org/JPCA

Validity and Limitations of the Bridged Annulene Model for Porphyrins Jun-ichi Aihara,*,†,‡ Yuto Nakagami,† Rika Sekine,† and Masakazu Makino‡ †

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: According to the bridged annulene model, macrocyclic aromaticity of a porphyrinoid species can be attributed to the annulene-like main macrocyclic conjugation pathway (MMCP). Macrocyclic aromaticity, however, is given theoretically as a sum of contributions from all macrocyclic circuits. We found that the aromaticity due to each macrocyclic circuit is determined formally but broadly by Hückel’s [4n + 2] rule of aromaticity. Nitrogen atoms in the pyrrolic rings effectively suppress the variation in the number of π electrons staying along each macrocyclic circuit. As a result, all or most macrocyclic circuits in oligopyrrolic macrocycles are made aromatic (or antiaromaitc) in phase with the MMCP. Thus, the MMCP is not a determinant of macrocyclic aromaticity but can be regarded as a good indicator of this quantity. This is why the bridged annulene model appears to hold for many porphyrins.

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INTRODUCTION The aromatic nature of free-base porphine and porphycene is commonly attributed to the presence of a diaza[18]annulene substructure.1−7 This annulene-like substructure is presumed to constitute a main macrocyclic conjugation pathway (MMCP) in the porphyrinoid macrocycle. Such a picture of porphyrins is called the bridged annulene model, in which the inner NH groups of pyrrole rings and the outer CHCH groups of 2H-pyrrole rings are regarded as inert bridges. The outer CHCH groups of 2H-pyrrole indeed have significant double bond character.8,9 The bridged annulene model states that the magnetotropicity and aromaticity of the porphyrinoid macrocycle can be predicted by formally applying Hückel’s [4n + 2] rule of aromaticity to the annulene substructure.1−7 Magnetotropicity and aromaticity of the Möbius-twisted porphyrinoid macrocycle can be predicted by applying the reverse of Hückel’s rule to the Möbius-twisted annulene substructure.10,11 However, one should remember that, in principle, Hückel’s rule should not be applied to polycyclic π-systems.12 Superaromatic stabilization energy (SSE) represents an extra stabilization energy due to macrocyclic aromaticity.13−15 Macrocyclic aromaticity is also called superaromaticity or porphyrinoid aromaticity. We have pointed out that the kinetic stability of a porphyrinoid molecule is determined primarily by the degree of macrocyclic aromaticity.14 As for porphyrinoid macrocycles, SSE has been equated to the bond resonance energy (BRE)16,17 for any of the π bonds that link pyrrolic and/or other small rings circularly.13−15 We found that, for porphyrinoids with positive SSEs, an MMCP can be determined by choosing a π bond with a larger BRE at every bifurcation in the π system.13 For those with negative SSEs, an MMCP is determined by choosing a π bond with a smaller BRE at every bifurcation in the π system.13 © XXXX American Chemical Society

One should, however, note that SSE is an aromatic stabilization energy (ASE) arising not from an MMCP alone but from all macrocyclic circuits.13,14 The bridged annulene model disregards the existence of macrocyclic circuits other than an annulenelike MMCP.1−7 Every macrocyclic circuit must contribute more or less to SSE. Therefore, it is very strange that macrocyclic aromaticity/antiaromaticity can be predicted by applying Hückel’s rule to the MMCP.1−7,13,14 Why is the bridged annulene model so useful and popular among porphyrin chemists? This must be the most fundamental problem to be solved in porphyrin chemistry.4,6,7 In this paper, we attempt to justify the bridged annulene model from some graph-theoretical points of view.

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THEORETICAL BACKGROUND Bond resonance energy (BRE) is defined as follows.16,17 A hypothetical π system, in which a given π bond (e.g., a π bond formed between conjugated atoms p and q) interrupts cyclic conjugation thereat, can be constructed by multiplying βp,q by i and βq,p by −i, where βp,q and βq,p represent the resonance integral between two conjugated atoms and i is the square root of −1. In this π system, no circulation of π electrons is allowed to occur in the circuits that pass through the p−q π bond. BRE for the p−q π bond is then given as a destabilization energy of this hypothetical π system. That is, BRE represents the contribution of all circuits that pass through the π bond to topological resonance energy (TRE).18,19 This quantity was originally introduced to justify the isolated pentagon rule for fullerenes.16 If the minimum BRE in a molecule (minBRE) is smaller than Received: August 18, 2012

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dx.doi.org/10.1021/jp310480d | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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Figure 1. Porphyrinoids and related species studied.

Table 1. Topological Resonance Energies (TREs) and Superaromatic Stabilization Energies (SSEs) for 12 Porphyrinoids Studied no. of π electronsa

species free-base porphine (1) porphycene (2) orangarin (3) sapphyrin (4) chlorin (5) bacteriochlorin (6) 21-deazaporphyrin (7) 21,23-dideazaporphyrin (8) 1,10-diaza[18]annulene (9) m-benziporphyrin (10) metal(II) complex of porphyrin (11) tetraaza[16]annulene dianion (12)

26 26 30 32 24 22 24 22 18 26 26 18

TREb/|β|

(18) (18) (20) (22) (18) (18) (18) (18) (18) (−) (18) (18)

0.4322 0.4862 0.5656 0.5904 0.3955 0.3171 0.3121 0.1760 0.0863 0.3866 0.4744 0.0941

t-SSEc/|β|

(0.3390) (0.3183) (0.6733) (0.4817) (0.3115) (0.2484) (0.2515) (0.1338) (0.0700) (0.3660) (0.3718) (0.0765)

0.0843 0.0779 −0.0696 0.0639 0.0793 0.0884 0.0762 0.0777 0.0863 0.0085 0.0795 0.0941

(0.0686) (0.0636) (−0.0871) (0.0530) (0.0661) (0.0720) (0.0630) (0.0633) (0.0700) (0.0084) (0.0649) (0.0765)

(TRE − t-SSE)/|β| 0.3499 0.4083 0.6352 0.5265 0.3162 0.2287 0.2359 0.0983 0.0000 0.3781 0.3949 0.0000

Values in parentheses indicate the nominal number of π electrons located along the main macrocyclic conjugation pathway (MMCP). bValues in parentheses indicate magnetic resonance energy (MRE). cValues in parentheses indicate magnetic SSE (m-SSE).

a

−0.100 |β|, the molecule will possibly be kinetically very unstable.16,17 The BRE concept can be used to estimate the extent of macrocyclic aromaticity in porphyrinoid species.13−15 Macrocyclic aromaticity arises from all possible macrocyclic circuits that enclose the inner cavity. As stated above, superaromatic stabilization energy (SSE) for a porphyrinoid macrocycle can be equated to the BRE for any of the π bonds that link pyrrolic and/or other small rings circularly.13 All macrocyclic circuits pass through these π bonds. Our graph-theoretical variant of Hückel−London theory for ring-current diamagnetism20−23 enables us to estimate the contributions of individual circuits to aromaticity and ring currents. According to this theory, circuit resonance energy (CRE), defined by eq 1, can be interpreted as an aromatic stabilization energy (ASE) arising from a given circuit in a polycyclic π system G:20,21 ri

occ

m>n

j

CREi = 4 ∏ km , n ∑

the jth largest zero of PG(X); a prime added to PG(X) indicates the first derivative with respect to X; and j runs over all occupied π molecular orbitals. If some of the occupied orbitals are degenerate in energy, eq 1 must be replaced by others.22,23 Some theoretical chemists effectively utilized the CRE concept to analyze the contribution of individual circuits to aromaticity and ring currents.24−26 A π-electron current induced magnetically in each circuit may be called a circuit current. The strength of the circuit current induced in the ith circuit is proportional to the CREi, multiplied by the area of the circuit, Si. Therefore, the strength of a π-electron current that flows through a peripheral π bond, Ip−q, is given by summing up all circuit currents that pass through the p−q π bond:20,21 Ip − q = 4.5

PG − ri(Xj) P′G (Xj)

I0 S0

p−q

∑ CREi × Si i

(2)

where I0 is the strength of a current induced in the benzene molecule; S0 is the area of the benzene ring; and i runs over all macrocyclic circuits that pass through the p−q π bond. Positive and negative CREi values point to diatropicity and paratropicity, respectively. Equation 2 is not applicable to π bonds other than peripheral ones. In fact, CRE is an ASE-like quantity derived from the magnetic response of the π system.20,21,27 In other words, the driving force for π circulation in a given circuit is presumed to be an ASE due

(1)

where ri refers to a set of conjugated atoms and π bonds that constitute the ith circuit, ci; km,n is the Hückel parameter for the resonance integral between conjugated atoms m and n, which run over all π bonds that belong to ri; G−ri is the subsystem of G, obtained by deleting ri from G; PG(X) and PG−ri(X) are the characteristic polynomials for G and G−ri, respectively; Xj is B



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Figure 2. Nonidentical circuits in free-base porphine (1).

Figure 3. Nonidentical circuits in porphycene (2).

difference between them can naturally be assigned to the aromaticity due to local five- and/or six-site circuits.13−15 Molecular geometries of free-base porphine (free-base porphyrin, 1), porphycene (2), orangarin (3), sapphyrin (4), m-benziporphyrin (10), and magnesium porphyrin (11) were calculated using the Gaussian 09 program at the B3LYP/6-311+G(d,p) level of theory.33 Cartesian coordinates of atoms in these porphyrinoids are reported in the Supporting Information, Tables S1−S6. Porphyrinoids 5−9 and 11 correspond to different substructures of 1 and so their geometries were equated to the corresponding substructures of 1. These molecular geometries were used to calculate the ring-current distributions in the π systems. Circuit Resonance Energies for Porphyrinoids. As has been seen above, the extent of global aromaticity can be estimated not only from TRE but also from the sum of CREs for all circuits (i.e., MRE),20,21 whereas the extent of macrocyclic aromaticity can be estimated not only from t-SSE but also from the sum of CREs for all macrocyclic circuits (i.e., m-SSE).28 To evaluate these quantities, we calculated CREs for all nonidentical circuits in 1−12. All nonidentical circuits in polycyclic porphyrinoids 1−8, 10, and 11 are shown respectively in Figures 2−10. CREs for all these circuits are listed in Tables 2−11, where asterisks indicate MMCPs. In these tables, Nπ represents the number of π electrons that resides along each circuit. MREs and m-SSEs obtained for 1−12 are added in Table 1. Figure 11 shows a good correlation found between TRE and MRE for 12 species, although MRE is systematically a bit smaller than TRE. Orangarin (3) alone seems to deviate appreciably from linearity. In general, species with antiaromatic circuits tend to deviate more or less from the linear relationship between TRE and MRE. We, however, cannot choose which of these two quantities is better, because both TRE and MRE are defined on physically or mathematically sound grounds. In fact, m-SSE is

to the circuit (i.e., CRE). This seems to be a reasonable interpretation of eq 2.20,21,27 The following approximate formula is then derived from the definitions of SSE and CRE:28 p−q

SSEp − q ≈

∑ CREi i

(3)

where i runs over all macrocyclic circuits that pass through the peripheral p−q π bond. To avoid confusion, quantities on the left- and right-hand sides of eq 3 will be referred to as topological SSE (t-SSE) and magnetic SSE (m-SSE), respectively.28 When one does not need to distinguish between t-SSE and m-SSE, the extent of macrocyclic aromaticity will be referred to simply as SSE. The sum of CREs for all circuits in the π system will become close to the TRE.20,21 This sum was termed the magnetic resonance energy (MRE) for the π system. A high correlation between TRE and MRE has been observed for many aromatic compounds.29−31 In this study, all porphyrinoids are assumed to be in a planar conformation and in a singlet electronic state. For simplicity, peripheral substituents and bond-length alternation are not taken into account. Van-Catledge’s Hückel parameters for heteroatoms32 are used. Nitrogen atoms coordinated to metal ions are dealt with as imine nitrogens. Metal(II) ions are assumed not to participate appreciably in π conjugation. All calculations but geometry optimization were carried out within the framework of simple Hückel molecular orbital theory.

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RESULTS AND DISCUSSION

All porphyrinoids studied (1−12) are presented in Figure 1, where MMCPs are shown in bold. TREs and t-SSEs for 1−12 are listed in Table 1. Because TRE and t-SSE represent the extents of global and macrocyclic aromaticity, respectively, the C



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Figure 4. Nonidentical circuits in orangarin (3). Sapphyrin (4) has two more methine carbon atoms but relative positions of five pyrrolic rings in 4 are the same as those in 3. Therefore, symbols a−w in this figure are also used as those representing nonidentical circuits in 4.

Figure 5. Nonidentical circuits in the chlorin π system (5).

better suited for the present purpose, because CREs are directly related to the senses and strengths of individual circuit currents.20,21 Figure 12 indicates a good correlation between t-SSE and m-SSE for 12 species; orangarin (3) again seems to deviate slightly from linearity. Bridged Annulene Model and Oligopyrrolic Macrocycles. All porphyrinoids but 3 and 10 exhibit macrocyclic aromaticity with appreciably large positive SSEs. Orangarin (3) alone exhibits macrocyclic antiaromaticity with a negative SSE.13,34−37 For all but m-benziporphyrin (10) and metal(II) porphyrin (11), the sign of SSE and the sense of macrocyclic π circulation can be predicted by formally applying Hückel’s rule to the annulene-like MMCP.1−7 That is, the sense of macrocyclic π circulation can be determined by the nominal number of π electrons on the MMCP. This must be why the bridged annulene model has been highly appreciated by porphyrin chemists. As will be seen, this model cannot be applied properly to cross-conjugated porphyrinoids with no MMCPs, such as 10.6,7,13,38,39

Figure 6. Nonidentical circuits in the bacteriochlorin π system (6).

We first focus on the macrocyclic aromaticity of oligopyrrolic macrocycles 1−4. These free-base porphyrins have distinct annulene-like MMCPs.13 However, the usefulness of the bridged annulene model never means that all macrocyclic circuits but the MMCP contribute little to macrocyclic aromaticity.14,36,40 There are many macrocyclic circuits in porphyrinoids. For example, there are 16 macrocyclic circuits in free-base porphine (1) and porphycene (2),40 whereas 32 macrocyclic circuits can be chosen from orangarin (3) and sapphyrin (4).36 Therefore, it is not a matter of course that macrocyclic aromaticity can be predicted from the nominal number of π electrons on the MMCP. D



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Figure 7. Nonidentical circuits in 21-deazaporphyrin (7).

of which is fully consistent with the positive SSE. In contrast, all macrocyclic circuits in 3 are antiaromatic with negative CREs (Table 4),14 the sign of which is consistent with the negative SSE. As far as oligopyrrolic macrocycles are concerned, CREs for all or most macrocyclic circuits have the same sign as that for the MMCP. Like synchronization of innumerable fireflies on mangrove trees, all or most macrocyclic circuits seem to be made aromatic (or antiaromaitc) in phase with the MMCP. From a different viewpoint, an aromatic MMCP, such as those in 1, 2, and 4, seems to make all other macrocyclic circuits aromatic simultaneously. Likewise, an antiaromatic MMCP, such as that in 3, seems to make all other macrocyclic circuits antiaromatic simultaneously. This aspect of macrocyclic circuits is apparently shared by many different porphyrinoids. The question is then reduced to why CREs for all or most macrocyclic circuits have the same sign as that for the MMCP. All five-site circuits in 1−4 are aromatic without exceptions. As in orangarin (3), aromaticity of aromatic circuits in a polycyclic π system are often intensified when aromatic and antiaromatic circuits coexist.43 Macrocyclic π Circulation in Porphyrinoids. Let us survey some aspects of macrocyclic π circulation in porphyrinoids. In experimental studies, the extent of macrocyclic aromaticity is inferred from the chemical shifts of protons attached to the macrocycle.1−7 In our theory,20,21,43 the ring-current map of a polycyclic π system is drawn by superposing π-electron currents induced in all circuits. The strengths of currents induced in individual circuits of 1−8, 10, and 11 are given respectively in

Figure 8. Nonidentical circuits in 21,23-dideazaporphyrin (8).

We attempted to associate macrocyclic aromaticity with individual macrocyclic circuits. For typical oligopyrrolic macrocycles 1−4, an MMCP exhibits the largest positive or negative CRE among the macrocyclic circuits, because it is the only macrocyclic conjugated circuit in the neutral species. Here, a conjugated circuit stands for a circuit consisting of alternating formal single and double bonds.41 An MMCP in any porphyrinoid, if any, are a macrocyclic conjugated circuit. For polycyclic aromatic hydrocarbons (PAHs), conjugated circuits are main contributors to aromaticity.41,42 However, one should note that the CRE for an MMCP never dominates the sign and magnitude of the sum of CREs for all macrocyclic circuits. For 1, 2, and 4 with positive SSEs, the sum of CREs for all macrocyclic circuits but the MMCP is much larger than the CRE for the MMCP. For 3 with a negative SSE, the sum of negative CREs for all macrocyclic circuits but the MMCP is much larger in absolute value than the negative CRE for the MMCP. As reported previously,14 all macrocyclic circuits in 1, 2, and 4 are aromatic with positive CREs (Tables 2, 3, and 5), the sign

Figure 9. Nonidentical circuits in m-benziporphyrin (10). E



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Figure 10. Nonidentical circuits in metal(II) porphyrin (11).

Table 2. Circuit Resonance Energies (CREs) and Circuit Currents (CCs) for Free-Base Porphine (1) circuit

no. of identical circuits

area/S0

Nπ

CRE/|β|

CC/I0

a b c d e f g h* i j k

2 2 1 2 2 1 4 1 2 2 1

0.6583 0.6602 5.1766 5.8368 5.8349 6.4970 6.4951 6.4932 7.1553 7.1534 7.8136

5.635 5.483 17.755 18.454 18.069 19.153 18.768 18.383 19.467 19.082 19.781

0.0780 0.0571 0.0050 0.0022 0.0082 0.0009 0.0035 0.0131 0.0013 0.0053 0.0018

0.2312 0.1698 0.1163 0.0588 0.2145 0.0250 0.1014 0.3837 0.0403 0.1710 0.0641

Table 4. CREs and CCs for Orangarin (3) circuit no. of identical circuits a b c d e f g h i j k l m n o p q* r s t u v w

Table 3. CREs and CCs for Porphycene (2) circuit


area/S0

Nπ

CRE/|β|

CC/I0

a b c d e f g h* i j k l

2 2 1 2 2 1 2 2 1 2 2 1

0.6560 0.6573 4.9580 5.6140 5.6152 6.2713 6.2713 6.2700 6.2725 6.9273 6.9286 7.5846

5.608 5.493 17.666 18.019 18.393 18.747 18.747 18.373 19.120 19.100 19.474 19.827

0.0757 0.0517 0.0050 0.0080 0.0020 0.0031 0.0031 0.0126 0.0007 0.0047 0.0010 0.0015

0.2234 0.1528 0.1121 0.2024 0.0517 0.0875 0.0875 0.3553 0.0202 0.1452 0.0317 0.0495

1 2 2 1 1 2 2 2 2 2 2 1 1 2 2 1 1 2 2 2 2 1 1

area/S0

Nπ

CRE/|β|

CC/I0

0.6497 0.6588 0.6536 5.6877 6.3374 6.3466 6.3414 6.9962 6.9911 7.0002 7.0002 7.0054 6.9950 7.6499 7.6499 7.6551 7.6447 7.6538 7.6590 8.3035 8.3087 8.3127 8.9624

5.920 5.591 5.656 19.349 19.871 20.159 19.731 20.681 20.253 20.541 20.541 20.969 20.114 21.064 21.064 21.492 20.636 20.924 21.352 21.446 21.874 21.734 22.257

0.1884 0.1344 0.1516 −0.0023 −0.0038 −0.0011 −0.0038 −0.0017 −0.0063 −0.0017 −0.0017 −0.0004 −0.0063 −0.0027 −0.0027 −0.0007 −0.0102 −0.0027 −0.0007 −0.0043 −0.0011 −0.0011 −0.0016

0.5508 0.3984 0.4460 −0.0579 −0.1079 −0.0300 −0.1079 −0.0533 −0.1968 −0.0533 −0.0533 −0.0142 −0.1969 −0.0934 −0.0934 −0.0237 −0.3518 −0.0935 −0.0237 −0.1606 −0.0402 −0.0402 −0.0662

or the nominal number of π electrons on the MMCP. At the level of individual circuits, an MMCP always sustains the strongest diamagnetic or paramagnetic π current (Tables 2−11), which is often comparable in strength to or stronger than the currents induced in local pyrrolic rings. Even from a magnetic standpoint, an MMCP is a representative of all macrocyclic circuits. Macrocyclic Aromaticity in Imperfect Free-Base Porphines. Some porphyrinoid π systems corresponding to different substructures of free-base porphine (1) have been prepared. Among such imperfect free-base porphines are the π systems of chlorin (5),44 bacteriochlorin (6),44 21-deazaporphyrin (7, also known as vacataporphyrin),45 and 21,23dideazaporphyrin (8)46 in Figure 1. All these species, including 1 and 1,10-diaza[18]annulene (9), have a diaza[18]annulenelike MMCP in common and so exhibit macrocyclic aromaticity with positive SSEs. As in the case of 1−4, MMCPs in 5−8 avoid the inner NH groups of the pyrrole rings and the outer CH CH groups of the 2H-pyrrole rings. CREs for all macrocyclic circuits in 5−9 have the same plus sign (Tables 6−9). Like those in 1, 2, and 4, the MMCPs in 5−8 exhibit the largest

Tables 2−11, where positive and negative values indicate diamagnetic and paramagnetic currents, respectively. All circuits in all porphyrinoids but 3 and 10, as well as all local circuits in 3, are diatropic, because CREs for all these circuits are positive in sign. On the other hand, all macrocyclic circuits in 3 are paratropic with negative CREs. Strengths of overall macrocyclic currents and related quantities in 1−12 are listed in Table 12. A macrocyclic current in Table 12 is nothing other than the sum of currents induced in all macrocyclic circuits. Equation 2 indicates that the strength of a π current induced in each circuit is proportional to the CRE, multiplied by the area of the circuit. Therefore, macrocyclic circuits with relatively large CREs contribute significantly to the overall macrocyclic current, because the areas of macrocyclic circuits are very large. Consequently, strong π circulation is often observed along the macrocycle, the sense of which is determined by the sign of SSE F



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Table 5. CREs and CCs for Sapphyrin (4) circuit

a


area/S0

Nπ

CRE/|β|

CC/I0

circuit


area/S0

Nπ

CRE/|β|

CC/I0

1 2 2 1 1 2 2 2 2 2 2 1 1 2 2 1 1 2 2 2 2 1 1

0.6616 0.6603 0.6563 7.3730 8.0346 8.0333 8.0293 8.6948 8.6909 8.6896 8.6896 8.6936 8.6857 9.3512 9.3512 9.3551 9.3473 9.3460 9.3499 10.0075 10.0115 10.0062 10.6078

5.696 5.541 5.695 21.586 21.908 22.302 21.949 22.623 22.270 22.665 22.665 23.017 22.312 22.986 22.986 23.339 22.633 23.027 23.380 23.349 23.701 23.743 24.064

0.0759 0.0773 0.0992 0.0012 0.0022 0.0006 0.0022 0.0011 0.0037 0.0011 0.0011 0.0002 0.0037 0.0016 0.0016 0.0006 0.0061 0.0016 0.0006 0.0028 0.0006 0.0006 0.0012

0.2258 0.2296 0.2929 0.0414 0.0800 0.0209 0.0799 0.0429 0.1431 0.0429 0.0429 0.0068 0.1430 0.0693 0.0693 0.0233 0.2585 0.0692 0.0233 0.1244 0.0273 0.0273 0.0565

a b* c d

2 1 2 1

0.6602 6.4932 7.1534 7.8136

5.423 17.818 18.529 19.240

0.0352 0.0358 0.0121 0.0034

0.1044 1.0446 0.3894 0.1180

a b c d e f g h i j k l m n o p q* r s t u v w a

Table 9. CREs and CCs for 21,23-Dideazaporphyrin (8)

Table 10. CREs and CCs for m-Benziporphyrin (10) circuit no. of identical circuits a b c d e f g h i j k l m n o

Refer to Figure 4.

2 1 1 1 2 1 2 1 1 1 2 1 2 1 1

area/S0

Nπ

CRE/|β|

CC/I0

0.6620 0.6636 1.0030 5.3486 6.0106 6.0122 6.6742 6.6726 7.3362 6.3517 7.0137 7.0152 7.6772 7.6757 8.3392

5.481 5.665 5.700 17.054 17.763 17.353 18.063 17.054 18.772 18.786 19.496 19.086 19.795 20.205 20.505

0.0719 0.0697 0.1440 0.0022 0.0015 0.0042 0.0025 0.0008 0.0012 −0.0011 −0.0007 −0.0021 −0.0012 −0.0004 −0.0006

0.2142 0.2082 0.6501 0.0537 0.0394 0.1146 0.0748 0.0230 0.0399 −0.0311 −0.0222 −0.0650 −0.0417 −0.0128 −0.0221

Table 11. CREs and CCs for Metal(II) Porphyrin (11)

Table 6. CREs and CCs for Chlorin (5) circuit


area/S0

Nπ

CRE/|β|

CC/I0

a b c d e f g* h

2 1 1 2 1 2 1 1

0.6583 0.6602 5.1766 5.8349 5.8368 6.4951 6.4932 7.1534

5.688 5.552 17.761 18.099 18.494 18.832 18.437 19.169

0.0794 0.0866 0.0063 0.0107 0.0031 0.0050 0.0176 0.0077

0.2352 0.2574 0.1472 0.2804 0.0819 0.1448 0.5153 0.2493

circuit


area/S0

Nπ

CRE/|β|

CC/I0

a b* c d e f g

4 1 4 4 2 4 1

0.6611 5.1723 5.8334 6.4945 6.4945 7.1556 7.8168

5.443 17.538 18.239 18.939 18.939 19.639 19.924

0.0767 0.0129 0.0066 0.0034 0.0034 0.0011 0.0006

0.2283 0.3011 0.1735 0.0990 0.0990 0.0368 0.0214

Table 7. CREs and CCs for Bacteriochlorin (6) circuit


area/S0

Nπ

CRE/|β|

CC/I0

a b c d*

2 1 2 1

0.6583 5.1766 5.8349 6.4932

5.767 17.799 18.181 18.562

0.0882 0.0097 0.0168 0.0286

0.2613 0.2251 0.4423 0.8364

Table 8. CREs and CCs for 21-Deazaporphyrin (7) circuit


area/S0

Nπ

CRE/|β|

CC/I0

a b c d* e f g h

1 2 1 1 2 2 1 1

0.6583 0.6602 5.8349 6.4932 6.4951 7.1534 7.1553 7.8136

5.585 5.455 17.802 18.112 18.514 18.824 19.226 19.536

0.0851 0.0517 0.0131 0.0205 0.0051 0.0076 0.0017 0.0024

0.2520 0.1536 0.3427 0.5989 0.1489 0.2457 0.0536 0.0832

Figure 11. Relationship between TRE and MRE for 12 porphyrinoids studied. Orangarin (3) is denoted by a filled circle.

It may be worth noting that t-SSEs for 1 and 5−9 lie in the narrow energy range 0.0762−0.0884 |β|, although TRE varies in the much wider range (0.3955−0.0863 |β|). This never means that the CRE for a diaza[18]annulene-like MMPC is preserved among these species. In reality, the CRE for an MMPC varies widely from molecule to molecule, so that it is obvious that the

positive CREs among the macrocyclic circuits (Tables 6−9). TRE for 1,10-diaza[18]annulene (9) is very close to that for carbocyclic [18]annulene (0.0877 |β|). G



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current must be roughly proportional to the sum of the CREs for the macrocyclic circuits. On the basis of eqs 2 and 3, we can say that m-SSE is roughly proportional to the strength of the macrocyclic current. Thus, the macrocyclic current of similar strength reflects the SSE of similar magnitude (see Table 1 for SSEs). Macrocyclic Aromaticity in Cross-Conjugated Porphyrinoids. An MMCP cannot be defined for cross-conjugated porphyrinoids, such as m-benziporphyrin (10), because there are no macrocyclic conjugated circuits.13 The bridged annulene model does not apply to such porphyrinoids. When macrocyclic conjugated circuits are lacking, SSE is very small in general. In fact, 10 has a very small t-SSE of 0.0085 |β|, exhibiting virtually no macrocyclic aromaticity.6,7,13,38,39 Interestingly, 10 has not only aromatic but also antiaromatic macrocyclic circuits; eight macrocyclic circuits exhibit positive CREs and eight others exhibit negative CREs (Table 10). This must be why 10 exhibits little macrocyclic aromaticity. All aromatic macrocyclic circuits in 10 (circuits j−o in Figure 9) pass the outer periphery of the benzene ring, whereas all antiaromatic macrocyclic circuits (circuits d−i in the same figure) pass the inner periphery of the benzene ring. As compared to those in porphyrinoids with distinct MMCPs, the CREs for all macrocyclic circuits in 10 are fairly small in magnitude and almost cancel each other out. All five- and six-site circuits in 10 are aromatic with large positive CREs. Macrocyclic Aromaticity in Metalloporphyrin. The bridged annulene model no longer holds if metals are coordinated to a porphyrinoid π system. This never means the lack of macrocyclic aromaticity. In the case of metal(II) porphyrin (11), an 18π conjugation pathway of a 16-membered inner ring, analogous to the tetraaza[16]annulene dianion (12), constitutes the MMCP.4 This MMCP is compatible with our definition of the MMCP based on the BRE concept.13 MMCPs in other metalloporphyrins are also chosen along the inner periphery of the macrocycle.13 All macrocyclic circuits in 11 are aromatic and contribute to diamagnetic π circulation around the macrocycle. The sense of the macrocyclic π circulation can be predicted formally from the nominal number of π electrons on the MMCP. All five-site circuits in 11 are aromatic with positive CREs. Macrocyclic Aromaticity in Other Porphyrinoids. We previously reported that, for a variety of free-base porphyrinoids with distinct MMCPs, the sign of t-SSE can be predicted from the nominal number of π electrons on the MMCP.13 Among these porphyrinoids are free-base porphyrins, fused porphyrins, confused porphyrins, expanded porphyrins, Möbius-twisted porphyrins, and porphyrinoids with nonpyrrolic rings.1−7,11,13 Such a wide applicability of the bridged annulene model strongly supports the view that, like oligopyrrolic macrocycles and related species 1−8, the CREs for all or most macrocyclic circuits must have the same sign as that for the MMCP. This is very true as long as an annulene-like MMCP can be defined for the π system. Macrocyclic Aromaticity in Porphyrinoid Molecular Ions. For many porphyrinoids, the sign of t-SSE changes on going from the neutral species to the molecular dianion or dication.14 Such a regular change in macrocyclic aromaticity was called the Hückel-like rule of macrocyclic aromaticity.49 t-SSEs for doubly charged molecular ions of 1−12 are listed in Table 13. All the porphyrinoids but 10 have distinct MMCPs and satisfy the Hückel-like rule of macrocyclic aromaticity we proposed previously.49 That is, porphyrinoids with a positive

Figure 12. Relationship between t-SSE and m-SSE for 12 porphyrinoids studied. Orangarin (3) is denoted by a filled circle.

Table 12. Strengths of Macrocyclic Currents and CREs for the Main Macrocyclic Conjugation Pathways (MMCPs) in the Porphyrinoids Studied species

no. of macrocyclic circuits

macrocyclic current strength/I0

CRE for the MMCP/|β|

1 2 3 4 5 6 7 8 9 10 11 12

16 16 32 32 8 4 8 4 1 16 16 1

1.964 1.749 −2.858 2.148 1.844 1.946 1.867 1.941 2.047 0.201 1.758 1.780

0.0131 0.0126 −0.0102 0.0061 0.0176 0.0286 0.0205 0.0358 0.0700 0.0129 0.0765

CRE value is not transferable among these species. As seen from Table 12, a CRE for an MMCP is larger for simpler species with a smaller number of macrocyclic circuits. It follows that essentially the same SSE value is preserved by the united efforts of all macrocyclic circuits; it does not matter however many macrocyclic circuits might be available. All five-site circuits in 5−8 are aromatic without exceptions. Steiner and Fowler estimated that the maximum current density in the macrocycle of 1 is about twice that of benzene.47 Fliegl and Sundholm recently calculated induced current densities for free-base porphine (1), chlorin (5), and bacteriochlorin (6) using a gauge including magnetically induced current method and reported that the strengths of macrocyclic ring currents in these species are 1.9−2.4 times stronger than that in benzene.48 Bröring noted that the chemical shifts of outer protons in 1, 7, and 8 are of comparable magnitude and that those of inner C-bound protons in 7 and 8 are also comparable in magnitude.4 These observations suggest that 1, 7, and 8 sustain macrocyclic π currents of comparable strength. Our simple Hückel−London calculations also indicate that the strengths of macrocyclic currents induced in 1 and 5−8 are of comparable magnitude, being 1.8−2.0 times as large as that in benzene (Table 12). Strengths of these macrocyclic currents can be associated with the SSEs in the following manner. As the areas of the macrocyclic circuits in 1 and 5−8 are large and of the same order of magnitude, the strength of the macrocyclic H



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molecular dianions of 1 and 2, respectively. These dihydroporphyrins likewise exhibit macrocyclic antiaromaticity.54−56 The molecular dication of metal(II) porphyrin (11) has been predicted to sustain a paramagnetic current along the macrocycle,57 the sense of which is again consistent with the negative SSE value (Table 13). Like the molecular dications of free-base porphyrinoids, all macrocyclic circuits in 112+ exhibit negative or negligibly small positive CREs (Table S16, Supporting Information). The molecular dianion of 11 must be an open-shell species because it has doubly degenerate HOMOs with two π electrons. Porphyrinoid Molecules and Hü ckel’s Rule. We proceed to the main problem of why macrocyclic aromaticity of a porphyrinoid species can be predicted from the nominal number of π electrons on the annulene-like MMCP. The socalled extended Hückel rule proposed by Hosoya et al.42 may give some clue to the apparent applicability of Hückel’s rule to porphyrin macrocycles. This rule states that [4n + 2]- and [4n]-site conjugated circuits are the main sources of aromaticity and antiaromaticity, respectively, in neutral conjugated hydrocarbons.42 Thus, Hückel’s original rule can be applied formally to individual conjugated circuits in neutral polycyclic conjugated hydrocarbons. Therefore, there is a large possibility that Hückel’s rule might formally be applied to the annulenelike conjugated circuit in a porphyrinoid macrocycle. The large contribution of the MMCP to macrocyclic aromaticity indeed is consistent with this rule, although the extended Hückel rule in its original form is not applicable to polycyclic heteroconjugated systems. We further noticed that something like Hückel’s rule is operative on various neutral and charged polycyclic π systems, including polycyclic heteroconjugated species.14,58 This finding is based on the following observation concerning polycyclic π systems formed by fusion of two or more rings of the same size. We examined the plots of the TRE against the number of π electrons for this type of π systems and noticed that these plots are very similar in appearance to that for a monocyclic π system of the same ring size.58 In these plots, the number of π electrons was varied from zero to twice the number of conjugated atoms. We found that these polycyclic π systems behave as if they are a collection of isolated monocyclic π systems of the same ring size. Hückel’s rule of aromaticity can of course be applied to monocyclic π systems. This finding clearly indicates that something like Hückel’s rule acts implicitly or explicitly on polycyclic π systems and may possibly support the formal applicability of the Hückel rule to individual circuits in polycyclic π systems. In this context, it is important to note that Hückel’s rule is a rule found between aromaticity and the number of π electrons. In the case of neutral alternant hydrocarbons, such as PAHs, the number of π electrons staying on each carbon atom is equal to unity. Therefore, the nominal number of π electrons counted along each conjugated circuit is always 4n + 2 or 4n, which is exactly the same as the actual number of π electrons on the circuit.42 Therefore, we can readily predict the aromaticity of each conjugated circuit from the number of π electrons on it. For porphyrinoids, the nominal number of π electrons counted along the MMCP is 4n + 2 or 4n, which, however, is not equal to the actual number of π electrons on it. Therefore, we need to examine the actual number of π electrons staying on each circuit (Nπ in Tables 2−11) in porphyrinoids 1−8, 10, and 11 and then to see if it is related to the sign of the CRE. The ranges where the Nπ values for aromatic and antiaromatic

Table 13. SSEs for Doubly Charged Molecular Ions of Porphyrinoids Studied t-SSE/|β| species 1 2 3 4 5 6 7 8 9 10 11 12

no. of π electrons in the neutral speciesa 26 26 30 32 24 22 24 22 18 26 26 18

(18) (18) (20) (22) (18) (18) (18) (18) (18) (−)b (18) (18)

dication

neutral

dianion

−0.0942 −0.1588 0.0707 −0.0705 −0.1624 −0.2667 −0.1062 −0.1183 −0.1641 −0.0135 −0.0574 −0.1234

0.0843 0.0779 −0.0696 0.0639 0.0793 0.0884 0.0762 0.0777 0.0863 0.0085 0.0795 0.0941

−0.1384 −0.0306 0.0482 −0.1627 −0.1729 −0.0707 −0.0987 −0.0692 −0.1690 0.0159 −0.2147 −0.2686

a Values in parentheses indicate the nominal number of π electrons on the MMCP. bNo MMCP.

SSE exhibit a negative SSE when they acquire or lose two π electrons. On the other hand, porphyrinoids with a negative SSE exhibit a positive SSE when they acquire or lose two π electrons. The same regularity has been observed in fully conjugated paracyclophanes consisting of four or more paraphenylene units linked circularly by an even number of ethylene fragments.50,51 CREs for neutral and doubly charged species of 1−8, 10, and 11 are listed respectively in the Supporting Information, Tables S7−S16. When a doubly charged molecular ion exhibits a negative SSE, CREs for all or most macrocyclic circuits, including the MMCP, are negative in sign. This applies to doubly charged ions of all species but 3 and 10. Doubly charged molecular ions of orangarin (3) exhibit a positive SSE and a positive CRE for most macrocyclic circuits. For porphyrinoids other than 10, the MMCP defined for the neutral species exhibits the largest positive or negative CRE even if they form doubly charged molecular ions. Therefore, the MMCP can still be regarded as a representative macrocyclic circuit in the doubly charged molecular ions. Some macrocyclic circuits in molecular dications of 1, 2, 7, and 8 exhibit small positive CREs, although these circuits never change the sign of SSE predicted by the Hückel-like rule of macrocyclic aromaticity. The existence of such exceptional circuits suggests that the Hückel-like rule of macrocyclic aromaticity cannot be proved exactly for macrocyclic π systems. Most five-site circuits in 1−8, 10, and 11 remain aromatic when they form molecular dianions and dications; only two types of five-site circuits in 32+ (circuits a and b in Figure 4) are exceptionally antiaromatic (Table S9, Supporting Information). Monocyclic species 9 and 12 simply obey Hückel’s original rule of aromaticity. Nucleus-independent chemical shift (NICS) values52 calculated for porphycene (2) and the molecular dianion (22−)53 are fully consistent with the signs of the SSEs for these species (Table 13) and those of the CREs for the macrocyclic circuits concerned (Table S8, Supporting Information). The NICS value at the center of the macrocycle is negative in sign for 2 but is positive for 22−. These values support that SSE is positive for 2 but is negative for 22−. NICS values at the centers of the pyrrolic rings are all negative in both 2 and 22−. Dihydroporphyrins in which two 2H-pyrrole rings in 1 and 2 are converted into pyrrole ones are iso-π-electronic with the I



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Table 14. Ranges of the Numbers of π Electrons (Nπ Values) on Aromatic and Antiaromatic Macrocyclic Circuits in 1−8 and 10−11 Nπ value(s) MMCPa

species

18.383 18.373 20.636 22.633 18.437 18.562 18.122 17.818

1 2 3 4 5 6 7 8 10 11

[18] [18] [20] [22] [18] [18] [18] [18]

17.538 [18]

aromatic region (I)b

antiaromatic regionb

aromatic region (II)b

17.755−19.781 (18.768) 17.666−19.827 (18.723) 19.349−22.257 (20.803) 21.586−24.064 (22.825) 17.761−19.169 17.799−18.562 17.802−19.536 17.818−19.240 17.054−18.772 17.538−19.924

(18.465) (18.181) (18.669) (18.529) (17.736) (18.913)

18.786−20.505 (19.646)

a

Values in square brackets indicate the nearest Hückel magic number. bValues in parentheses indicate the average of the Nπ values on macrocyclic circuits concerned.

Table 15. Ranges of the Numbers of π Electrons (Nπ Values) on Aromatic and Antiaromatic Macrocyclic Circuits in the Divalent Molecular Ions of 1−8 and 10−11 Nπ values species 2+

1 12− 22+ 22− 32+ 32− 42+ 42− 52+ 52− 62+ 62− 72+ 72− 82+ 82− 102+ 102− 112+ a

MMCP 16.536 19.743 16.400 19.973 18.896 22.237 20.825 24.016 16.581 20.412 16.562 20.562 16.177 19.590 15.819 19.418

a

[16] [20] [16] [20] [18] [22] [20] [24] [16] [20] [16] [20] [16] [20] [16) [20]

15.588 [16]

antiaromatic region (I)

b

aromatic region (I)b

15.989−17.434 (12)

17.786−18.334 (4)

16.009−17.302 (12)

17.813−18.205 (4)

antiaromatic region (II)b

antiaromatic region (III)b

19.100−21.637 (16) 18.865−21.827 (16) 18.111−20.357 (32) 20.864−23.905 (32) 19.899−22.534 (32) 22.936−25.886 (32) 15.985−17.510 (8) 19.217−21.035 (8) 16.167−16.562 (4) 19.300−20.562 (4) 15.905−17.077 (6)

17.704−17.977 (2)

15.819−16.720 (3)

17.620 (1)

15.235−17.206 (8)

17.179−19.150 (8) 18.212−19.957 (8) 18.224−19.103 (5)

19.269−21.438 (8) 19.418−21.240 (4)

15.588−17.345 (11)

20.557−22.302 (8)

Values in square brackets indicate the nearest Hückel magic number. bValues in parentheses indicate the number of macrocyclic circuits concerned.

macrocyclic circuits are nonconjugated ones. For free-base porphyrinoids, macrocyclic circuits corresponding the inner and outer peripheries carry the largest and the smallest numbers of π electrons, respectively. For example, Nπ values for the macrocyclic circuits corresponding to the inner and outer peripheries of 1 are 17.755 and 19.781, respectively. Although the Nπ values for macrocyclic circuits in this π system are fairly different from each other, CREs for all of them have the same sign. The same is true for 2−8 and 11, with the Nπ regions for aromatic or antiaromatic macrocyclic circuits being fairly wide for each π system. The most important point in these observations is that all or most of the macrocyclic circuits in each porphyrinoid have the same aromaticity as the MMCP. It follows that the macrocyclic aromaticity of a porphyrinoid macrocycle can be predicted formally by the sign of the CRE for the MMCP. The dominant role of the MMCP may be associated with the fact that the Nπ value for the MMCP is close to the average of Nπ values for all macrocyclic circuits. It is instructive to examine the Nπ values for individual macrocyclic circuits of m-benziporphyrin (10), even though it is

macrocyclic circuits lie in individual porphyrinoids are summarized in Table 14. We then explore possible relationships of the aromaticity of an MMCP (i.e., the sign of the CRE) in each porphyrinoid with the Nπ value for it. Free-base species 1, 2, and 5−8 have a diaza[18]annulene-like MMCP in common. The Nπ values for the MMCPs of these species lie in the relatively narrow range 17.818−18.562 (Table 14). These Nπ values are all very close to one of the Hückel magic number of aromaticity (18). The MMCPs of orangarin (3) and sapphyrin (4) carry 20.636 and 22.633 π electrons, respectively, which are rather close to the Hückel magic numbers of antiaromaticity (20) and aromaticity (22), respectively. Thus, the MMCPs with Nπ values close to 4n + 2 and 4n exhibit positive and negative CREs, respectively. Therefore, we can say that the aromaticity of the MMCP is consistent with the Nπ value for it, formally conforming to Hückel’s rule of aromaticity. When it comes to macrocyclic circuits other than the MMCP, the Nπ value for each circuit, however, varies widely. These J



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(URFs) for the corresponding heteroconjugated systems.60 He stated that nature prefers to place heteroatoms of greater electronegativity in those positions where π electrons tend to accumulate in the URF. Many porphyrinoids indeed conform to this rule.59−61 TREs and SSEs for 13−16 are summarized in Table 16. As seen from this table, these hydrocarbons obviously reflect the

a cross-conjugated species. As seen from Table 10, the Nπ values for macrocyclic circuits of 10 lie very widely in the range 17.054−20.505. This brings about the coexistence of aromatic and antiaromatic macrocyclic circuits in 10. The Nπ values for aromatic (CRE > 0) and antiaromatic (CRE < 0) macrocyclic circuits lie in the ranges 17.054−18.722 and 18.786−20.505, respectively. It is noteworthy that both Nπ ranges in 10 do not overlap with each other. These two Nπ regions correspond roughly to the Hückel magic numbers of aromaticity (18) and antiaromaticity (20), respectively. As has been seen, it is very likely that something like Hückel’s rule acts on the MMCP and all other macrocyclic circuits in porphyrinoids.14 This aspect of macrocyclic aromaticity will further be confirmed by examining the Nπ values for macrocyclic circuits in the porphyrinoid molecular ions. Porphyrinoid Molecular Ions and Hückel’s Rule. In the porphyrinoid molecular dications and dianions, the Nπ values for macrocyclic circuits likewise range fairly widely. The Nπ regions for aromatic (CRE > 0) and antiaromatic (CRE < 0) macrocyclic circuits in doubly charged molecular ions of 1−8, 10, and 11 are summarized in Table 15. Molecular dications of six porphyrinoids 1, 2, 7, 8, 10, and 11 have both aromatic and antiaromatic macrocyclic circuits (Supporting Information, Tables S7, S8, and S13−S16). As in the case of the neutral species of 10, the Nπ region for CRE > 0 does not overlap with that for CRE < 0 in the same π system. However, there seem to be no common or definite boundaries between the Nπ regions for aromatic and antiaromatic circuits. All molecular dianions but the cross-conjugated species 102− have either aromatic or antiaromatic macrocylic circuits but do not have both. The Nπ values for the MMCPs in the doubly charged molecular ions of 1−8 and 11 are added in Table 15. For all these species, the MMCP defined for the neutral species makes the largest contribution to macrocyclic aromaticity. It is noteworthy that the Nπ value for each MMCP is again close to one of the Hückel magic numbers. The sign of CRE for each MMCP reflects the nearest Hückel magic number. The MMCP always belongs to a major group of macrocyclic circuits with the CREs of the same sign; the Nπ value for it lies near the average of the Nπ values for all macrocyclic circuits that belong to the same group (see Table 15). Thus, even in the doubly charged molecular ions, the aromaticity of the MMCP can be regarded as an indicator of macrocyclic aromaticity. Macrocyclic aromaticity of the doubly charged species can be predicted from the Hückel magic number closest to the Nπ value for the MMCP. Role of Nitrogen Atoms in Oligopyrrolic Macrocycles. To further confirm that Hückel’s rule of aromaticity holds formally but broadly for circuits in macrocyclic π systems, we examined the applicability of this rule to porphyrinoid hydrocarbons, such as 13−16 in Figure 13. These are hypothetical hydrocarbons isostructural and iso-π-electronic with typical oligopyrrolic macrocycles 1-4.59−61 Gimarc referred to these conjugated hydrocarbons as the uniform reference frames

Table 16. TREs and SSEs for Four Porphyrinoid Uniform Reference Frames (Porphyrinoid URFs) species URF URF URF URF

for for for for

1 2 3 4

(13) (14) (15) (16)

TRE/|β|

t-SSE/|β|

0.3941 0.3794 0.7812 0.6044

0.0598 0.0542 −0.0106 0.0336

global and macrocyclic aromaticity of their respective parent compounds. However, conjugated circuits are missing in these hydrocarbons, because they are charged species. CREs for all nonidentical circuits in 13−16 are listed respectively in the Supporting Information, Tables S17−S20. Among the macrocyclic circuits in these URFs, the innermost macrocyclic circuit always exhibits the largest positive or negative CRE and so can be regarded as an MMCP. The URFs are similar in this sense to metalloporphyrins. We found that the URFs for oligopyrrolic macrocycles are characterized by the larger variation in the Nπ value for macrocyclic circuits. Hereafter, the difference between the largest (Nπmax) and smallest (Nπmin) numbers of π electrons on macrocyclic circuits will be denoted by ΔNπ. As can be seen from Table 17, the ΔNπ value for each URF is more than Table 17. Difference between the Largest (Nmax π ) and Smallest (Nmin π ) Numbers of π Electrons on Macrocyclic Circuits for the Oligopyrrolic Macrocycles (1−4) and the Corresponding URFs (13−16) species

Nmax − Nmin π π

species

min Nmax π −Nπ

1 2 3 4

2.026 2.161 2.908 2.478

13 14 15 16

3.785 3.880 5.509 4.898

1.8 times as large as that for the parent porphyrinoid. The large ΔNπ value brings about the variation of the sign of CRE among the macrocyclic circuits, which did not occur in the neutral oligopyrrolic macrocycles. The sign of CRE changes once or twice on going from the macrocyclic circuit with the Nmin π value to that with the Nmax value, broadly obeying Hückel’s rule of π aromaticity (Table 18). We can now assuredly say that Hückel’s rule is opeartive broadly on macrocyclic circuits of neutral and charged porphyrinoids. One should again remember that the ΔNπ value for each URF is much larger than that for the parent porphyrinoid. In other words, the vatiation in Nπ among the macrocyclic circuits was found to be fairly small at least for oligopyrrolic macrocycles, such as 1−4. The smaller ΔNπ value is one of the very desirable conditions for making all or most macrocyclic circuits aromatic or for making all or most macrocyclic circuits antiaromatic. Because the Nπ value for an MMCP is close to the average of the Nπ values for all macrocyclic circuits, all or most macrocyclic circuits in the same porphyrinoid π system

Figure 13. Four porphyrinoid hydrocarbons. K



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Table 18. Ranges of the Numbers of π Electrons (Nπ Values) on Aromatic and Antiaromatic Macrocyclic Circuits in the URFs for 1−4 Nπ values

a

species

aromatic region (I)a

antiaromatic region (I)a

13 14 15 16

17.323−19.215 (11) 17.219 (1)

20.161−21.108 (5) 18.189−20.129 (14) 18.597−20.869 (15)

aromatic region (II)a

antiaromatic region (II)a

21.099 (1) 21.834−23.036 (16) 20.942−23.022 (16)

24.106 (1) 23.740−25.840 (16)

Values in parentheses indicate the number of macrocyclic circuits concerned.

suppressing the variation in the number of π electrons that each macrocyclic circuit carries, helping to keep the numbers of π electrons on many macrocyclic circuits within the same aromatic (or antiaromatic) region. This way of justification for the bridged annulene model must be the best one, because any exact proof for it is not imaginable in principle. In brief, the annulenelike MMCP is not a determinant of macrocyclic aromaticity but can be regarded as a convenient indicator of this quantity. The bridged annulene model cannot be applied to crossconjugated porphyrinoids, because an MMCP cannot be defined for them. For most porphyrinoids for which an MMCP can be defined, SSEs for the molecular dication and dianion have a different sign from that for the neutral molecule. As in the case of neutral species, CREs for most macrocyclic circuits in these molecular ions have the same sign as the MMCP.

necessarily exhibit CREs of the same sign. Thus, nitrogen atoms in the porphyrinoids play two important roles; one is to form a stable heteroconjugated system in Gimarc’s sense,59−61 and the other is to enhance macrocyclic aromaticity/antiaromaticity by suppressing the Nπ variation among the macrocyclic circuits. Justification for the Bridged Annulene Model. We summarize the reason why the bridged annulene model holds for a variety of porphyrinoid molecules. Here, porphyrinoid molecules are limited to free-base ones with a distinct MMCP, because this model is usually applied to these species. The Nπ value for an MMCP was found to be fairly close to one of the Hückel magic numbers, which is identical to the nominal number of π electrons on the MMCP. Therefore, the aromaticity of the MMCP (i.e., the sign of the CRE) can be predicted from the Hückel magic number. The Nπ value for the MMCP is also close to the average of the Nπ values for all macrocyclic circuits. If the MMCP is aromatic with a positive CRE, the Nπ range for aromatic macrocyclic circuits must be fairly wide and so all or most of the macrocyclic will be more or less aromatic with positive CREs. Conversely, if the MMCP is antiaromatic with a negative CRE, the Nπ range for antiaromatic macrocyclic circuits must be fairly wide and so all or most of the macrocyclic circuits will be more or less antiaromatic with negative CREs. As a result, the sign of SSE is adjusted to that of the MMCP with a high probability. In fact, 1−8 exhibit SSEs that have the same sign as their respective MMCPs. This is why the bridged annulene model holds for many different porphyrinoids. The bridged annulene model has also been applied to many expanded porphyrins larger than orangarin (3) and sapphyrin (4).2,5 For these enlarged porphyrins, the Nπ values on macrocyclic circuits may be distributed in wider ranges and some of them may exhibit CREs the sign of which is different from that of the MMCP. Even in these species, most of the macrocyclic circuits must still exhibit CREs of the same sign as the MMCP, because nitrogen atoms tend to level the Nπ values on macrocyclic circuits to their averaged value. We noted previously that the macrocyclic aromaticity of typical hexaphyrins and octaphyrins can likewise be predicted from the nominal number of π electrons on the MMCP.11,13 At present, expanded free-base porphyrins that are not compatible with the bridged annulene model are not known.3,5,62,63

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ASSOCIATED CONTENT

S Supporting Information *

Cartesian coordinates of constituent atoms in 1−4, 10, and 11 calculated at the B3LYP/6-311+G(d,p) level of theory, circuit resonance energies for nonidentical circuits in doubly charged molecular ions of 1−8, 10, and 11, circuit resonance energies for nonidentical circuits in 13−16, and the complete description of ref 33. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Prof. Timothy D. Lash, Illinois State University, for critically reading the manuscript of this paper. Computations were carried out at the Information Processing Center, Shizuoka University, and the Research Center for Computational Science, Okazaki National Research Institutes.

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REFERENCES

(1) Vogel, E. Pure Appl. Chem. 1993, 65, 143. (2) Franck, B.; Nonn, A. Angew. Chem., Int. Ed. Engl. 1995, 34, 1795. (3) Vogel, E. J. Heterocycl. Chem. 1996, 33, 1461. (4) Bröring, M. Angew. Chem., Int. Ed. 2011, 50, 2436. (5) Stepień, M.; Sprutta, N.; Latos-Grażyński, L. Angew. Chem., Int. Ed. 2011, 50, 4288. (6) Lash, T. D. J. Porphyrins Phthalocyanines 2011, 15, 1093. (7) Lash, T. D. J. Porphyrins Phthalocyanines 2012, 16, 423. (8) Crossley, M. J.; Harding, M. M.; Sternhell, S. J. Am. Chem. Soc. 1992, 114, 3266. (9) Liu, D.; Lash, T. D. J. Org. Chem. 2003, 68, 1755. (10) Yoon, Z. S.; Osuka, A.; Kim, D. Nat. Chem. 2009, 1, 113.

CONCLUDING REMARKS Macrocyclic aromaticity and π circulation in an oligopyrrolic macrocycle can be predicted from the nominal number of π electrons that stay on the annulene-like MMCP. We have shown that this bridged annulene model can be justified well in terms of CREs for macrocyclic circuits. Hückel’s rule of aromaticity holds formally but broadly for macrocyclic circuits of porphyrinoids and makes all or most of them exhibit CREs of the same sign as the MMCP. Nitrogen atoms are effective in L



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Validity and Limitations of the Bridged Annulene Model for Porphyrins

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