J. Phys. Chem. A 2010, 114, 1093–1097
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Unusually High Aromaticity and Diatropicity of Bond-Alternate Benzene Jun-ichi Aihara* Department of Chemistry, Faculty of Science, Shizuoka UniVersity, Oya, Shizuoka 422-8529, Japan
Toshimasa Ishida Fukui Institute for Fundamental Chemistry, Kyoto UniVersity, Takano-Nishibirakicho, Kyoto 606-8103, Japan ReceiVed: September 23, 2009; ReVised Manuscript ReceiVed: December 2, 2009
Enormous effort has been devoted to the elucidation of possible effects of bond-length alternation on the benzene π-system. Benzene tends to stay highly aromatic and highly diatropic even if strong bond-length alternation is introduced artificially into the π-system. Such peculiar aromatic and magnetic characters of benzene were justified consistently and unambiguously within a single theoretical framework. From all physically sound points of view, bond-alternate benzene is highly aromatic with a large aromatic stabilization energy. We confirmed that in the annulene family benzene is least sensitive in aromaticity to bond-length alternation. Introduction Benzene is the only neutral conjugated hydrocarbon in which all CC π bonds have equal lengths. Possible effects of bondlength alternation (BLA) on the electronic and magnetic properties of benzene have hence been the target of numerous theoretical and experimental investigations.1-19 It has been suggested repeatedly that the aromatic stabilization energy (ASE) of benzene might be rather insensitive to BLA.3,5 In 1990, we evaluated Dewar resonance energies (DREs)20 for benzene and hypothetical bond-alternate D3h benzene (Kekule´ benzene) with short and long CC distances fixed at 1.3517 and 1.4627 Å, respectively;5 these fixed bond lengths are those chosen from the innermost CC bonds in long linear polyenes. We found that the DRE for bond-alternate benzene (15.95 kcal/mol) is only 4.08 kcal/mol smaller than that for regular D6h benzene (20.03 kcal/mol).5 Therefore, the bond-alternate benzene molecule retains ca. 80% of DRE for symmetric D6h benzene. The ability to sustain a diamagnetic ring current in an external magnetic field is characteristic of many aromatic species.21 In 1996, Schleyer et al. proposed a new magnetic criterion for aromaticity: a nucleus-independent chemical shift (NICS), which is defined as the negative of the magnetic shielding at selected points in space, for example, at a ring center.10 Negative (i.e., shielded) and positive (i.e., deshielded) NICS values denote local aromaticity and antiaromaticity, respectively. They justified the NICS value as an indicator of aromaticity by showing a nice correlation between the NICS value and the heat of homodesmotic reaction, a commonly used ASE, for a series of fivemembered heterocycles.10 Benzene is a typical aromatic species with a NICS value of -9.7 ppm. Interestingly, the NICS value for Kekule´ benzene with CC bond lengths fixed at 1.350 and 1.449 Å (the central CC distances in 1,3,5-hexatriene) was only 0.8 ppm less than that for D6h benzene.10,12 This also implies that the aromaticity of benzene is relatively insensitive to BLA.9 It now seems quite likely that benzene tends to preserve not only aromaticity but also diatropicity, even if the molecule is artificially deformed. In this article, we apply our analytical theory of aromaticity and ring-current diamagnetism22-29 to bond-alternate * Corresponding author.
benzenes and related cyclic species to interpret consistently many theoretical and experimental studies so far made on the aromaticity and diatropicity of deformed benzene molecules.1-19 Theory In Hu¨ckel molecular orbital theory, resonance integrals associated with short and long CC π bonds in bond-alternate benzene are denoted by k1β and k2β, respectively, where β is the resonance integral for CC bonds in D6h benzene; k1 and k2 are adjustable parameters. Heilbronner has once argued that π distortivity can be deduced from the standard Hu¨ckel treatment and properly interpreted.4,13 He assumed that alternate CC bonds in bond-alternate benzene carry resonance integrals of (1 ( δ)β. The rationale is that a decrease in resonance integral of one CC bond radiating from a given carbon atom would entail an increase in β for the other CC bond. The range in which this model is acceptable corresponds roughly to the interval -0.5 < δ < 0.5.4 On this theoretical basis, we assume that k1 ) 1.00 + δ and k2 ) 1.00 - δ for bond-alternate benzene. This simple model π-system reproduces well the fact that the negative total π-electron energy increases with enhanced BLA.1,2,4,13 Traditionally, “aromaticity” describes molecules that benefit energetically from the delocalization of π electrons in closed circuits.20-23 In this sense, topological resonance energy (TRE)22,23 is one of the typical indicators of aromaticity. This quantity is given as the difference in total π-binding energy between an actual π system and the graph theoretically defined polyene reference. Let the characteristic and matching polynomials for a π system G be denoted by PG(X) and RG(X), respectively, and the roots of the equations PG(X) ) 0 and RG(X) ) 0 represent energies of the π-molecular orbitals in G and the polyene reference, respectively.22,23 A characteristic polynomial for the π system of bond-alternate benzene can be expressed in the form
PG(X) ) X6 - (3k21 + 3k22)X4 + (3k41 + 3k21k22 + 3k42)X2 (k61 + k62 + 2k31k32)
(1)
Here G represents the π system of bond-alternate benzene. When it is placed in an external magnetic field, the characteristic polynomial is modified to the form28
10.1021/jp909141t 2010 American Chemical Society Published on Web 12/28/2009
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Aihara and Ishida
PG(X, H) ) X6 - (3k21 + 3k22)X4 + (3k41 + 3k21k22 + 2πeSH 3k42)X2 - k61 + k62 + 2k31k32 cos hc
[
(
)]
(2)
Here S is the area enclosed by the deformed benzene ring, H is the magnitude of the component of the magnetic field perpendicular to the molecular plane, and e, h, and c are the standard physical constants with these symbols. A matching polynomial for G is expressed as22,23
simplicity, all bond-alternate benzenes are assumed to have the same ring area as that of D6h benzene (i.e., S ≈ S0). All antiaromatic and large aromatic annulenes undergo BLA to a varying extent.30,31 The harmonic oscillator model of aromaticity (HOMA) index is one of the well-established structural indices of aromaticity,12,16 which reflect the variation of bond lengths in a given cyclic π system. If a given species is highly aromatic, then HOMA value will be close to unity. The HOMA indices for regular and deformed benzenes are given as12,16
RG(X) ) X6 - (3k21 + 3k22)X4 + (3k41 + 3k21k22 + 3k42)X2 -
6
(k61 + k62) (3) The equations PG(X) ) 0, PG(X,H) ) 0, and RG(X) ) 0 can be solved analytically because all polynomials concerned can readily be reduced to cubic ones.13 Ring currents induced in bond-alternate benzene can also be derived analytically from eq 2. First, the A value is defined as24-27 occ
A ) 4k31k32
∑ j
1 PG′ (Xj)
(4)
Here Xj is the jth largest root of the equation PG(X) ) 0; a prime added to PG(X) indicates the first derivative with respect to X; and j runs over all occupied π-molecular orbitals (three in benzene). When there are degenerate π molecular orbitals, eq 4 must be replaced by others.28,29 We previously pointed out that the A value can be interpreted as an ASE arising from the circuit concerned.26,27 In the case of monocyclic π systems, the sole A value is equal to the magnetic resonance energy (MRE) for an entire molecule,26,27 a kind of ASE estimated from the magnetic response of the π system. MRE is roughly proportional to TRE.26,27 The intensity of the ring current induced in a monocyclic π-system can then be expressed in the form24-28
I ) 4.5I0A
S S0
(5)
where I0 is the intensity of a π current induced in regular D6h benzene; and S and S0 are the areas of bond-alternate and regular benzene rings, respectively. Positive and negative A values represent diatropic and paratropic currents, respectively. For
HOMA ) 1 -
∑
a (R - Ri)2 6 i)1 0
(6)
where R ) 257.7 and R0 ) 1.388 Å, both of which are the constants characteristic of CC bonds in a hydrocarbon π system, and Ri is the length of the ith CC bond in the benzene ring. Results and Discussion TREs calculated for bond-alternate annulenes, including benzene, with the ranges 0 e δ e 0.2 are summarized in Table 1. All annulene π systems are assumed to be planar and in the singlet electronic state. Three typical geometries for benzene are presented as 1-3 in Figure 1. One can see from Table 1 that the aromaticity of benzene is indeed much less sensitive to BLA than that of any other annulene. Even markedly bondalternate D3h benzene (2) with k1 ) 1.20 and k2 ) 0.80 retains 69% of TRE for D6h benzene (1). In 1989, Glukhovtsev et al.3,21 estimated the TRE value for D3h benzene with CC bond lengths of 1.3400/1.4627 Å to be 0.220|β|, which corresponds roughly to the set of resonance integrals k1 ) 1.15 and k2 ) 0.85. They utilized the molecular geometry that Shaik et al.1,2 had designed in such a manner that deformation keeps the nuclear repulsion between carbon atoms constant. TRE for bond-alternate benzene is plotted in Figure 2 as a function of δ. In contrast with highly aromatic benzene, highly antiaromatic cyclobutadiene is greatly stabilized by the introduction of BLA. Bond-alternate cyclobutadiene with k1 ) 1.20 and k2 ) 0.80 loses 50% of the negative TRE for the square-planar species. Therefore, cyclobutadiene is effectively stabilized by BLA. All other antiaromatic [4n]annulenes are more sensitive to BLA, in that a greater percentage of negative TRE is lost by BLA. Aromatic [4n+2]annulenes larger than benzene are also more
TABLE 1: Topological Resonance Energies (TREs) for Annulenes in the Singlet Electronic State, Each as a Function of the Degree of Bond-Length Alternation (A) Bond-Alternate [4n]Annulenes Hu¨ckel parameters
TRE/|β|
k1
k2
cyclobutadiene
cyclooctatetraene
[12]annulene
[16]annulene
1.00 1.05 1.10 1.15 1.20
1.00 0.95 0.90 0.85 0.80
-1.226 -1.038 -0.873 -0.731 -0.608
-0.595 -0.420 -0.293 -0.202 -0.139
-0.394 -0.232 -0.134 -0.077 -0.044
-0.295 -0.145 -0.070 -0.034 -0.016
(B) Bond-Alternate [4n+2]Annulenes Hu¨ckel parameters
TRE/|β|
k1
k2
benzene
[10]annulene
[14]annulene
[18]annulene
1.00 1.05 1.10 1.15 1.20
1.00 0.95 0.90 0.85 0.80
0.273 0.266 0.247 0.219 0.188
0.159 0.149 0.123 0.092 0.065
0.113 0.099 0.070 0.043 0.024
0.088 0.071 0.042 0.021 0.010
Aromatic and Diatropic Bond-Alternate Benzene
J. Phys. Chem. A, Vol. 114, No. 2, 2010 1095 TABLE 2: Intensity of the Ring Current Induced in Benzene as a Function of the Degree of Bond-Length Alternation
Figure 1. Benzene (1) and hypothetical bond-alternate benzenes (2,3).
Figure 2. TRE for bond-alternate benzene (b) and the intensity of the ring current induced in it (O), each as a function of δ. Note that k1 ) 1 + δ and k2 ) 1 - δ.
sensitive to BLA than benzene is. All these annulenes retain a smaller fraction of the positive TRE, when BLA is introduced in the π system. There is no doubt that benzene is least sensitive to BLA. Note, however, that BLA lowers the total π-electron energy of any annulene molecule.4 The reason NICS can be used as an indicator of aromaticity for monocyclic π-systems, such as bond-alternate benzenes, is as follows. If monocyclic π-systems have essentially the same ring radii, the NICS value at the center of the ring will be proportional to the intensity of the induced ring current, which then is proportional to MRE.27 This justifies the fact that the NICS value correlates well with ASE for a series of fivemembered heterocycles.10,32 However, NICS values for such five-membered heterocycles must not be compared with those for bond-alternate benzenes because both have fairly different ring radii. As pointed out previously,33,34 NICS cannot be safely used as a descriptor of global or local aromaticity for polycyclic π-systems. The intensity of the current induced in bond-alternate benzene (2) as a function of the degree of BLA as well as the TRE, is listed in Table 2 and graphically shown in Figure 2. Here the area of the bond-alternate benzene nucleus was set equal to that of D6h benzene. Therefore, the NICS value at the center of the benzene ring must be proportional to the current intensity. The MRE in Table 2 can be obtained by dividing the intensity of the ring current in units of I0 by 4.5.27 It is noteworthy that 84% of the π current induced in D6h benzene is retained in bondalternate benzene with k1 ) 1.15 and k2 ) 0.85. The near
k1
k2
TRE/|β|
MRE/|β|
ring current/I0
1.00 1.05 1.10 1.15 1.20
1.00 0.95 0.90 0.85 0.80
0.273 0.266 0.247 0.219 0.188
0.222 0.218 0.206 0.187 0.164
1.000 0.981 0.926 0.842 0.739
retention of a ring current in bond-alternate benzene supports the occurrence of a large negative NICS value at the ring center.10,12 The NICS value for D3h benzene with CC bond lengths of 1.350/1.449 Å, calculated by Schleyer et al.,10 amounts to 92% of that for D6h benzene. In fact, it has been postulated that the out-of-plane component of the NICS value at 1 Å above the center of the ring, NICS(1)zz, is much better as an indicator of aromaticity than the NICS value at the ring center.35,36 The NICS(1)zz values for D6h benzene and Kekule´ benzene in which long CC bonds are 0.10 Å longer than short ones are -29.248 and -27.471 ppm, respectively.16 It seems that ca. 94% of the current intensity is retained in typical bond-alternate benzene. Fortunately, at least for benzenes, this percentage is comparable to that estimated using the NICS values at the ring centers.16 A diamagnetic or paramagnetic ring current induced in an annulene π system is more or less weakened by the introduction of BLA into it. If one bends all benzene HCC bond angles to 90° or by annelation with three cyclobuteno clamping groups, BLA can naturally be introduced into the benzene ring.11,14-16 Such an angle-constrained benzene molecule (3) leads, via optimization at the B3LYP/6-31G* level of theory,37,38 to a bond-alternate geometry with CC bond lengths of 1.5298/1.3482 Å.11 The NICS value at the center of this constrained benzene is only 0.84 ppm less than that for D6h benzene itself (-9.65 ppm). This fact is also consistent with the insensitivity of the current intensity to BLA. Soncini et al. pointed out that the ring current in benzene is resilient under clamping by saturated species.14 Havenith et al. also noted the robustness of a diatropic ring current in distorted benzene.15 We reported in 1980 that the following relationship holds for aromatic annulenes without BLA39,40
TRE ≈ constant ×
I S
(7)
Likewise, for all monocyclic π-systems, MRE is exactly proportional to the current intensity divided by the ring area27
MRE ) constant ×
I S
(8)
This relationship can readily be derived from eq 5 and is applicable not only to all bond-alternate annulenes but also to charged annulenes. It then follows that as far as monocyclic aromatic systems are concerned MRE is nearly proportional to TRE and that the current intensity, divided by the ring area, can be used reasonably as a measure of aromaticity. Equation 8 indicates that if MRE was sensitive to BLA, the current intensity would also be so. As has been seen, however, MRE for benzene is rather insensitive to BLA and so is the intensity of the ring current; highly aromatic bond-alternate benzene will necessarily be highly diatropic in nature. Feixas et al. calculated the HOMA values for bond-alternate benzene molecules at the B3LYP/6-311++G(d,p) level of theory.16 They designed bond-alternate benzenes, in such a
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TABLE 3: HOMA and NICS Values for Bond-Alternate Benzene Calculated at the B3LYP/6-311++G(d,p) Level of Theorya
a
∆RCC/Å
HOMA
NICS/ppm
0.00 0.05 0.10 0.15 0.20 0.25
0.989 0.827 0.344 -0.461 -1.588 -3.038
-8.045 -7.913 -7.541 -6.996 -6.373 -5.782
Ref 16.
manner that the difference in lengths between long and short CC bonds (∆RCC) is varied with consecutive steps of 0.05 Å. The HOMA values they calculated for such bond-alternate species16 are listed in Table 3. It is noteworthy that the HOMA value decreases significantly as BLA is enhanced. The HOMA value is already negative in sign when ∆RCC is 0.15 Å. Therefore, HOMA overestimates the loss of aromaticity due to BLA. Considering that such bond-alternate benzene is still highly aromatic with a large negative NICS value, it is evident that the HOMA value cannot be used as an indicator of aromaticity for such deformed or strained species.16,41 The HOMA concept should be applied to realistic or strain-free geometries. Aromaticity and ring-current diamagnetism of benzene are likewise insensitive to the out-of-plane deformation of the molecule.15,16,42-46 In 1986, van Zijl et al. noted that [5]metacyclophane is highly diatropic despite its strongly bent benzene ring.42,44 In fact, the observed CC bond lengths (1.393 ( 0.007 Å) in the benzene ring are uniform within an experimental error.42 On the basis of the magnetic criterion of aromaticity, Rice et al. reported that [n]paracyclophanes (N ) 5-7) should be classified as aromatic notwithstanding the considerable nonplanar distortions of the benzene ring.43 Remember that planarity has been a structural feature of many aromatic molecules. Dijkstra and van Lenthe estimated DRE for boatshaped benzene as a function of the bending angle.45 They then found that the DRE is 20.30 kcal/mol at the start and still as large as 17.39 kcal/mol at 55°. After 55°, a nonaromatic Dewar structure takes over completely. The transition state of a Diels-Alder reaction between ethylene and butadiene is formally iso-π-electronic with bond-alternate benzene.47,48 This pericyclic transition state also exhibits a large negative NICS value at the central region of the species (
