The Breakdown of the Minimum Polarizability Principle in Vibrational Motions as an Indicator of the Most Aromatic Center

In previous works, we have shown that bond length alternation vibrational modes in πconjugated organic molecules may break the minimum polarizability principle (MPP). To arrive at this conclusion, we have developed a method that diagonalizes the polarizability Hessian matrix with respect to the vibrational nontotally symmetric normal coordinates. The vibrational motions that disobey the MPP in π-conjugated molecules are distortions of the equilibrium geometry that produce a reduction of the polarizability due to the localization of π-electrons. For aromatic species, this electronic localization is responsible for the subsequent reduction of the aromaticity of the system. In the present work, we apply our methodology to calculate the nontotally symmetric distortions that produce the maximum breakdown of the MPP in a series of twenty polycyclic aromatic hydrocarbons. It is shown that the nuclear displacements that break the MPP have larger components in those rings that possess the highest local aromaticity. Thus, these vibrational motions can be use as an indicator of local aromaticity.


I. INTRODUCTION
Benzene exhibits aromaticity in all its structural and chemical manifestations. All energetic, geometric, magnetic, and reactivity based criteria of aromaticity highlight this particular property of benzene, and, consequently, this species is considered as the quintaessential and archetypical aromatic molecule. [1][2][3][4][5][6][7] Among the different vibrational modes of benzene, the bond length alternation (BLA) mode of b2u symmetry (see Scheme 1), which transforms symmetric D6h benzene in a Kekulé-like D3h symmetry structure, has singular characteristics.

SCHEME 1
The first particular feature of this b2u vibrational mode we mention here is its surprisingly low frequency of 1309 cm -1 determined by gas phase two-photon spectroscopy measurements. [8] Assuming that both the σ and π-electronic systems of benzene oppose resistance to this BLA distortion (D6h to D3h), one would predict a larger frequency of about 1600 cm -1 . [9] In addition, there is a remarkable up-shift of this low frequency upon excitation to the first 1 B2u excited state from 1309 to 1570 cm -1 . [8,10,11] This result, which has been confirmed theoretically through coupled cluster calculations, [12] is totally unexpected in the context of the π*←π nature of the electronic transition involved in the transit from the 1 A1g ground state to the first 1 B2u excited state. Considering that the πsystem is weakened in the first 1 B2u excited state, one may expect a reduction in the frequency of this b2u vibrational mode upon excitation. [13] The solution to these two paradoxes came from the work of Hiberty, Shaik, and co-workers, [9,[14][15][16][17][18] among others, [19][20][21] who demonstrated by means of different procedures that the π-electrons of benzene possess a distortive tendency away from the D6h symmetry structure. Now it is accepted that the properties attributed to aromaticity derive from the π-delocalization, but that are the σ-electrons which are responsible for the symmetric D6h framework. [22,23] This conclusion was further substantiated by calculations of the second derivatives of orbital energies with respect to this b2u normal coordinate performed by Gobbi et al. [24] According to these authors, the second derivative of the total energy with respect to normal coordinates can be expressed in a good approximation as the sum of the second derivatives of the orbital energies, thus allowing for a σ/π-separation of the force constants. [24] For the BLA b2u vibration mode of benzene, it was found that the π-force constant is negative while the σforce constant is positive. [24] This result reinforced the conclusion that the π-system of benzene is distortive and π-delocalization is not the driving force of bond equalization in benzene.
The second particular aspect of this b2u vibrational mode is that the deformation of benzene along this mode induces a partial localization of the π-electrons into localized πbonds, reducing the aromaticity of the six-membered ring (6-MR). The effect of this distortion on the delocalization of the π-electrons in benzene was studied by Bader and co-workers [25] considering an unsymmetrical distortion obtained by alternately increasing and decreasing the equilibrium C-C bond length of 1.42 Å bond lengths to 1.54 and 1.34 Å, respectively. Contour maps of the Fermi-hole density indicated that there is a significant decrease in the delocalization of the π electrons between para carbons with the distortion. [25] Accordingly, the value of the recently defined paradelocalization index (PDI) [26] of aromaticity is reduced and this means that the movement along this vibrational mode produces a significant reduction of aromaticity.
This conclusion completely agree with the harmonic oscillator model of aromaticity (HOMA) [6,[27][28][29] and the nucleus independent chemical shift (NICS) [5,30] results obtained by Cyrański and Krygowski, [31] although the NICS values do not sharply differentiate between the aromaticity of benzene (-9.7 ppm) and a Kekulé structure (-8.6 ppm) with bond lengths as in ethane and ethene. [31] Finally, another interesting property is that the b2u BLA vibrational mode breaks the maximum hardness (MHP) and minimum polarizability principles (MPP). [32] These two principles together with the hard and soft acids and bases principle (HSAB) [33] are among the most important chemical reactivity principles that have been rationalized within the framework of conceptual density functional theory (DFT). [34,35] The MHP affirms that, at a given temperature, molecular systems evolve to a state of maximum hardness. [33,[36][37][38][39][40] The MPP was formulated on the basis of the MHP and an inverse relation between hardness and polarizability. [41] This principle states that the natural direction of evolution of any system is towards a state of minimum polarizability. [42,43] A formal proof of the MHP based on statistical mechanics and the fluctuation-dissipation theorem was given by Parr and Chattaraj [38] under the constraints that the chemical potential and the external potential must remain constant upon distortion of molecular structure. However, relaxation of these constraints seems to be permissible, and in particular, it has been found that in most cases the MHP still holds even though the chemical and external potentials vary during the molecular vibration, internal rotation or along the reaction coordinate. [42,[44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63] Hereafter, we will refer to the generalized MHP or MPP (GMHP or GMPP) as the maximum hardness or minimum polarizability principles that do not require the constancy of chemical and external potentials during molecular change.
For nontotally symmetric molecular motions and using symmetry arguments, Pearson and Palke [44] showed that the values of the average external potential ( en ν ), hardness (η ), polarizability (α ), and chemical potential ( µ ) for the positive deviation are -5-the same as those for the negative deviation from equilibrium. Thus, ( ) = 0 at the equilibrium geometry, where Q is a nontotally symmetric normal mode coordinate. In conclusion, the chemical and external potential are roughly constant [32], [64] for small distortions along nontotally symmetric normal modes, thus nearly following the two conditions of Parr and Chattaraj. [38] As a consequence, the MHP and MPP are expected to be obeyed for nontotally symmetric vibrations, as confirmed by most numerical calculations of hardness and polarizability along nontotally symmetric normal modes performed so far. [37,[44][45][46]65] For this reason, the breakdown of the MHP and MPP for the nontotally symmetric b2u mode of benzene is particularly relevant. In contrast, for totally symmetric distorsions, the situation is completely different. In this case, starting from the equilibrium geometry en ν and µ keep increasing or decreasing as the nuclei approach each other; and then, the GMPP can not be applied. Therefore, the equilibrium structure is not a maximum/minimum of hardness/polarizability for displacements along totally symmetric normal modes.
We have found in previous works [32,66,67] small polycyclic aromatic hydrocarbons (PAHs) having nontotally symmetric BLA modes that disobey the GMHP and GMPP. The breakdown of these two principles for nontotally symmetric vibrational motions in PAHs has been extended later to molecules without π-conjugated structure or even without πbonds. [68] To arrive at these results, we have developed a method that diagonalizes the second derivative of the polarizability with respect to the nontotally symmetric normal coordinates (α"). [32] This method provides the distortions that produce the largest polarizability changes, which correspond to nuclear displacements that have a more marked GMPP or anti-GMPP character than the original vibrational modes.
One can expect that BLA modes in PAHs that break the GMHP and GMPP will have similar properties to the b2u vibrational mode of benzene, which is, among the different vibrational modes of benzene, the one that likely produces the largest reduction of aromaticity. Indeed, the nontotally symmetric distortion that produces the maximum breakdown of the GMPP implies a distortion of the equilibrium geometry that produces the largest reduction of polarizability. In aromatic systems, this diminution of polarizability is usually related to the localization of π-electrons, which is expected to be especially important in the region with a more delocalized electron cloud. Thus, it is likely that the nontotally symmetric distortion that produces the maximum failure of the GMPP may have largest components in the most aromatic ring(s) of the system. To investigate this hypothesis, we analyze in this paper a series of PAHs having well-defined local aromaticities to determine whether nontotally symmetric vibrational distortions that disobey the MHP and MPP are mainly located in the most aromatic ring(s). Related to our study, we mention briefly that it has been found [69] that the energy of out-of-plane deformations correlates well with changes in the degree of aromaticity of the conjugated system of whole molecules as well as specific rings. For this reason, Zhigalko and coworkers have proposed to use the frequency of the lowest out-of-plane vibration and ring deformation energy as aromaticity indexes. [69] Finally, we must note that the GMPP is usually more restrictive that the GMHP. Then, the nontotally symmetric distortions that disobey the GMPP normally break also the GMHP (our experience indicates that a failure of the GMHP does not imply unavoidably a breakdown of the GMPP).

II. COMPUTATIONAL DETAILS
In this work, we have used the GAUSSIAN 98 package [70] to perform the geometry optimizations and polarizability, frequency, and NICS calculations. The diagonalization of α" has been carried out at the Hartree-Fock (HF) [71] level using the Pople standard 6-31G basis set [72] for the twenty aromatic molecules studied in this work. To analyze the basis set and electron correlation effects, HF/6-311G(d) and B3LYP/6-311G(d) [73][74][75] calculations have been performed for the smallest eight PAHs analyzed. The NICS calculations have been done at the HF/6-31+G(d) level using the HF/6-31G optimized geometry.
The elements of the Hessian matrix of the polarizability with respect to the nontotally symmetric normal coordinates are calculated as αkl" where α is the isotropic average polarizability where H is the Hessian matrix of the energy with respect to the nontotally symmetric normal coordinates.
To evaluate αkl" we have used two methods both of them equally correct, although they have different computational requirements. In the first method, the αkl"  [32] III. RESULTS AND DISCUSSION Figure 1 depicts the molecules studied in this work, while Table 1 contains the diagonal terms and eigenvalues of α" corresponding to the nontotally symmetric distortions that before or/and after diagonalization break the GMPP for the eight PAHs evaluated at the HF/6-31G, HF/6-311G(d), and B3LYP/6-311G(d) levels. The same information for the rest of studied PAHs computed at the HF/6-31G level is collected in Table 2. Finally, Figure 2 depicts the postdiagonalization nuclear distortions that show the largest negative eigenvalues, which correspond to the eigenvectors with the most marked anti-GMPP character, for each aromatic system analyzed.

FIGURE 1
In a previous study, [32] we have established a set of simple rules to a priori predict, without calculations, whether a given π-conjugated molecule will show nontotally symmetric vibrations with an anti-GMPP character. According to these rules, nontotally Applying these guidelines to the twenty systems of the Figure 1, we obtain that all the molecules, except benzocyclobutadiene and benzo[g]quinoline, at least present a nontotally symmetric vibration mode that refuses to comply the GMPP. This fact is corroborated by the results collected in Tables 1 and 2. Benzocyclobutadiene and benzo[g]quinoline show special characteristics and they will be separately analyzed at the end of this section.

TABLES 1 and 2
The results of the Table 1 show that, while the diagonal elements of the α" matrix are dependent on the methodology used to compute them, the eigenvalues of the diagonalization of α" are almost totally basis set and electron correlation independent. This fact is confirmed by the displacement vectors corresponding to the postdiagonalization nuclear distortions evaluated at the HF/6-31G level and depicted in the Figure 2, which are identical to those obtained with the HF/6-311G(d) and B3LYP/6-311G(d) methods. The eigenvectors obtained from the diagonalization of α" indicate the linear combinations of nontotally symmetric vibrational modes (for a given eigenvector all implicated vibrational modes belong to the same symmetry species) that produce the largest polarizability changes.
As can be seen in Tables 1 and 2, the diagonalization of α" is required to show all the symmetry species that violate the GMPP (e.g., the b1u in anthracene or the b3g, b1u, and b3u in pentacene appear after diagonalization). The diagonalization of α" also reduces the number of vibrational displacements that break the GMPP and simultaneously increases the absolute value of their negative eigenvalue. This concentration of information facilitates the analysis of our results. In the case of the acenaphtylene at the B3LYP/6-311G(d) level, the diagonalization of α" is essential to find molecular distortions that do not follow the GMPP, displaying the utility of this method to detect this kind of molecular distortions.

FIGURE 2
We shall begin our discussion on the relation between the breakdown of GMPP and aromaticity by analyzing first the two molecules (benzene and naphthalene) that have a unique type of ring. In these two systems, the postdiagonalization nuclear distortions of the C atoms display an evident BLA distortion (see Figure 2). On the other hand, biphenylene, acenaphthylene, fluorene, and anthracene-9,10-dione show two types of rings -one is an aromatic 6-MR (rings A of the Figure 1) with a negative value of NICS and the other is a non-aromatic center (rings B) with a positive value of NICS. The differentiation between the two rings is also reflected in the Figure 2, where only the most aromatic ring A presents the expected BLA distortion. Pyracylene, a controversial aromatic system, [26,76,77] presents a similar structure with two different rings. It contains a 5-MR (center B) that is clearly non-aromatic according to magnetic (ring currents and NICS), [77] geometric (HOMA), [26] and electronic (aromatic fluctuation index, FLU) criteria of aromaticity. [76] Moreover it presents a 6-MR (center A) with an intermediate aromaticity as indicated by HOMA, ring currents, PDI, and FLU results. [76,77] However, according to NICS values this 6-MR possesses only a slight aromaticity. A recent work [77] has shown that the magnetic field It is worth nothing that perylene is the only studied molecule with two large negative eigenvalues (-3.273 and -3.106 with b3g and b2u symmetry, respectively). As can be seen in the Figure 2, the postdiagonalization distortion of b3g symmetry shows a quasi-BLA movement for ring A (see Figure 1), while the center B presents a sequence of shortlong-long-short-long distances, indicating that the most aromatic center is ring A. In contrast, the b2u distortion displays a BLA in the six rings, without making an obvious differentiation between centers A and B.
Finally, the [5]helicene, pentacene, and picene contain three kinds of aromatic 6-MRs with negative values of NICS. The method of diagonalization of α" in the pentacene molecule allows by only looking at the b2u distortion of Figure 2 to determine the relative aromaticity order of the three rings. While ring A contains fix atoms, rings B and C show significant displacements of the carbon atoms, although only the most aromatic ring C displays a BLA distortion. This result is consistent with the NICS values indicating that ring A is more aromatic than B, and this, in turn, more aromatic than C. On the other hand, the postdiagonalization displacements in [5]helicene and picene simply show the largest components of the BLA mode in the most aromatic ring.
At the beginning of this section, we mention that the benzocyclobutadiene and benzo[g]quinoline show special characteristics. The diagonalization of α" with respect to the nontotally symmetric normal coordinates and the direct application of the set of simple rules based on the symmetry of the possible BLA movements lead to the same conclusion, that is, the GMPP is obeyed by all nontotally symmetric distortions of these molecules.
Thus, with the present methodology is not possible to ascertain the most aromatic center in these molecules. As an alternative way, we have investigated whether the diagonalization of α" with respect to the totally symmetric normal coordinates can provide information that can not be obtained in this case from the nontotally symmetric vibrations. It is important to remark that totally symmetric distortions at the equilibrium geometry are neither a maximum nor minimum of properties such as α, µ, or en ν . Notwithstanding, their eigenvalues apprise the curvature of the polarizability along these symmetric displacements and indicate whether we are near or far from a polarizability maximum or minimum.
As can be seen in the Figure 2, the application of this method to the benzocyclobutadiene and benzo[g]quinoline helps to determine the relative aromatic character of the different rings. In benzo[g]quinoline, this method only points out the most aromatic ring; nevertheless, the difference of aromaticity between centers A and C is small.
Certainly, it is possible to apply the diagonalization of the polarizability Hessian along totally symmetric modes to the rest of the molecules, although the results do not provide much more information than that obtained from the diagonalization along nontotally symmetric modes. For instance, using this methodology to systems like naphthalene, biphenylene, or fluorene one obtains large negative eigenvalues with postdiagonalized totally symmetric movements, which give exactly the same information than the diagonalization with respect to the nontotally symmetric displacements. In contrast, systems like benzene, pyracylene, or perylene only show positive eigenvalues or few small negative eigenvalues that have postdiagonalization distortions without any relation with BLA motions and aromaticity. Nevertheless, in some cases the diagonalization along totally symmetric modes can be useful to complement the information obtained from the nontotally symmetric modes. As an example, the totally symmetric postdiagonalization movements of [5]helicene and picene (see Figure 3) differentiate the two most aromatic rings (centers A and C), which show BLA distortion from the less aromatic ring (center B).
However, at variance with the nontotally symmetric, totally symmetric postdiagonalization displacements assign similar aromatic character to centers A and C.

IV. CONCLUSIONS
The aromaticity is a property associated with the cyclic delocalization of π-

Figure 1
Tetracene D 2h  b1u -0.013 -0.012 -0.002 a The diagonal terms and eigenvalues of the Hessian matrices of the polarizability (α") are with respect to the totally symmetric modes. -0.137 a The diagonal terms and eigenvalues of the Hessian matrices of the polarizability (α") are with respect to the totally symmetric modes.