Aromaticity of Nonplanar Fully Benzenoid Hydrocarbons - The Journal

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Aromaticity of Non-Planar Fully Benzenoid Hydrocarbons Marija Antic, Boris Furtula, and Slavko Radenkovi# J. Phys. Chem. A, Just Accepted Manuscript • Publication Date (Web): 25 Apr 2017 Downloaded from http://pubs.acs.org on April 29, 2017

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The Journal of Physical Chemistry A is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

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The Journal of Physical Chemistry

Aromaticity of Non-Planar Fully Benzenoid Hydrocarbons Marija Antić, Boris Furtula*, Slavko Radenković* Faculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, Serbia

ABSTRACT: The Clar aromatic sextet theory can provide the qualitative description on the dominant modes of cyclic π-electron conjugation in benzenoid molecules, and on the relative stability among a series of isomeric benzenoid systems. In a series of non-planar fully benzenoid hydrocarbons, the predictions of the Clar theory were tested by means of several different theoretical approaches: topological resonance energy (TRE), energy effect (ef), harmonic oscillator model of aromaticity (HOMA) index, six centre delocalization index (SCI) and nucleus independent chemical shifts (NICS). To assess deviations from planarity in the examined molecules, four different planarity descriptors were employed. It was shown how the planarity indices can be used to quantify the effect of non-planarity on the local and global aromaticity of the studied systems.

1. INTRODUCTION Benzenoid hydrocarbons belong to a class of polycyclic aromatic hydrocarbons (PAHs), the compounds that continuously attract the scientists’ attention for more than a century.1,2 Research in PAHs chemistry has experienced a surge during the last decades, as these compounds are of importance in many diverse ﬁelds, such as materials, astrochemistry and environmental

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chemistry.3–7 Higher members of benzenoid hydrocarbon’s family display unusual electronic and optical properties, making these compounds as candidates for applications in different areas such as organic electronics, photonics, spintronics, and energy storage.8–10 Theoretical studies of π-electron properties of benzenoid molecules started as early as in the 1930s.11,12 Much of these studies were done within the framework of the topological theory, in which the Clar’s aromatic sextet theory1,13 and Kekulé structures2,11 play an important role. In the Clar theory, the so-called Clar formulas are constructed by placing the maximal possible number of “aromatic sextets” into non-adjacent rings, so that the rest of the rings in the given benzenoid molecule have a Kekulé structure.14 The original Clar aromatic sextet theory was a qualitative method, but eventually several quantitative approaches were put forward.2,15–17 The Clar formulas can provide the information on the dominant modes of cyclic π-electron conjugation in a benzenoid molecule. Besides, the Clar theory can be employed to predict the relative stability among a series of isomeric benzenoid hydrocarbons. The so-called fully benzenoid (or allbenzenoid) molecules have a single Clar formula, in which all π-electrons are grouped into aromatic sextets. Examples of Clar formulas of fully benzenoid hydrocarbons are given in Figure 1. The rings in which aromatic sextets are located (indicated by circles) are said to be “full” whereas the other rings are “empty”. According to the Clar theory, fully benzenoid hydrocarbons are the most stable benzenoids, and a number of these molecules has been synthesized.3,18


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Figure 1. Set of benzenoid hydrocarbons and the molecular nomenclature used. The capital letters denote symmetry-unique rings in these molecules.


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In all theories based on Kekulé structures, including the Clar theory, it is (implicitly) assumed that a given benzenoid molecule is planar. On the other hand, it is well known that a large part of the known benzenoid hydrocarbons are non-planar.19 Recently, several new non-planar benzenoid systems have been obtained and characterized.20–24 Bays, coves, and fjords are the structural features that can be found on the perimeter of a underlined benzenoid system.12 Benzenoid hydrocarbons possessing coves and fjords have a large steric strain and non-planar geometry. On the other hand, benzenoid systems possessing only bays, but no coves and fjords, are perfectly planar. In the recent studies, it has been shown that although such molecules are planar, they have a non-negligible steric strain.25,26 It is well known that the concept of aromaticity has no unique definition. The evaluation of the aromaticity in a whole molecule or parts thereof can be performed by means of several criteria.27 Recall, that according to the Clar formulas, one can expect that the full rings have a significant aromatic character, whereas the empty rings have non-aromatic or moderately aromatic character.28 In this paper, the Clar-formula-based aromaticity predictions for the set of the studied molecules were compared with several different aromaticity indices. The local aromaticity in the examined molecules was measured by means of the energy effect (ef),29,30 harmonic oscillator model of aromaticity (HOMA) index,31,32 six centre delocalization index (SCI)33,34 and nucleus independent chemical shifts (NICS).35,36 Topological theories were successfully applied to assess π-electron properties of numerous planar PAHs. On the other hand, the applicability of the topological approaches to non-planar systems has not been much studied in the past. In this work, a series of isomeric non-planar fully benzenoid systems was considered (1-5 in Figure 1). By replacing the near-lying hydrogen atoms


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with carbon-carbon bonds in the fjord regions of 1-5, the new series of fully benzenoid molecules were obtained, in which planarity of the carbon-atom skeleton is successively increased. As an example, one can consider the series 1, 1a, 1b and 1c (Figure 1). In order to provide a quantitative information on the deviation form planarity in a given molecule, in the present work several different planarity indices were employed. In what follows it is shown that the planarity descriptors can provide a deeper understanding on how the deviations from planarity influence the local and global aromaticity in benzenoid hydrocarbons.

2. METHODOLOGY Planarity descriptors The planarity of the studied molecules and their hexagonal rings was quantified by means of four different indices. As it is shown below, the three of them (brute-force planarity index (BFPI), least-square plane index (LSPI) and covariance matrix based planarity index (CMPI)) are based on finding a plane that minimizes the mean distance of a set of atoms from it. An advantage of these indices is that each of them can be calculated for all atoms or for any subset of atoms in the given molecule. Thus, the BFPI, LSPI and CMPI can be applied for the whole molecule or parts thereof. Obviously, their common feature is that their values are directly proportional to the extent of planarity deviations. The minimal value that they can take is 0. In what follows, atoms of a molecule can be considered as a set of points in the space. An equation of a plane Π can be derived from the coordinates of three non-collinear points that belong to Π. Then, the mean of distances of all atoms in a molecule from Π will be the measure of planarity of the given molecule. In the present study, the hydrogen atoms were omitted. However, there are as many planes (and planarity indices) as there are combinations of triples of


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points from a given set of atoms. The minimum in the obtained basket of planarity indices is taken to be the brute-force planarity index (BFPI). The second descriptor of planarity is based on the least-square technique for obtaining the best fitted plane in the 3D space. Namely, the explicit form of an equation of plane can be rewritten as: Z = aX + bY + c

(1)

Then, knowing the coordinates of all points, the coefficients a, b, and c are calculated using the following equations:

a=

Cov( X , Y ) ⋅ Cov(Y , Z ) − Cov(Y , Y ) ⋅ Cov( X , Z ) Cov( X , Y ) 2 − Cov( X , X ) ⋅ Cov(Y , Y )

(2)

b=

Cov( X , Y ) ⋅ Cov( X , Z ) − Cov( X , X ) ⋅ Cov(Y , Z ) Cov( X , Y ) 2 − Cov( X , X ) ⋅ Cov(Y , Y )

(3)

c = z − ax − by

(4)

where x is the mean of X-set of coordinates and Cov(X,Y) is the covariance between X-set and Yset of coordinates. The least-square plane index (LSPI) is the mean of distances of the obtained plane. Matito et al.37 have proposed the planarity descriptor based on constructing covariance matrix (CM) from known coordinates of atoms in a molecule. The CM is a 3×3 matrix. The CM eigenvalues correspond to the variances of data. It is well known that the standard deviation is


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equal to the square root of the variance. This was our motivation to use the square root of the smallest eigenvalue of CM as the covariance matrix based planarity index (CMPI). In the recent paper,38 as the measure of the ring non-planarity, Dobrowolski and Lipiński have employed the root mean square deviation of all ring CCCC dihedral angles:

T=

1 n i 2 ∑ (τ CCCC ) n i=1

(5)

where summation goes over all n different CCCC dihedral angles (τ) in the given ring. As a measure of non-planarity of the underlined polycyclic molecule, one can consider the sum of the T values of all rings ( ∑ T ).

Aromaticity indices The topological resonance energy (TRE)39,40 is a quantifier of the global aromatic stabilization. The TRE was independently introduced by two research groups.39,40 TRE is defined as the difference between the total π-electron energy (Eπ) and appropriate reference energy (Eref): TRE = E π − E ref

(6)

The reference energy is defined in a manner fully analogous to Eπ, except that any effect resulting from the presence of cycles is disregarded. TRE found numerous applications in studies on aromaticity of conjugated compounds.27 The energy effect of a given ring R (ef(R))29,30 provides the energy contribution of R to the total π-electron energy of the considered molecule. Within the framework of chemical graph theory, the ef(R) can be calculated as follows:


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ef ( R ) =

2

∞

φ (G, ix)

dx ln π ∫ φ (G, ix) + 2φ (G − R, ix)

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(7)

0

where G is the molecular graph representing the π-electron system considered, φ (G ) is its characteristic polynomial and G-R is the subgraph obtained by deleting from G the ring R. The TRE and ef values are expressed in the units of the carbon-carbon resonance integral β, whose

recommended value is ≈-137kJ/mol. Since β is negative-valued, positive TRE and ef values mean a stabilizing energy-effect caused by cyclic π-electron conjugation. The six centre index (SCI) as a member of the class of multicentre delocalization indices, can be defined within different partition schemes.41–43 Using the Mulliken partition scheme, the six centre index (SCI) can be calculated as follows:33,34

SCI =

6! 1 L ∑∑∑ ∑∑ Γi [(PS)µν (PS)νσ L(PS)ξµ ] 32 µ∈A ν ∈B σ ∈C ξ∈F i =1

(8)

where P is the density matrix and S is the overlap matrix. The summation goes over all basis functions centred at the atoms A-F involved in a given six-membered ring, and Γi is the permutation operator which for the given set of six basis functions produces all possible permutations (in total 6! permutations). Details of the underlying theory, as well as an exhaustive bibliography can be found in the recent review.44 It is worth mentioning that in the case of benzenoid hydrocarbons there is a good concordance between the SCI and topological π-electron ring currents.45 The harmonic oscillator model of aromaticity (HOMA) index is the geometry-based measure of aromaticity, and can be calculated as:31,32


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HOMA = 1 −

1 n α ( Ropt − Ri ) 2 ∑ n i =1

(9)

where n is the number of bonds of the ring considered, α is a normalization constant, Ropt is the optimal bond length for a fully delocalized π-electron system, and Ri stands for an actual bond length. The parameters needed for the HOMA calculations were taken from ref 31. The nucleus independent chemical shift (NICS) index is among the most popular aromaticity indices.46 Originally, the NICS was defined as the negative value of the isotropic shielding constant calculated at the ring centre. In the recent paper,38 it has been pointed out that for nonplanar rings the NICS should be calculated separately, 1Å above (NICS(1)) and 1Å below the ring centre (NICS(-1)). As a good probe of aromaticity in non-planar molecular systems, the NICS(1)av has been suggested: NICS(1)av = [NICS(-1)+ NICS(1)]/2.

(10)

Note that in the case of planar molecules, the NICS(1)av value coincides with the NICS(1) value.

3. COMPUTATIONAL METHODS Geometries of the studied molecules (Figure 1) were optimized at the BP8647/def2-SVP48 level of theory with the D3 dispersion correction suggested by Grimme et al.49 using the Gaussian 09 program.50 Harmonic frequency calculations were carried out to verify that the optimized structures correspond to minima on the potential energy surface. The computational method used in the present work has been successfully applied in the recent study of large benzenoid molecules.51 The NICS were calculated through the gauge-including atomic orbital (GIAO) method52,53 at the same level of theory. The BP86-D3/def2-SVP optimized structures were employed for the HOMA indices calculations. The SCI were calculated from the BP86-D3/def2-


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SVP density matrices. Calculations of SCI and ef were performed using our own Fortran routines, and for planarity indices we made in-house Python programs. It was tested that BFPI can be efficiently calculated for small molecules (up to 100 atoms).

4. RESULTS AND DISCUSSION Structural formulas of the studied benzenoid hydrocarbons (Figure 1) represents only the carbonatom skeleton, whereas the hydrogen atoms are not indicated. Although, this is a standard way of representing such molecules, one may miss to immediately recognize that in the fjord regions the respective hydrogen atoms are so close to each other that the molecule necessarily becomes nonplanar. Figure 2 shows the optimized geometries of 1-5. The optimized structures of the other studied molecules can be found in Figure S1 (Supporting Information). The molecule 1 has the most distorted optimized geometry among the considered isomers (Figure 2). The obtained D3symmetric propeller conformation of 1 is in agreement with the X-ray structure.54 From Figure 2 it is evident that the extent of the deviation from planarity is directly related to the number of the fjord regions. According to the values of BFPI and LSPI, 1 with three fjord regions has the largest deviation from planarity in the series of isomers 1-5 (Table 1). On the other hand, the CMPI and ΣT predicts that 3 has the most significant planarity distortion. Having two fjords, 2

and 3, have very similar degrees of non-planarity, and the values of all planarity indices for 3 are somewhat larger than for 2. The smallest deviation from planarity was found in 4 and 5, which have only one fjord region. For 4 and 5, based on the employed planarity indices, it is not possible to uniquely determine which one has more distorted geometry (Table 1). The values of planarity indices for all other molecules from Figure 1 are given in Table S1 (Supporting Information). The cross-correlation analysis provides a deeper understanding of the relation between the planarity indices in the series of the studied molecules. As can be seen from Table 2


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and Figure S2 (Supporting Information), between the employed planarity indices there are linear, but not particularly good correlations. Besides, if the planar molecules were discarded from the analysis the quality of the correlations is significantly reduced (Table 2 and Figure S3 in the Supporting Information). In some cases, for instance, between the CMPI and ΣT, and LSPI and ΣT, there is practically no correlation. The best correlation was found between LSPI and BFPI.

Thus, the LSPI and BFPI provide a very similar description of the planarity deviation in the considered benzenoid molecules. The studied isomeric molecules 1-5 are fully benzenoid systems, and based on the Clar formulas one is not able to distinguish between their stability. The TRE values of 1-5 are very close, and this way the TRE supports the predictions based on the Clar theory (Table 1). According to the TRE values there are negligibly small differences in aromatic stability of the isomers 1-5. The largest difference, based on the TRE, was found between 3 and 5, which is still smaller than 3 kJ/mol. The problem with the topological approaches, such as the Clar structures and TRE, is that beside the connectivity, any other geometrical features of the given molecular system are not taken into account. On the other hand, the relative free energies (∆G°) obtained at the BP86-D3/def2-SVP level of theory showed that between the studied isomers 1-5 there are significant differences in their relative stability (Table 1).


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a)

b)

c)

d)

e)

Figure 2. Optimized geometries obtained at the BP86-D3/def2-SVP level of theory of: a) 1; b) 2; c) 3; d) 4 and e) 5.


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Table 1. Relative free energies, ∆G° (in kJ/mol), TRE (in β units), BFPI (in Å), LSPI (in Å), CMPI (in Å) and ΣT (in degrees) for the studied molecules 1-5.

compound

∆G°

TRE

BFPI

LSPI

CMPI

ΣT

1

78.88

1.6799

8.908

8.944

10.807

88.08

2

45.55

1.6736

7.682

8.395

10.897

82.15

3

61.23

1.6871

7.951

8.451

11.515

92.66

4

7.39

1.6660

5.040

5.715

7.948

50.05

5

0.00

1.6662

4.802

5.665

8.428

56.40

Table 2. Correlation coefficients (R) between the planarity indices for the molecules examined (see Figure S2 in the Supporting Information). The correlation coefficients obtained by considering only the non-planar molecules from Figure 1 are given in the brackets (see Figure S3 in the Supporting Information). BFPI

LSPI

CMPI

BFPI

1.0000(1.0000)

LSPI

0.9960(0.9870)

1.0000(1.0000)

CMPI

0.9851(0.9537)

0.9948(0.9758)

1.0000(1.0000)

ΣT

0.9711(0.8345)

0.9694(0.7736)

0.9739(0.7766)


ΣT

1.0000(1.0000)

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Based on the values of ∆G°, 1 was found to be the most destabilized in the studied isomeric series 1-5. It was shown that the relative free energies of 1-5 can be expressed as a function of the values of the corresponding planarity descriptors (Figure 3). Although, it was shown that between some of the employed planarity indices there exist good linear correlations (Table 2), only the LSPI and BFPI can properly describe the order of stability in the series 1-5. The obtained results indicate that the relative stabilization in the series of the isomeric fully benzenoid molecules is directly related to the extent of planarity deviation. Thus, the BFPI and LSPI indices were found to be the most proper measures of planarity in the studied systems.

The values of local aromaticity indices for the benzenoids 1-5 are presented in Table 3. Results for all other molecules from Figure 1 are given in Table S2 (Supporting Information). The Clar structures can provide qualitative description of π-electron conjugation modes in the given benzenoid molecule. The limitation of the qualitative Clar-formula-based approach is that it cannot assess the differences in aromaticity among the full rings, and among the empty rings in the underlined molecule. For instance, rings A and C in 1 are both full rings, but according to the employed indices their aromaticity is very different (Table 3). It should be noted, that such differences in aromaticity between the full rings were also found in planar fully benzenoid molecules. For instance, in 1c all indices, except NICS(1)av, predict that ring A is significantly more aromatic than ring C. The data from Table 3 and Table S2 (Supporting Information) support the conclusion coming from the Clar formulae that the full rings are more aromatic than the empty ones. Again, the only exception was found in 1 where, according to the NICS(1)av values, the empty ring B is slightly more aromatic than the full ring C.


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a)

b)

c)

d)

Figure 3. Dependence of the values of the relative free energies (∆G°) for the isomeric fully benzenoid molecules 1-5 on the different planarity indices: a) BFPI; b) LSPI; c) CMPI and d) ΣT.


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In a series of previous papers,55–58 it has been shown the NICS results do not always agree with other criteria of aromaticity. It was argued that NICS does not reflect only the aromaticity of the ring considered, because the NICS values are influenced by delocalization in all cycles involving the given ring.59–62 In addition, it has been found that the aromaticity indices based on magnetic response properties are not necessarily correlated with the ones based on ground-state properties.57,58,63 Anyway, based on the data presented in Table 3 and Table S2 (Supporting Information), it is evident that the Clar formulas can be used to qualitatively assess the local aromaticity distribution even in non-planar benzenoid systems.

Table 3. ef, HOMA, SCI, and NICS(1)av, BFPI, LSPI, CMPI and T values of the symmetryunique rings in 1-5; the labeling of the rings is indicated in Figure 1. Molecules

1

Ring

ef

HOMA

SCI

NICS(1)av

BFPI

LSPI

CMPI

T

A

0.1822

0.7855

0.0299

-9.34

0.083

0.108

0.128

1.96

B

0.0296

-0.0803

0.0076

-5.46

0.929

1.050

1.297

19.49

C

0.0910

0.4204

0.0159

-4.81

0.647

0.650

0.712

17.85

A

0.1866

0.7900

0.0302

-9.94

0.064

0.073

0.092

1.45

B

0.0272

-0.0766

0.0074

-5.45

0.619

0.781

0.938

14.06

C

0.1852

0.7718

0.0297

-9.54

0.165

0.196

0.241

3.70

D

0.1295

0.5984

0.0214

-8.20

0.506

0.581

0.739

11.79

E

0.0302

-0.0095

0.0080

-5.28

0.535

0.703

0.911

15.55

F

0.1287

0.5571

0.0201

-7.32

0.640

0.754

0.965

15.56

G

0.1815

0.7530

0.0285

-9.41

0.119

0.149

0.179

2.72

H

0.0271

-0.0393

0.0080

-5.63

0.559

0.698

0.836

12.52

I

0.1864

0.7833

0.0300

-10.01

0.069

0.079

0.098

1.55

J

0.1854

0.7701

0.0293

-9.25

0.142

0.171

0.210

3.26

2


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A

0.1865

0.7911

0.0303

-9.94

0.066

0.075

0.094

1.49

B

0.1854

0.7740

0.0298

-9.37

0.168

0.199

0.246

3.80

C

0.0271

-0.0645

0.0075

-5.83

0.691

0.849

1.026

15.37

D

0.1295

0.5746

0.0207

-7.84

0.633

0.731

0.904

14.38

E

0.1803

0.7300

0.0276

-8.72

0.453

0.514

0.635

14.38

F

0.0302

0.0027

0.0083

-5.31

0.464

0.466

0.519

12.83

A

0.1941

0.7902

0.0308

-9.62

0.002

0.003

0.003

0.05

B

0.0227

-0.1747

0.0070

-4.68

0.009

0.011

0.014

0.23

C

0.1942

0.7910

0.0309

-9.68

0.003

0.004

0.004

0.07

D

0.1353

0.7326

0.0274

-9.84

0.061

0.073

0.090

1.39

E

0.1888

0.7813

0.0300

-9.09

0.145

0.176

0.215

3.32

F

0.0253

-0.1002

0.0069

-4.42

0.605

0.745

0.898

13.46

G

0.1345

0.6049

0.0217

-7.73

0.568

0.657

0.815

13.17

H

0.1860

0.7765

0.0298

-9.44

0.144

0.172

0.212

3.26

I

0.0269

-0.0718

0.0075

-5.58

0.617

0.759

0.914

13.67

J

0.1870

0.7905

0.0303

-9.90

0.063

0.073

0.090

1.43

A

0.1871

0.7876

0.0302

-9.82

0.077

0.088

0.109

1.72

B

0.0269

-0.0777

0.0075

-5.30

0.595

0.769

0.923

13.95

C

0.1859

0.7744

0.0295

-8.69

0.200

0.243

0.298

4.62

D

0.1345

0.6011

0.0218

-7.74

0.544

0.632

0.783

12.67

E

0.1942

0.7929

0.0307

-10.03

0.051

0.065

0.079

1.24

F

0.0253

-0.0959

0.0072

-4.71

0.552

0.671

0.814

12.24

G

0.1345

0.7344

0.0275

-9.55

0.179

0.209

0.267

4.59

H

0.0226

-0.1605

0.0068

-4.74

0.150

0.184

0.218

3.21

I

0.1900

0.7931

0.0305

-9.59

0.056

0.065

0.081

1.30

J

0.1940

0.7866

0.0308

-9.68

0.039

0.044

0.055

0.85

3

4

5


17


Page 18 of 31

A detailed analysis of the calculated values of planarity indices showed that the empty rings suffer more significant deviation from planarity than the full rings (Table 3 and Table S2 in the Supporting Information). For instance, the sum of the BFPI values of the three empty rings in 1 is 2.787Å, whereas for the seven full rings is 1.145Å. This finding can be understood as a response of a given molecule in order to preserve its global aromaticity. Among the studied molecules, the only exception from this regularity was found for 3. The correlation between the local aromaticity indices for planar benzenoid hydrocarbons have been much examined in the past.63–66 The fact that between different aromaticity criteria there is poor or even a lack of correlation, has resulted in the so-called multidimensional concept of aromaticity.67,68 The cross correlations between all indices employed were examined and the obtained results are summarised in Table 4. It can be seen, that between some aromaticity indices there are very poor correlations (Table 4 and Figure S4 in the Supporting Information). The obtained data showed that HOMA is much better correlated with ln ef than with ef. The worst agreement was found between the NICS(1)av and other indices. On the other hand, the best correlations were found to exist between the SCI and ef, and HOMA and ln ef. It should be pointed out, that the ef and all other topology-based indices does not take into account nonplanarity and any other molecular geometric features.69 Thus, the obtained correlations between the ef and the other indices coming from DFT calculations suggest that the ring non-planarity does not significantly influence the local aromaticity in the studied systems. This is in agreement with the results that the aromaticity of benzene,70 linear polyacenes71 and some other polycyclic conjugated systems72 can be preserved under drastic geometry changes. Along these lines, it can be shown that the correlation between the SCI and ef can be slightly improved by explicitly


18

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including the measure of planarity through the planarity indices PI (PI = BFPI, LSPI, CMPI and T):

SCI = a1ef + b1 PI + c1

(11)

In an analogous way, the correlation between HOMA and ln ef can be improved as:

HOMA = a2 ln ef + b2 PI + c2

(12)

The coefficients in Eqs. 11 and 12 were determined by means of the least-square fitting procedure (Table 5). From the data in Table 5 one can come to the important conclusions. First, the parameters b1 and b2 in Eqs. 11 and 12 are negative, which means that the ef values, having no influence of non-planarity, predict somewhat higher values of SCI and HOMA. In other words, the planarity deviation of the hexagonal rings decreases the local aromaticity. The second observation concerns the relative values of the parameters b1 and b2 in Eqs. 11 and 12. It can be seen that the planarity deviations only moderately decrease the local aromaticity of benzenoid rings. The methodology presented in this work enable us to quantitatively assess the effects of planarity distortions. From the data given in Table 5, it is evident that the HOMA is more sensitive to the planarity deviations than the SCI index. The correlations between the SCI and SCI* obtained through Eq. 11 are presented in Fig. 4, and the analogous correlations between the HOMA and HOMA* values obtained through Eq. 12 are presented in Fig. 5.


19


Page 20 of 31

Table 4. Correlation coefficients (R) between all aromaticity indices of 142 symmetry-unique hexagonal rings in the molecules examined (Figure 1). The corresponding coefficients of determination (R2, in %) are given in the brackets. ef

ln ef

HOMA

SCI

NICS(1)av

ef

1.0000(100.00)

ln ef

0.9817(96.37)

1.0000(100.00)

HOMA

0.9518(90.59)

0.9820(96.43)

1.0000(100.00)

SCI

0.9804(96.11)

0.9638(92.88)

0.9591(91.98)

1.0000(100.00)

NICS(1)av

-0.7894(62.31)

-0.8631(74.49)

-0.8668(75.13)

-0.7818(61.11)

1.0000(100.00)

Table 5. Parameters in Eqs. 11 and 12 and the respective correlation coefficients (R) obtained for different non-planarity indices PI (PI = BFPI, LSPI, CMPI and T). BFPI

LSPI

CMPI

T

a1

0.123±0.002

0.123±0.002

0.123±0.002

0.123±0.002

b1

-0.0033±0.0006

-0.0028±0.0005

-0.0023±0.0004

-0.00014±0.00003

c1

0.0070±0.0003

0.0070±0.0003

0.0070±0.0003

0.0070±0.0003

R

0.9840

0.9842

0.9841

0.9840

a2

0.380±0.006

0.379±0.006

0.380±0.006

0.380±0.006

b2

-0.15±0.02

-0.13±0.02

-0.10±0.01

-0.0066±0.0009

c2

1.45±0.01

1.45±0.01

1.45±0.01

1.46±0.01

R

0.9873

0.9874

0.9873

0.9873

Eq. 11

Eq. 12


20

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a)

b)

c)

d)

Figure 4. Correlation between the SCI and SCI* for the symmetry-nonequivalent rings in the studied benzenoid molecules (Figure 1). The SCI* were calculated by means of Eq. 11 using the ef values and the corresponding planarity indices: a) PI = BFPI; b) PI = LSPI; c) PI = CMPI and

d) PI = T. The respective correlation coefficients were found to be: a) 0.9840; b) 0.9842; c) 0.9841 and d) 0.9840.


21


Page 22 of 31

a)

b)

c)

d)

Figure 5. Correlation between the HOMA and HOMA* for the symmetry-nonequivalent rings in the studied benzenoid molecules (Figure 1). The HOMA* were calculated by means of Eq. 12 using the ln ef values and the corresponding planarity indices: a) PI = BFPI; b) PI = LSPI; c) PI = CMPI and d) PI = T. The respective correlation coefficients were found to be: a) 0.9873; b) 0.9874; c) 0.9873 and d) 0.9873.


22

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The relation between different planarity indices for the hexagonal rings in the studied molecules was examined. The results of this analysis are summarized in Table 6 and Figure S5 (Supporting Information). It was found that the correlations between the planarity indices in the case of individual rings are significantly improved compared to the analogous correlations for the molecules (Table 2). Thus, the employed planarity indices in a very similar way describe the planarity deviation of the individual benzenoid rings.

Table 6. Correlation coefficients (R) between different planarity indices of 142 symmetry-unique hexagonal rings in the molecules examined (Figure 1). BFPI

LSPI

CMPI

BFPI

1.0000

LSPI

0.9966

1.0000

CMPI

0.9957

0.9993

1.0000

T

0.9933

0.9919

0.9908

T

1.0000

5. CONCLUSIONS In this work, the global and local aromaticity of the series of fully benzenoid systems were examined. The Clar aromatic sextet theory predicts that all molecules within the series of isomeric fully benzenoid systems should have about the same aromatic stabilization. The relative free energies (∆G°) obtained at the BP86-D3/def2-SVP level of theory showed that among the studied non-planar fully benzenoid isomers there are significant differences in their stability. It was revealed that the found differences between the relative free energies are directly related to


23


Page 24 of 31

the extent of deviation from planarity of the molecules examined. In order to quantify the nonplanarity of the studied systems, four different indices were employed, namely: BFPI, LSPI, CMPI and ΣT. It was shown that the BFPI is the most proper measure of the non-planarity

effects on the relative stability of the considered benzenoid molecules. Based on the results of several different aromaticity indices it was shown that the Clar formulas can be used to qualitatively assess the local aromaticity distribution even in non-planar benzenoid systems. The analysis of the local aromaticity distribution in the studied fully benzenoids revealed that the ring deviations from planarity only slightly decrease the local aromaticity. The employed indices of planarity in a very similar way characterized planarity deviations of the hexagonal rings in the examined molecules. It was shown, based on the values of the employed planarity descriptors, that the geometry distortion of the empty rings is more favorable than the one in the full rings. This way, the global aromaticity of a given benzenoid molecule is reduced to the smallest possible extent. ASSOCIATED CONTENT Supporting Information. Figure S1: Optimized geometries of the molecules which are not included in Figure 2; Figures S2, S3 and S5: Correlations between different planarity indices; Figure S4: Correlations between different aromaticity indices; Tables S1 and S2: Free energies, TRE, aromaticity and planarity indices for the molecules which are not included in Tables 1 and

3; complete ref 50. This information is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Author


24

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*E-mail address: [email protected]; [email protected]. Notes The authors declare no competing financial interest.

ACKNOWLEDGMENT This work is supported by the Ministry of education, science and technological development of Serbia (grant no. 174033).

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