Effect of Benzo-Annelation on Local Aromaticity in Heterocyclic

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Effect of Benzo-Annelation on Local Aromaticity in Heterocyclic Conjugated Compounds Slavko Radenkovi#, Jelena Koji#, Jelena Petronijevi#, and Marija Anti# J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/jp507309m • Publication Date (Web): 21 Nov 2014 Downloaded from http://pubs.acs.org on November 28, 2014

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The Journal of Physical Chemistry

Effect of Benzo-annelation on Local Aromaticity in Heterocyclic Conjugated Compounds Slavko Radenković*, Jelena Kojić, Jelena Petronijević, Marija Antić Faculty of Science, University of Kragujevac, 12 Radoja Domanovića, 34000 Kragujevac, Serbia KEYWORDS: Electron delocalization, DFT calculations, multicentre delocalization index, HOMA, NICS ABSTRACT: The effect of benzo-annelation on the local aromaticity of the central ring of acridine (1), 9H-carbazole (2), dibenzofuran (3) and dibenzothiophene (4) was analyzed by means of the energy effects (ef), pairwise energy effects (pef), multicentre delocalization index (MCI), electron density at ring critical points ( ρ (rC ) ), harmonic oscillator model of aromaticity (HOMA) and nucleus independent chemical shifts (NICS). According to energetic, electron delocalization and geometrical indices, angular benzo-annelation increases, whereas linear benzo-annelation decreases the extent of the local aromaticity of the central ring containing heteroatoms. The local aromaticity of the central heterocyclic ring in the examined molecules can significantly vary by applying different modes of benzo-annelation. The NICS values do not always support the results obtained by the other aromaticity indices, and in some cases lead to completely opposite conclusions.

1. INTRODUCTION Aromaticity is one of the most fundamental concepts in chemistry. The concept of aromaticity was originally introduced in order to describe distinctive stability, reactivity, geometric and

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magnetic properties of benzene and structurally related polycyclic conjugated hydrocarbons.1,2 Shortly thereafter, it was recognized that similar “aromatic” properties can be found in many of heterocyclic conjugated molecules.3,4 Heterocyclic aromatic compounds have a significant importance in several diverse fileds. These molecules play a major part in biochemical processes as constituents of DNA, RNA, hemoglobin and many other important biomolecules.5 Heterocyclic aromatic compounds represent very useful reactants for organic synthesis.6 In addition, aromatic heterocycles are employed as core structures of novel electronic materials that can be used as organic field-effect transistors7–9 and organic photovoltaics.10–12 In a series of recent papers, 13–20 the effects of benzo-annelation on the local aromaticity distribution in benzenoid13–16 and non-benzenoid polycyclic aromatic hydrocarbons17–19 have been thoroughly studied. Two ways of attaching a benzene ring to a polycyclic conjugated compound, namely benzo- and benzocyclobutadieno-annelation, have been examined so far.15 The effect of benzo-annelation on the local aromaticity of the five- and six-membered rings is found to be the most intriguing. The obtained results can be summarized as the following regularities:13 Regularity 1: If the annelated benzo-ring is in linear position with regard to the considered ring R, then the extent of local aromaticity of R is decreased compared to the same ring in the parent non-annelated molecule. Regularity 2: If the annelated benzo-ring is in angular position with regard to the considered ring R, then the extent of local aromaticity of R is increased compared to the same ring in the parent non-annelated molecule.


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It should be noted that by the presence of linear (respectively angular) benzo-annleations the structure of the annelated molecules becomes more straight (respectively kinked) compared to the structure of the parent non-annelated molecule. The observation that straight linear polyacenes are less stable than the kinked ones has been thoroughly studied.20–23 The regularities stated above were first recognized within the studies of the energy-effect (ef) values calculated at the level of the simple Hückel molecular orbital theory.16 Eventually, these regularities were confirmed in a number of other theoretical studies using geometrical and electron delocalization indices of aromaticity.13,15,18,19,24 The obtained regularities have never been tested for heterocyclic aromatic compounds. This is the main subject of this paper: can regularities 1 and 2 be extended to heterocyclic conjugated systems as well? To this end, a detailed study of the local aromaticity in the series of benzo-annelated acridine (1), 9H-carbazole (2), dibenzofuran (3), dibenzothiophene (4) was performed (Figure 1). Figure 1 comes about here

The aromatic character of a given molecule can be quantified through different indices, namely structural,25,26 magnetic,27 energetic,26 electronic,28 and reactivity-based indices of aromaticity.29 In this paper, the local aromaticity in the examined molecules was quantified by means of the energy effect (ef),30,31 multicenter delocalization index (MCI),32,33 electron density at ring critical points ( ρ ( rC ) ),34 harmonic oscillator model of aromaticity (HOMA) index25,35 and nucleus independent chemical shifts (NICS).27,36 The ef-value of a cycle quantifies the energyeffect caused by cyclic conjugation of π-electrons along the given cycle.31 In a recent work,37 a new method, aimed at assessing the influence of an individual ring on the energy effect of some other ring present in a molecule, was introduced. The quantity that measures such an effect is the pairwise energy effect (pef) of cyclic conjugation. In the present study, the pef method was used


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to provide a deeper insight into the effect of benzo-annelations. More details on the ef and pef methods are presented in Appendix. The MCI32,33 measures the extent of the electron delocalization among a set of n atoms. This index is an extension of the so-called multicenter bond index, firstly introduced by Giambiagi.32,33 In this work, the electron density ( ρ ( rC ) ) at the ring critical points (RCP) was employed as another electron delocalization index of aromaticity. Howard and Krygowski first proposed to use RCP properties as local aromaticity indices.34 Several properties related to RCPs have been tested as aromaticity indices.38,39 The HOMA index,26 as the most popular aromaticity index measuring the “geometry” effects of aromaticity (see ref40), was also employed in this study. The NICS, as a measure of magnetic manifestations of aromaticity, is one of the most widely employed indicators of aromaticity.27 There are several NICS indices in common use, and because the NICS is a tensor, different components of the tensor are considered as appropriate indices of aromaticity.27 The analysis of the correlation between different aromaticity indices showed that aromaticity is a multidimensional property.41,42 Although the multidimensional concept of aromaticity has been much criticized43–45 it is widely accepted that a proper characterization of the aromatic character of a given molecule can be achieved only by using a set of different aromaticity indices.

2. COMPUTATIONAL METHODS The structures of the studied molecules were optimized by means of the B3LYP/6-311G(d,p) 46,47

method using the Gaussian 09 program.48 Hessian calculations showed that the obtained

geometries of the examined molecules correspond to minima on the potential energy surface at this level of theory. NICS values were calculated at the B3LYP/6-311+G(d,p) level through the gauge-including atomic orbital (GIAO) method.49,50 In the present study, NICS values calculated 1 Å above the ring centre, NICS(1) were employed. The B3LYP/6-311G(d,p) optimized


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structures were employed for the HOMA index calculations. The parameters needed for the HOMA calculations proposed by Krygowski26 were used. The MCI were calculated at the B3LYP/6-311G(d) level of theory using the Mulliken partitioning scheme.33 It has been shown that MCI can be calculated using several different partitioning schemes,51 but the Mulliken approach is computationally the most efficient. Although the Mulliken partitioning scheme has some limitations,52 the MCI calculated using this scheme has been successfully applied in a number of aromaticity studies.33,45,53–55 ρ ( rC ) were computed at the B3LYP/6-311G(d) level of theory. The calculated ef and pef values were obtained at the level of the Hückel molecular orbital theory using the chemical graph theory formalism.24,37,54 In these calculations the parametrization scheme for the heteroatoms proposed by Van-Catledge56 was used. The ef and pef values are expressed in units of the HMO carbon−carbon resonance integral β. Since β is a negative valued quantity, positive ef and pef values indicate thermodynamic stabilization, whereas negative ef and pef values indicate thermodynamic destabilization of the considered molecule. Calculations of ef, pef, MCI were performed using in house FORTRAN routines.

ρ (rC ) were calculated by means of the Multiwfn program.57

3. RESULTS AND DISCUSSION The optimized geometries of all examined molecules were found to be perfectly planar, with the exception of benzo-annelated derivatives of 2-4 having two annelated rings at positions a1 and a4. Optimized geometries of all molecules studied are given in Supporting information. It should be noted that the effect of benzo-annelation has been initially examined in polycyclic conjugated hydrocarbons having planar geometry.13,15 Thereafter, it has been shown that the found effects of benzo-annelation (regularities 1 and 2) remain the same in the systems with a


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significant deviation from planarity.18,58 This finding goes along with some previous studies showing that the local aromaticity of a ring resists significant out-of-plane distortions.59–63 In order to examine the effect of different modes of benzo-annelation all possible benzoannelated 1-4 were considered. In order to simplify notation, the sites of annelation in the examined molecules are labeled as indicated in Figure 1.The examples presented in Figure 2 illustrate the nomenclature employed in this paper. It should be noted that in the previous studies three different modes of benzo-annelations have been considered, namely linear, angular and germinal benzo-annelation. Two benzene rings are in germinal position with regard to the central ring, if these benzene rings are both angularly annelated to the same benzene ring. For instance, in 4-a1a2a3 (Figure 2) two benzo-rings annelated at positions a1 and a2 are in germinal arrangement relative to the central five-membered ring. Results of our investigation of the local aromaticity of the central heterocyclic ring in the examined molecules are presented in Tables 14. The local aromaticity in the studied molecules can be compared with the local aromaticity of their benzenoid analogs, i.e. the central ring in 1 can be compared with the central ring in anthracene, whereas the central rings in 2, 3, and 4 can be compared with the central ring in phenanthrene. The values of aromaticity indices for the central ring in benzo-annelated derivatives of anthracene and phenanthrene are tabulated in Tables S1 and S2 (Supporting information). It should be noted that the effect of benzo-annelation on the local aromaticity in anthracene15 and phenanthrene58 have been previously studied. As seen from Tables S1 and S2, according to the values of ef, MCI, ρ ( rC ) and HOMA, regularities 1 and 2 are obeyed. The change of aromaticity in the central ring in benzo-derivatives of anthracene and phenanthrene fallows the same trend, regardless the fact that some phenanthrene derivatives are non-planar.58 According to the NICS values, in the case of phenanthrene derivatives regularities 1 and 2 are


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obeyed, with the exception of tetrabenzophenanthrene (two germinal annelations) in which the central ring is less aromatic that the analogous ring in phenanthrene. In addition, in the series of benzo-annelated phenanthrenes the changes of NICS values of the central ring are not always proportional to the number of annelated rings. In the series of benzo-annelated anthracenes, according to the NICS values, the opposite effects are found: linear benzo-annelation increases, whereas angular benzo-annelation decreases the aromatic character of the central ring. Figure 2 comes about here

Tables 1-4 come about here

A detailed inspection of the data given in Tables 1-4 revealed that, according to the values of ef, MCI, ρ ( rC ) and HOMA indices, regularities 1 and 2 hold for benzo-annelated 1-4: angular benzo-annelation increases, whereas linear benzo-annelation decreases the extent of the local aromaticity of the central ring in the examined molecules. This can be seen, for instance, from the following data (Figure 3): for 1 (no annelations) ef(R) = 0.0626, MCI(R) = 0.0159, ρ(R) = 0.0222, HOMA(R) = 0.7325; for 1-a1 (one angular annelation) ef(R) = 0.0766, MCI(R) = 0.0165, ρ(R) = 0.0224, HOMA(R) = 0.7685; for 1-l1 (one linear annelation) ef(R) = 0.0508, MCI(R) = 0.0129, ρ(R) = 0.0219, HOMA(R) = 0.6436. In addition, according to the ef-, MCI- and ρ- values, the effects of linear and angular anelations are proportional to the number of benzo-annelated rings. In the case of HOMA, it was found that the effects of different anelations are not always proportional to the number of benzo-annelated rings. For instance, for 1-a1a2a3 and for 1a1a2a3a4 HOMA(R) has the same value, or for 3-a2a3 HOMA(R) > HOMA(R) for 3-a1a2a3a4. From the data given in Tables 1-4 one can see that different aromaticity indices predict very similar changes of the local aromaticity of the examined heterocycles upon benzo-annelation.


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This is not surprising, since in the case of polycyclic aromatic hydrocarbons it has been shown that between ef and MCI,19 and between MCI and ρ ( rC ) 64 exist good correlations. On the other hand, the NICS results reveal somewhat different regularities. It can be seen that in the series of benzo-annelated derivatives of 1, the same trend is observed as the one found in the series of anthracene derivatives: linear annelations increase, whereas angular benzo-annelation decrease the aromatic character of the central ring. Anyway, the aromatic character of ring R in 1-l1 is practically unchanged (by the presence of one linear benzo-annelation) compared to the aromatic character of ring R in 1 (Figure 3). In the series of benzo-annelated 2, 3 and 4, which can be considered as the analogs of phenanthrene, it is hard to find some certain regularities. In most cases of mono- and dibenzo-annelated molecules angular benzo-annelation increases the aromaticity, and linear benzo-annelation decreas the aromaticity of central ring. These effects are not always proportional to the number of benzo-annelated rings. A complete analysis of the origin of these differences found for the NICS results is beyond the scope of the present work. It should be noted that there is the widely accepted notion that the NICS-values can provide the information on the current density distribution in a given molecule. It has been shown that the NICS and ring current results do not always agree.65 On the other hand, NICS does not reflect only the aromaticity of the ring considered, because NICS computed at the ring center is influenced by delocalization in all circuits involving the given ring.44,45,55 A good example is anthracene for which the NICS for the inner ring is the most negative.66 This is a consequence of the fact that circulation in the central six-membered ring has contributions from circulations along the ten- and fourteen-membered circuits (the so-called anthracene effect). 44,45,55,67 Figure 3 comes about here


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In previous studies regularities 1 and 2 have never been analyzed by means of Clar’s sextet model.68 In the Clar theory, so called “Clar’s formulas” are constructed by placing “aromatic sextets” into some rings of a benzenoid molecule, obeying certain formal rules.69 Clar’s formulas can be used to elucidated the local aromaticity in benzenoid hydrocarbons as follows: rings in which “aromatic sextets” are located are predicted to have a high extent of aromatic character, whereas rings with the localized double bonds, or rings with no π-electrons (“empty rings”) have a significantly lower level of aromatic character. Clar’s qualitative approach has been corroborated in a number of theoretical studies.70–72 There were several attempts aimed at quantifying the Clar’s aromatic sextet theory.73,74 Clar’s model can be used in order to rationalize regularities 1 and 2, found in the case of benzo-derivatives of antaracene and phenanthrene (Figure 4). As seen from Figure 4, in one of three Clar’s formulas of anthracene aromatic sextet is placed in the central ring (1/3 of full aromatic sextet). In angularly benzo-annelated anthracene (benz[a]anthracene) in one of two Clar’s formulas ring R has aromatic sextet (1/2 of full aromatic sextet), whereas in linearly annelated anthracene (tetracene) ring R has aromatic sextet in one of four Clar’s structures (1/4 of full aromatic sextet). In phenanthrene ring R has the fixed double bound in the Clar’s formula. In angularly benzo-annelated phenanthrene (cryzene) ring R contains aromatic sextet in one of three Clar’s formulas (1/3 of full aromatic sextet), whereas in linearly annelated phenanthrene (benz[a]anthracene) ring R contains the fixed double bond in two Clar’s formulas. Originally, Clar’s concept was introduced to characterize the properties of polycyclic aromatic hydrocarbons. Nevertheless, the arguments based on Clar’s model given above can be easily extended to the heterocyclic aromatic systems studied in this work. This way, the obtained values of ef, MCI, ρ ( rC ) and HOMA go along with the predictions of Clar’s


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concept. The present work support findings that NICS results are not always in agreement with Clar’s model.71 Figure 4 comes about here

Regularities stated above do not provide any information about the magnitude of these effects. Tables 1-4 also collect the numbers of benzo-annelated rings in angular, linear and germinal position. Those numbers are denoted by A, L and G, respectively. It can be observed that the values of the aromaticity indices employed in this work differ slightly for isomers with the same numbers A, L, and G. In ref 13 it has been shown that between the ef-values and the position and number of annelated rings exists a quantitative relation. By applying the same methodology, it is now found that the value of a given aromaticity index (I(R)) of the central ring R in the series of the studied molecules can be expressed as a linear function of A, L and G numbers: I ( R) = l ⋅ L + a ⋅ A + g ⋅ G + I 0

(1)

The parameters a, l, g and I0 that appear in Eq. 1 were determined by least-squares fitting for the series of benzo-annelated 1-4 and for all aromaticity indices employed in this work (i.e. I can be ef, MCI, ρ ( rC ) , HOMA or NICS(1)). The values of the parameters a, l, g and I0 and the corresponding correlation coefficients are collected in Table 5. Table 5 comes about here

The observed regularities indicate that the local aromatic characters of the central ring R and the annelated rings in the examined systems are mutually correlated. Using the standard methodologies in the analysis of the local aromaticity26–28 it is not possible to assess the interactions between the local aromaticity of different rings present in the given molecule. The


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pef method is aimed at measuring the effect of cyclic conjugation of a ring on the cyclic conjugation of another ring in the same molecule. The pef values were calculated for all pairs of the central ring R and the benzo-annelated rings in all examined benzo-derivatives of 1-4. The results obtained are given in Tables 6-9. The calculated pef values are in complete agreement with the observed regularities: in the case of linear arrangement of the central ring and the annelated ring, the values of pef are negative, indicating that the aromatic character of the central ring is diminished; in the case of angular arrangement of the central rings and the annelated ring, the values of pef are positive, indicating that the aromatic character of the central ring is increased by the presence of the annelated ring. A mathematical analysis of the obtained pef results can be found in Appendix. Tables 6-9 come about here

As an example, in Figure 5 are plotted the values of different aromaticity indices of the central ring R in benzo-derivatives of 3 and the sum of all pef values of the respective central ring and the annelated benzo-rings. Analogous plots for the other studied molecules are given in Supporting information. As can be seen, the correlations are good, but evidently non-linear. The worst correlation was found for the NICS values. The existence of these correlations implies that the changes of the local aromaticity of R in the series of the examined molecules can be assessed by means of the interaction of cyclic conjugations in the ring R and in the annelated rings. Figure 5 comes about here

4. CONCLUSIONS The analysis of the local aromaticity of the central heterocyclic ring of acridine (1), 9H-carbazole (2), dibenzofuran (3) and dibenzothiophene (4), assessed by means of the energy effects (ef),


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pairwise energy effects (pef), multicentre delocalization index (MCI), electron density at ring critical points ( ρ ( rC ) ), harmonic oscillator model of aromaticity (HOMA) and nucleus independent chemical shifts (NICS), showed that the effects of benzo-annelation obey the same regularities as the ones previously found for polycyclic conjugated hydrocarbons.13 According to energetic, electron delocalization and geometrical indices, it was revealed that angular benzoannelation increases, whereas linear benzo-annelation decreases the extent of the local aromaticity of the central ring in the examined molecules. This way, it was shown that these regularities hold for heterocyclic conjugated systems as well, regardless the size of heterocyclic ring and the nature of heteroatoms. On the other, the NICS do not always support the results obtained by the other employed aromaticity indices. The found results are in agreement with Clar’s aromatic sextet theory. It was shown that the effect of benzo-annelation can be well represented by a linear function of the number of angular, linear and geminal annelations. The effect of different annelation modes can be used for tuning the local aromaticity to the desired degree.

5. APPENDIX: Mathematical Analysis of the pefs of Benzoannelated Heterocyclic Systems Within the framework of chemical graph theory, a heterocyclic molecule is represented by a vertex- and edge-weighted graph (GVEW).75 The energy effect (ef)30,31 of a cycle Za in the considered polycyclic conjugated system is defined as the difference between the total π-electron energy and the appropriate reference energy having no contributions from the given cycle Za:

ef ( GVEW ; Z a ) =

1

φ (G

∞

∫ ln π −∞

VEW

, ix )

( )

dx

a

φ (G

, ix ) + 2 VEW

∏k

i

φ (G

VEW

(2)

− Z a , ix )

i


12

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where GVEW is the molecular graph, GVEW – Za is the subgraph obtained by deleting the cycle Za a

from GVEW, ϕ stands for the characteristic polynomial of the considered graphs, and ∏ ki is the i

product of the Hückel parameters k of all weighted-edges involved in the cycle Za. The pef(Za,Zb)-value 37 measures the effect of the cycle Zb on the ef value of the cycle Za and can be calculated as the difference between ef(GVEW; Za) and the reference energy effect efref(GVEW\Zb; Za): pef ( GVEW ;Z a ,Z b ) = pef ( Z a ,Z b ) = ef ( GVEW ;Z a ) − ef ref ( GVEW \ Z b ;Z a )

(3)

The reference energy effect efref(GVEW\Zb; Za) corresponds to the ef of the cycle Za without contributions coming from the cycle Zb and can be calculated as follows:

ef

1

π

ref

(GVEW

\Z

; Za ) =

b

 

∞

∫ ln −∞

b

 

φ (GVEW , ix ) + 2  ∏ k j  φ (GVEW − Z b , ix )

  , ix ) + 2  ∏ k  φ (G   b

φ (GVEW

j

(∏ )

j

a

VEW

− Z b , ix ) + 2

j

k i φ ( GVEW − Z a , ix ) + 4

i

(∏ ) a

ki

i

dx

 k  φ (G ∏    b

j

− Z b − Z a , ix )

VEW

j

(4) Combining Eqs. (2), (3) and (4) we arrive at the following expression for pef: pef (GVEW ; Z a , Z b ) = 1

∞

∫ π

−∞

( )

∏ k  ∏ k a

ln[1 + 4

b

i

i

j

j

   [φ (G

VEW

φ (GVEW , ix )φ (GVEW − Z a − Z b , ix ) − φ (GVEW − Z a , ix )φ (GVEW − Z b , ix )

(∏ ) a

, ix ) + 2

k i φ (GVEW − Z a , ix )][φ (GVEW

i

  , ix ) + 2  ∏ k  φ (G   b

j

VEW

]dx

− Z b , ix )]

j

(5)


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Consider now the special case of Eq. (5) when GVEW is the molecular graph of a benzoannelated acridine-like system, Za is its central ring, and Zb is one of the annelated benzo rings (Figure 4). The number n of vertices of GVEW is even and the polynomials occurring in Eq. (5) can be written as:

φ (G , ix) = i n ( P( x) + iQ( x) ) φ (G − Z a , ix) = i n −6 Pa ( x) φ (G − Z b , ix) = i n −6 ( Pb ( x) + iQb ( x) )

(6)

φ (G − Z a − Z b , ix) = i n −12 Pab ( x) Our numerical results showed that as a reasonable approximation we can adopt:

φ (G, ix) ≈ i n P( x) (7)

φ (G − Z b , ix) ≈ i n −6 Pb ( x)

Note that this approximation is equivalent to the case h ≈ 0.0 . From Eqs. (5)- (7) one can easily obtain:

pef (GVEW ; Z a , Z b ) ≈

1

π

∞

∫ ln[1 + Γ(G

VEW

; Z a , Z b , x )] dx

(8)

−∞

where, for the sake of brevity, the auxiliary function is introduced:

Γ( x ) = Γ(GVEW ; Z a , Z b , x ) = 4k 2

P ( x ) Pab ( x ) − Pa ( x ) Pb ( x ) [ P ( x ) − 2k 2 Pa ( x)][ P ( x ) − 2k 2 Pb ( x )]

(9)

From Eq. (8) it can be seen that the sign of pef depends on the sign of the function Γ ( x ) . In particular, if Γ ( x ) ≥ 0 for all x,

−∞ ≤ x < ∞ , then pef will necessarily be positive-valued,


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and if Γ( x ) ≤ 0 for all x,

−∞ ≤ x < ∞ , then pef < 0 . In the general case, Γ( x)

may change

sign, and then it is less straightforward to predict the sign of pef. Our numerical studies showed that for all examined molecules, the sign of the function Γ( x ) is same in the entire interval −∞ ≤ x < ∞ . The sign of pef then can be predicted as follows: the sign of the function

Γ (x )

(i. e., the sign of pef) is determined

P ( x) Pab ( x) − Pa ( x) Pb ( x) ,

occurring

by the sing of the polynomial

in

Eq.

(9),

because

[ P( x) − 2k 2 Pa ( x)][ P( x) − 2k 2 Pb ( x)] > 0 for all values of the variable x, −∞ ≤ x < ∞ . Based on the Sachs theorem75 it is known know that the polynomials defined above by Eq. (6), have the following form:

P ( x ) = x n + ( m − 2 + 2 k 2 ) x n − 2 + [ q (GVEW ) + k 2 (2 m − 8)] x n − 4 + L , Pa ( x ) = x n − 6 + ( m − 10) x n −8 + q (GVEW − Z a ) x n −10 + L , Pb ( x ) = x n − 6 + ( m − 10 + 2 k 2 ) x n −8 + [ q (GVEW − Z b ) + k 2 (2 m − 24)] x n −10 + L Pab ( x ) = x n −12 + ( m − 17) x n −14 + q (GVEW − Z a − Z b ) x n −16 + L if Zb is in angular position, and

Pab ( x ) = x n −12 + ( m − 18) x n −14 + q (GVEW − Z a − Z b ) x n −16 + L if Zb is in linear position. m denotes the number of edges, and q is the number of pairs of disjoint unweighted edges in the given graph. Consider the set E(GVEW) of unweighted edges in the molecular graph GVEW. We can partition E(GVEW) into subsets Ea(GVEW), Eb(GVEW), and Ex(GVEW), where Ea(GVEW) contains the unweighted edges belonging or incident to Za, Eb(GVEW) contains the unweighted edges belonging or incident to Zb, and Ex(GVEW) contains the unweighted edges


15


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Page 16 of 54

that neither belong nor are incident to Za or Zb. In Figure 4 the unweighted edges belonging to Ea(GVEW) are indicated by heavy lines, and the unweighted edges belonging to Eb(GVEW) by dashed lines. It should be noted that in case of angular annelation a single edge (marked by an arrow) belongs both to Ea(GVEW) and Eb(GVEW). Consider now pairs of disjoint unweighted edges e, f in the graph GVEW and its subgraphs. Denote the number of such pairs by: qaa if e ∈ Ea (GVEW ) , f ∈ Ea (GVEW ) qbb if e ∈ Eb (GVEW ) , f ∈ Eb (GVEW ) qxx if e ∈ Ex (GVEW ) , f ∈ E x (GVEW ) qab if e ∈ Ea (GVEW ) , f ∈ Eb (GVEW ) qax if e ∈ Ea (GVEW ) , f ∈ E x (GVEW ) qbx if e ∈ Eb (GVEW ) , f ∈ Ex (GVEW ) The number of pairs of disjoint unweighted edges in the considered graphs can be expressed as:

q (GVEW ) = qaa + qbb + q xx + qax + qbx + qab , q (GVEW − Z a ) = qbb + q xx + qbx , q (GVEW − Z b ) = qaa + q xx + qax , and q (GVEW − Z a − Z b ) = q xx .

Figure 6 comes about here

By direct calculation the following expressions can be obtained:

P ( x) Pab ( x) − Pa ( x) Pb ( x) = 1 ⋅ x 2 n −14 + ( qab + m + 2k 2 − 66) x 2 n −16 +L

(10)

if Zb is in angular position, and


16

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


P ( x) Pab ( x) − Pa ( x) Pb ( x) = (qab − 64) x 2 n −16 + L = −2 ⋅ x 2 n −16 + L

(11)

if Zb is in linear position, because qab =| Ea (GVEW ) | ⋅ | Eb (GVEW ) | −2 = 8 ⋅ 8 − 2 . From Eqs. (10) and (11) directly follows that for large positive and negative values of x,

P ( x) Pab ( x) − Pa ( x) Pb ( x) is positive for angular, and negative for linear annelation. Since P ( x) Pab ( x) − Pa ( x) Pb ( x) does not change sign, we arrive at our final conclusions: in the case of angularly (respectively linearly) annelated benzenoid rings the pairwise energy effect is positive (respectively negative).

AUTHOR INFORMATION

Corresponding Author *E-mail address: [email protected].

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENT This work is supported by the Ministry of Science of Serbia, (Grant No. 174033). ASSOCIATED CONTENT

Supporting Information. Tables S1-S3: values of aromaticity indices for the central ring in benzo-annelated anthracenes and phenanthrenes; Figures S1-S3: dependence of the values of different aromaticity indices of the central ring in benzo-derivatives of 2-4 on the sum of all pef


17


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Page 18 of 54

values of the respective central ring and the annelated benzo-rings; Cartesian coordinates of the studied molecules and complete ref 48. This information is available free of charge via the Internet at http://pubs.acs.org.

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FIGURE AND TABLE CAPTIONS Table 1. ef, MCI, ρ (rC ) HOMA, and NICS(1) values of the central ring R in benzo-annelated derivatives of 1. L, A and G indicates the numbers of benzo-rings annelated in linear, angular and geminal position, respectively.

Table 2. ef, MCI, ρ (rC ) HOMA, and NICS(1) values of the central ring R in benzo-annelated derivatives of 2. L, A and G indicates the numbers of benzo-rings annelated in linear, angular and geminal position, respectively.



Table 5. The coefficients in Eq. 1 and correlation coefficient (R) for I ≡ ef, I ≡ MCI, I ≡ ρ (rC ) , I ≡ HOMA and I ≡ NICS(1) calculated by least squares fitting for the series of benzo-annelated derivatives of 1−4.

Table 6. pef (in β ×102 units) of the central and annelated rings in benzo-annelated derivatives of 1. Table 7. pef (in β ×102 units) of the central and annelated rings in benzo-annelated derivatives of 2. Table 8. pef (in β ×102 units) of the central and annelated rings in benzo-annelated derivatives of 3.


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Table 9. pef (in β ×102 units) of the central and annelated rings in benzo-annelated derivatives of 4.

Figure 1. The polycyclic heteroconjugated molecules whose benzo-annelated derivatives were studied in this work: acridine (1), 9H-carbazole (2), dibenzofuran (3) and dibenzothiophene (4), and the labeling of the sites of annelation (a = angular; l = linear); R denotes the central heterocyclic ring in these molecules.

Figure 2. Examples illustrating the labeling of benzo-annelated derivatives of the molecules depicted in Figure 1.

Figure 3. ef, MCI, ρ (rC ) HOMA, and NICS(1) values of the central ring R in 1, 1-a1 and 1-l1. Figure 4. Clar formulas of anthracene, benz[a]anthracene (angularly benzo-annelated anthracene), tetracene (linearly benzo-annelated anthracene), phenanthrene, crysene (angularly benzo-annelated phenanthrene), benz[a]anthracene (linearly benzo-annelated phenanthrene).

Figure 5. The dependence of the values of different aromaticity indices of the central ring R in benzo-annelated derivatives of 3 on the sum of pef values of the central and the annelated benzorings ( ∑ pef ): a) ef vs. ; e) NICS(1) vs.

∑ pef ; b) MCI vs. ∑ pef ; c) ρ (r ) vs. ∑ pef ; d) HOMA vs. ∑ pef C

∑ pef .

Figure 6. An angularly and a linearly benzo-annelated acridine-like molecule, A and L. The unweighted edges belonging to or incident to the ring Za are marked by heavy lines and form the set Ea (GVEW ) ; the edges belonging to or incident to the ring Zb are marked by dashed lines and form the set Eb (GVEW ) . The edge indicated by an arrow (which occurs only in the case of angular annelation) belongs to both Ea (GVEW ) and Eb (GVEW ) .


26

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Compound

ef( ×102 )

MCI( ×102 )

ρ (rC ) ( ×102 )

HOMA

1

6.26

1.59

2.22

0.7325

-13.35

0

0

0

1-l1

5.08

1.29

2.19

0.6436

-13.30

1

0

0

1-a1

7.66

1.65

2.24

0.7685

-12.60

0

1

0

1-a2

7.70

1.78

2.23

0.7698

-12.51

0

1

0

1-l1l2

4.25

1.11

2.18

0.5952

-14.03

2

0

0

1-a1l2

6.04

1.26

2.21

0.6546

-12.01

1

1

0

1-a2l2

6.08

1.38

2.21

0.6560

-11.86

1

1

0

1-a1a2

8.79

1.82

2.26

0.8081

-11.47

0

0

2

1-a1a3

9.70

1.96

2.26

0.8284

-12.17

0

2

0

1-a1a4

9.67

1.81

2.26

0.8276

-12.25

0

2

0

1-a2a3

9.75

2.10

2.26

0.8320

-12.09

0

2

0

1-a1a2l2

6.82

1.37

2.23

0.6939

-10.82

1

0

2

1-a1a2a3

11.36

2.21

2.29

0.8706

-11.15

0

1

2

1-a1a2a4

11.32

2.06

2.31

0.9064

-11.18

0

1

2

1-a1a2a3a4

13.45

2.34

2.32

0.9064

-10.24

0

0

4


NICS(1) L A

G

27


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Page 28 of 54

Table 2. ef, MCI, ρ (rC ) HOMA, and NICS(1) values of the central ring R in benzo-annelated derivatives of 2. L, A and G indicates the numbers of benzo-rings annelated in linear, angular and geminal position, respectively. Compound

ef( ×102 )

MCI( ×102 )

ρ (rC ) ( ×102 )

HOMA

NICS(1)

L

A

G

2

3.86

1.17

4.74

0.6093

-9.17

0

0

0

2-l1

3.09

0.69

4.67

0.5127

-7.71

1

0

0

2-a1

5.06

1.45

4.78

0.6317

-9.68

0

1

0

2-a2

5.03

1.55

4.80

0.6784

-9.55

0

1

0

2-l1l2

2.59

0.33

4.60

0.4186

-6.06

2

0

0

2-a1l2

3.90

0.86

4.71

0.5252

-8.25

1

1

0

2-a2l2

3.87

0.97

4.73

0.5812

-8.18

1

1

0

2-a1a2

6.12

1.69

4.83

0.6528

-9.19

0

0

2

2-a1a3

6.75

1.89

4.85

0.7050

-10.07

0

2

0

2-a1a4

6.77

2.09

4.82

0.6350

-10.24

0

2

0

2-a2a3

6.72

2.00

4.87

0.7448

-9.96

0

2

0

2-a1a2l2

4.60

1.04

4.76

0.5447

-8.03

1

0

2

2-a1a2a3

8.31

2.19

4.90

0.7262

-9.43

0

1

2

2-a1a2a4

8.32

2.44

4.88

0.6778

-9.82

0

1

2

2-a1a2a3a4

10.35

2.83

4.93

0.7189

-8.74

0

0

4


28

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Table 3. ef, MCI, ρ (rC ) HOMA, and NICS(1) values of the central ring R in benzo-annelated derivatives of 3. L, A and G indicates the numbers of benzo-rings annelated in linear, angular and geminal position, respectively. Compound

ef( ×102 )

MCI( ×102 )

ρ (rC ) ( ×102 )

HOMA

NICS(1)

L

A

G

3

2.09

1.14

4.94

-0.0048

-7.59

0

0

0

3-l1

1.73

0.81

4.87

-0.0908

-6.41

1

0

0

3-a1

2.70

1.31

5.00

0.0459

-7.91

0

1

0

3-a2

2.71

1.37

5.00

0.0423

-7.71

0

1

0

3-l1l2

1.50

0.56

4.80

-0.1647

-5.18

2

0

0

3-a1l2

2.15

0.89

4.93

-0.0499

-6.70

1

1

0

3-a2l2

2.16

0.95

4.93

-0.0514

-6.55

1

1

0

3-a1a2

3.28

1.50

5.04

0.0434

-7.49

0

0

2

3-a1a3

3.61

1.56

5.05

0.1025

-8.08

0

2

0

3-a1a4

3.58

1.67

5.03

0.0703

-8.34

0

2

0

3-a2a3

3.63

1.56

5.05

0.1016

-8.08

0

2

0

3-a1a2l2

2.54

1.02

4.97

-0.0577

-6.44

1

0

2

3-a1a2a3

4.48

1.87

5.10

0.1040

-7.58

0

1

2

3-a1a2a4

4.43

1.88

5.08

0.0824

-7.97

0

1

2

3-a1a2a3a4

5.56

2.22

5.13

0.0954

-7.15

0

0

4


29


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Page 30 of 54

Table 4. ef, MCI, ρ (rC ) , HOMA, and NICS(1) values of the central ring R in benzo-annelated derivatives of 4. L, A and G indicates the numbers of benzo-rings annelated in linear, angular and geminal position, respectively. Compound

ef( ×102 )

MCI( ×102 )

ρ (rC ) ( ×102 )

HOMA

NICS(1)

L

A

G

4

2.84

1.36

3.48

0.4268

-7.55

0

0

0

4-l1

2.23

1.03

3.43

0.3185

-6.29

1

0

0

4-a1

3.78

1.62

3.54

0.4301

-7.86

0

1

0

4-a2

3.77

1.62

3.54

0.4905

-7.54

0

1

0

4-l1l2

1.85

0.81

3.37

0.0468

-4.98

2

0

0

4-a1l2

2.86

1.23

3.48

0.2955

-6.60

1

1

0

4-a2l2

2.85

1.18

3.48

0.3766

-6.28

1

1

0

4-a1a2

4.64

1.83

3.58

0.4466

-7.53

0

0

2

4-a1a3

5.16

1.95

3.60

0.5083

-7.82

0

2

0

4-a1a4

5.15

2.05

3.61

0.4941

-8.25

0

2

0

4-a2a3

5.16

2.00

3.59

0.5562

-7.50

0

2

0

4-a1a2l2

3.43

1.36

3.52

0.3198

-6.42

1

0

2

4-a1a2a3

6.45

2.26

3.63

0.5261

-7.43

0

1

2

4-a1a2a4

6.43

2.31

3.65

0.5203

-7.49

0

1

2

4-a1a2a3a4

8.11

2.67

3.68

0.5434

-6.47

0

0

4


30

Page 31 of 54

Table 5. The coefficients in Eq. 1 and correlation coefficient (R) for I ≡ ef, I ≡ MCI, I ≡ ρ (rC ) , I ≡ HOMA and I ≡ NICS(1) calculated by least squares fitting for the series of benzo-annelated derivatives of 1−4. series

l

±∆l

a

±∆a

g

±∆g

I0

±∆I0

R

1-

-1.3

0.3

1.8

0.3

1.7

0.3

6.0

0.4

0.9796

2-

-0.9

0.3

1.5

0.2

1.5

0.1

3.7

0.4

0.9761

3-

-0.4

0.2

0.8

0.1

0.80

0.08

2.0

0.2

0.9754

4-

-0.7

0.2

1.2

0.2

1.2

0.1

2.7

0.3

0.9747

1-

-0.31

0.07

0.19

0.06

0.17

0.04

1.6

0.1

0.9485

2-

-0.50

0.09

0.43

0.08

0.38

0.05

1.1

0.1

0.9754

3-

-0.36

0.06

0.24

0.05

0.25

0.03

1.11

0.08

0.9773

4-

-0.34

0.06

0.33

0.05

0.30

0.03

1.31

0.08

0.9811

1-

-0.024 0.005 0.024 0.005 0.026 0.003

2.21

0.01

0.9771

2-

-0.069 0.007 0.055 0.006 0.048 0.003

4.74

0.01

0.9921

3-

-0.070 0.004 0.053 0.003 0.048 0.002

4.942

0.005 0.9973

4-

-0.058 0.003 0.058 0.003 0.049 0.002

3.484

0.005 0.9979

1-

-0.09

0.01

0.05

0.01

0.045 0.007

0.73

0.02

0.9736

2-

-0.10

0.02

0.04

0.01

0.025 0.008

0.61

0.02

0.9555

3-

-0.087 0.006 0.047 0.005 0.024 0.003 -0.004 0.008 0.9927

4-

-0.16

0.02

0.05

0.02

0.03

0.01

0.43

0.03

0.9676

1-

0.0

0.3

0.6

0.2

0.8

0.1

-13.3

0.3

0.9120

HOMA

ρ(rC)

a

MCIa

efa

I

NI CS( 1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

a

2-

1.5

3-

1.18

4-

1.3

0.1

Page 32 of 54

-0.47

0.09

0.04

0.05

-9.2

0.1

0.9882

0.008 -0.29

0.07

0.07

0.04

-7.6

0.1

0.9890

0.1

0.18

0.08

-7.6

0.2

0.9571

0.1

-0.1

The values of parameters l, a, g nad I0 are multiplied by 100.


32

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Table 6. pef (in β ×10−2 units) of the central and annelated rings in benzo-annelated derivatives of 1. Rings Compound R,a1 R,a2 R,a3 R,a4

R,l1

R,l2

1-l1

-

-

-

-

-0.24

-

1-a1

0.94

-

-

-

-

-

1-a2

-

0.96

-

-

-

-

1-l1l2

-

-

-

-

1-a1l2

0.60

-

-

-

-

-0.33

1-a2l2

-

0.62

-

-

-

-0.33

1-a1a2

0.75 0.75

-

-

-

-

1-a1a3

1.38

-

1.38

-

-

-

1-a1a4

1.39

-

-

1.39

-

-

1-a2a3

-

-

-

-

-

-

-0.40

-

-

-

1.77

-

-

1.53 1.53 1.53 1.53

-

-

1.41 1.41

1-a1a2l2

0.48 0.48

1-a1a2a3

1.16 1.20 1.79

1-a1a2a4

1.16 1.18

1-a1a2a3a4

-

-

-0.18 -0.18


33


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Page 34 of 54


R,l1

R,l2

2-l1

-

-

-

-

-0.18

-

2-a1

0.78

-

-

-

-

-

2-a2

-

0.76

-

-

-

-

2-l1l2

-

-

-

-

2-a1l2

0.49

-

-

-

-

-0.27

2-a2l2

-

0.47

-

-

-

-0.27

2-a1a2

0.77 0.77

-

-

-

-

2-a1a3

1.17

-

1.17

-

-

-

2-a1a4

1.16

-

-

1.16

-

-

2-a2a3

-

-

-

-

-

-

-0.35

-

-

-

1.54

-

-

1.52 1.52 1.52 1.52

-

-

1.16 1.16

2-a1a2l2

0.48 0.48

2-a1a2a3

1.16 1.14 1.53

2-a1a2a4

1.14 1.14

2-a1a2a3a4

-

-

-0.12 -0.12


34

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



R,l1

R,l2

3-l1

-

-

-

-

-0.09

-

3-a1

0.37

-

-

-

-

-

3-a2

-

0.38

-

-

-

-

3-l1l2

-

-

-

-

3-a1l2

0.23

-

-

-

-

-0.14

3-a2l2

-

0.24

-

-

-

-0.14

3-a1a2

0.37 0.37

-

-

-

-

3-a1a3

0.58

-

0.58

-

-

-

3-a1a4

0.57

-

-

0.57

-

-

3-a2a3

-

-

-

-

-

-

-0.18

-

-

-

0.77

-

-

0.77 0.77 0.77 0.77

-

-

0.59 0.59

3-a1a2l2

0.23 0.23

3-a1a2a3

0.58 0.60 0.79

3-a1a2a4

0.57 0.59

3-a1a2a3a4

-

-

-0.06 -0.06


35


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Page 36 of 54


R,l1

R,l2

4-l1

-

-

-

-

-0.15

-

4-a1

0.62

-

-

-

-

-

4-a2

-

0.61

-

-

-

-

4-l1l2

-

-

-

-

4-a1l2

0.38

-

-

-

-

-0.22

4-a2l2

-

0.38

-

-

-

-0.22

4-a1a2

0.61 0.61

-

-

-

-

4-a1a3

0.94

-

0.94

-

-

-

4-a1a4

0.93

-

-

0.93

-

-

4-a2a3

-

-

-

-

-

-

-0.28

-

-

-

1.24

-

-

1.24 1.24 1.24 1.24

-

-

0.94 0.94

4-a1a2l2

0.38 0.38

4-a1a2a3

0.94 0.94 1.25

4-a1a2a4

0.92 0.93

4-a1a2a3a4

-

-

-0.10 -0.10


36

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Figure 1. The polycyclic heteroconjugated molecules whose benzo-annelated derivatives were studied in this work: acridine (1), 9H-carbazole (2), dibenzofuran (3) and dibenzothiophene (4), and the labeling of the sites of annelation (a = angular; l = linear); R denotes the central heterocyclic ring in these molecules.


37


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Page 38 of 54

Figure 2. Examples illustrating the labeling of benzo-annelated derivatives of the molecules depicted in Figure 1.


38

Page 39 of 54

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Figure 3. ef, MCI, ρ (rC ) HOMA, and NICS(1) values of the central ring R in 1, 1-a1 and 1-l1.


39


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Page 40 of 54

Figure 4. Clar formulas of anthracene, benz[a]anthracene (angularly benzo-annelated anthracene), tetracene (linearly benzo-annelated anthracene), phenanthrene, crysene (angularly benzo-annelated phenanthrene), benz[a]anthracene (linearly benzo-annelated phenanthrene).


40

Page 41 of 54

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


a)

b)

c)

d)

e)

Figure 5. The dependence of the values of different aromaticity indices of the central ring R in benzo-annelated derivatives of 3 on the sum of pef values of the central and the annelated benzorings ( ∑ pef ): a) ef vs. ; e) NICS(1) vs.

∑ pef ; b) MCI vs. ∑ pef ; c) ρ (r ) vs. ∑ pef ; d) HOMA vs. ∑ pef C

∑ pef . ACS Paragon Plus Environment

41


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Page 42 of 54

Figure 6. An angularly and a linearly benzo-annelated acridine-like molecule, A and L. The unweighted edges belonging to or incident to the ring Za are marked by heavy lines and form the set Ea (GVEW ) ; the edges belonging to or incident to the ring Zb are marked by dashed lines and form the set Eb (GVEW ) . The edge indicated by an arrow (which occurs only in the case of angular annelation) belongs to both Ea (GVEW ) and Eb (GVEW ) .


42

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Table of Contents Image


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The polycyclic heteroconjugated molecules whose benzo-annelated derivatives were studied in this work: acridine (1), 9H-carbazole (2), dibenzofuran (3) and dibenzothiophene (4), and the labeling of the sites of annelation (a = angular; l = linear); R denotes the central heterocyclic ring in these molecules. 134x91mm (300 x 300 DPI)


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Examples illustrating the labeling of benzo-annelated derivatives of the molecules depicted in Figure 1. 172x119mm (300 x 300 DPI)



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ef, MCI,

HOMA, and NICS(1) values of the central ring R in 1, 1-a1 and 1-l1. 170x67mm (300 x 300 DPI)


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Clar formulas of anthracene, benz[a]anthracene (angularly benzo-annelated anthracene), tetracene (linearly benzo-annelated anthracene), phenanthrene, crysene (angularly benzo-annelated phenanthrene), benz[a]anthracene (linearly benzo-annelated phenanthrene). 129x247mm (300 x 300 DPI)



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226x183mm (150 x 150 DPI)


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239x183mm (150 x 150 DPI)



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237x181mm (150 x 150 DPI)


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245x186mm (150 x 150 DPI)



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The dependence of the values of different aromaticity indices of the central ring R in benzo-annelated derivatives of 3 on the sum of pef values of the central and the annelated benzo-rings (∑pef ): a) ef vs. ∑pef; b) MCI vs. ∑pef; c) ρ(rc) vs. ∑pef; d) HOMA vs. ∑pef; e) NICS(1) vs. ∑pef. 279x215mm (150 x 150 DPI)


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An angularly and a linearly benzo-annelated acridine-like molecule, A and L. The unweighted edges belonging to or incident to the ring Za are marked by heavy lines and form the set Ea(Gvew); the edges belonging to or incident to the ring Zb are marked by dashed lines and form the set Eb(Gvew). The edge indicated by an arrow (which occurs only in the case of angular annelation) belongs to both Ea(Gvew) and Eb(Gvew). 170x69mm (300 x 300 DPI)



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179x155mm (300 x 300 DPI)


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Effect of Benzo-Annelation on Local Aromaticity in Heterocyclic

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