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7 Benzenoids with More Than One Geometric Kekulé Structure Corresponding to the Same Algebraic Kekulé Structure

7 Benzenoids with More Than One Geometric Kekulé Structure Corresponding to the Same Algebraic Kekulé Structure

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A.T. Balaban and M. Randic´



182



5

3



3

5



Fig. 8.16 For pyrene, two different geometric Kekule´ structures (from the six possible Kekule´

structures) correspond to the one and the same algebraic Kekule´ structure shown between them



(Gutman et al. 2004). However, there are a few exceptions, and they are mentioned in

this section, but for more details readers should consult Part 4 of the above-mentioned

series entitled “Partitioning of p-electrons in rings of polycyclic benzenoid

hydrocarbons. Part 4. Benzenoids with more than one geometric Kekule´ structure

corresponding to the same algebraic Kekule´ structure” (Vukicˇevic´ et al. 2004). The

most obvious case is benzene itself, which has two (symmetry equivalent) Kekule´

valence structures, having necessarily the same p-electron ring partition.

Whereas all benzenoid catafusenes and most benzenoid perifusenes show a oneto-one correspondence between geometric and algebraic Kekule´ structures

(as shown in Fig. 8.5 for anthracene), some perifusenes are exceptions, the smallest

being pyrene. As can be seen from Fig. 8.16, there are two pyrene Kekule´ structures

(albeit symmetry related) that have identical p-electron ring partitions for all rings.

A one-to-one correspondence between the “geometric” and “numerical” (or “algebraic”) individual Kekule´ valence structures signifies that no loss of information is

associated with the numerical Kekule´ valence structures, which therefore allow a

full reconstruction of Kekule´ valence structure from the “algebraic” form. In the

case of pyrene, however, there is no longer a one-to-one correspondence between

the two types of Kekule´ valence structures, but nevertheless there is no loss of

information accompanying the “numerical” Kekule´ valence structures. That is, in

the case of pyrene a single numerical structure yields two solutions.

If one condenses other benzenoid moieties on both sides of pyrene, the resulting

systems also show the same kind of degeneracy that we have seen in pyrene, as

shown on the three algebraic Kekule´ structures of dibenzopyrene in Fig. 8.17.

Interestingly, it is not necessary for the moieties attached to the left and right side

of pyrene to be identical, as shown in the next lower part of Fig. 8.17. All these types

of degeneracy were examined in Part 4 of the above-mentioned series (Vukicˇevic´

et al. 2004). This observation can be further generalized to the class of benzenoids

schematically represented by the diagram at the bottom of Fig. 8.17, which is

discussed again for non-alternant systems that present a similar degeneracy.

In Fig. 8.18 at the top we show two special structures among 200 geometric

Kekule´ structures of kekulene that correspond to one and the same algebraic Kekule´

structure. Again these two Kekule´ structures are symmetry related, but in this case

all rings have the same EC value, even though the distribution of C¼C double

bonds in different rings is different. In the lower part of Fig. 8.18 we show a

fourfold degeneracy of a perifusene combining pyrene and perylene moieties.



Structural Approach to Aromaticity and Local Aromaticity. . .



8



5

5



2



2



5



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5



5



2



2



1



5



2



6



6



1



5



5



6



2



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3



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1



6



1



6



5

1



6



6



1

5



5

2



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1

5



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2

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183



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1

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Fig. 8.17 The first three rows display some of the Kekule´ structures of one and the same

benzenoid perifusene: one and the same algebraic Kekule´ structure on the left corresponds to the

pair of geometric Kekule´ structures on the right. The fourth row shows that a similar degeneracy

occurs even when the moieties condensed to the left- and right-hand bonds of pyrene are different.

The bottom line shows the generalized feature of degeneracy that holds even for nonalternants



8.8



Alternant Non-benzenoids



The EC values of alternant non-benzenoids have not been discussed till now.

We shall limit the discussion to cata-fused non-benzenoids having two or three

condensed rings with ring sizes equal to 4, 6, or 8, and to peri-condensed nonbenzenoids with the above ring sizes having at most four rings. Figure 8.19 contains

structures of systems that are discussed in this section.

Observe that among the 35 structures of Fig. 8.19 the first five are bicyclic.

As we see the systems with two condensed rings have integer EC values. Recall that

both rings of naphthalene also have integer EC values. The EC ¼ 5 p-electrons in

naphthalene is the average of three resonance (Kekule´) structures, two of which are

symmetry related having partitions 4:6 and 6:4 and the Fries structure with a C¼C

double bond in the middle with partition 5:5. It was postulated by Fries that among



A.T. Balaban and M. Randic´



184



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4



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4



4

4



4



5



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5

1

4



4

2

4



4

1



5



5



Fig. 8.18 Top row: one algebraic Kekule´ structure (left) corresponds (right) to a pair of geometric

Kekule´ structures of kekulene (a coronafusene) – note that these are not EC counts, and that there

are many other geometric Kekule´ structures. Bottom row: one algebraic Kekule´ structure (left)

corresponds to four geometric Kekule´ structures (right) for the perifusene with 11 benzenoid rings

combining pyrene and perylene moieties (left)



all Kekule´ resonance structures, the most important one(s) is (are) that (those) with

a maximal number of benzenoid rings with three double bonds (Fries 1927).

An algorithm for finding such Fries structures is available (Ciesielski et al. 2010).

But observe that the average of the first two structures gives 5:5, thus the partition of

the Fries resonance structures give also the overall partition for a molecule as a

whole. By contrast, all systems with three cata-condensed rings discussed here have

non-integer partitions. Interestingly, on fusing a benzenoid ring to one bond of a

bicyclic system such as 4/19 one obtains one of the following two situations:

(i) when the extra benzenoid ring is fused to a double bond of the “Fries-type”

structure, its EC value is 5.2 p-electrons, the ring to which it is fused loses 1.4

p-electrons, and the last ring gains 0.2 p-electrons; (ii) when the extra benzenoid

ring is fused to a single bond of the “Fries-type” Kekule´ structure, its EC value is 4.75

p-electrons, the ring to which it is fused loses 0.5 p-electrons, and the last

ring loses 0.25 p-electrons. This is a general situation, which was encountered

also in the annelation of naphthalene yielding either phenanthrene (1/7) À case(i) À,

or anthracene (3/6) À case (ii); and in the annelation of perylene to form either

12/15 or 11/15, corresponding to the above two cases, respectively. Even more



8



Structural Approach to Aromaticity and Local Aromaticity. . .



3



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7



3



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5



185



7



7



3



7



4



5

5.2



3.2



3.2 1.6 3.2



3.2 1.6



6



2.75 4.50 2.75



5.2



7



2.75 4.50



3.6



3.2



9



8



3.2



5.2



12



11

3.2



2.75



7.2



3.2



7.2



3.2



13



6.50



2.75



14



3.2 1.6



15



7.2



5.2



5.2



5.6



3.2



19



18



16



17



21



24

4.75

6.50



7.2



22



5.2

6.50



7.2



3.6



6.75



6.75



25



5.2



5.6



5.2



23



4.75



2.75 4.50



2.75



5.6



7.2



1.6



3.2



20



3.2



7.2



6.50



4.75



5.6



5.2



2.75



1.6



7.2



3.6



3.2



10



3.2



5.2 1.6



4.75



5.6



26



27



5.2

3.6



28



7.2



4.75



4.50



29



6.75



6.75



4.50



30



6.75



31



6.75

5.2



5.6



32



7.2



5.2



5



6.50



33



6.75

7.2



5.6



34



7.2



6.75



6.50



35



Fig. 8.19 Alternant non-benzenoids their EC values. Note that for 13 and 14 the EC value for the

eight-membered ring is 5.6. and not 7.2



generally, on inspecting Fig. 8.19, one may see that the same scenario is valid

for annelating a polycyclic system with definite single/double bonds by fusing

an even-membered ring (having, instead of 5 as for a six-membered ring, an EC

value in the bicyclic system, denoted by V) on one of the bonds of the bicyclic



186



A.T. Balaban and M. Randic´



system: the extra ring has after annelation an EC value of either V + 0.2, or

V À 0.25, whereas the two other rings have exactly the values discussed in the

two cases mentioned above.



8.9



Non-alternant Conjugated Hydrocarbons



Part 5 of the series was entitled “Partitioning of p-electrons in rings of polycyclic

conjugated hydrocarbons. 5. Nonalternant compounds” (Balaban and Randic´

2004b) and discussed in detail non-alternant systems, providoing a (presumed)

complete list of all 189 possible systems with two, three, or four condensed rings

having ring sizes equal to 5, 6, or 7. Table 8.2 contains data about all these systems,

but we include here formulas only for a small part of these structures (the absent

structures are indicated by a long dash in Table 8.2, and/or may be found by

difference from the columns “Compound” and “Isomer count”). The first two

columns in this Table indicate ring sizes and their sum. Whereas cata-condensed

systems have no internal vertices and no odd-numbered sums of ring sizes, pericondensed systems have either even-membered or odd-numbered sums of ring

sizes. In the latter case, the system may be tricyclic with one internal vertex (and

three odd-numbered ring sizes or one odd-numbered and two even-numbered ring

sizes) or tetracyclic (with one internal vertex, one or three odd-membered rings and

the remaining rings with even ring sizes). In the former case (peri-condensed

systems with even-numbered sums of ring sizes) there are two internal vertices,

and 0, 2, or 4 odd-membered ring sizes.

In the present review we shall not aim at such exhaustive enumerations of all

possible systems, but we shall expand the ring sizes to include also four- and eightmembered rings, which were also included in the preceding section. Structures of

the systems that will be discussed in this section are presented in Fig. 8.20 (bicyclic

and tricyclic cata-condensed systems), Fig. 8.21 (tetracyclic cata-condensed

systems) and Fig. 8.22 (peri-condensed systems).

It was noted in Part 5 (Balaban and Randic´ 2004b) that among the 189

nonalternants (cata- and peri-condensed systems) with two, three, or four

condensed rings having ring sizes equal to 5, 6, or 7, more than half had integer

EC values. In fact all tricicyclic catacondensed systems considered have integer

p-electron partitions. A simple rule accounts for the EC values of Fig. 8.20 that

have ring sizes ranging from 4 to 8 sp2-hybridized carbon atoms: for catacondensed

tricyclic systems composed of r, s, and t-membered rings, the EC values are r À 1,

s À 2, and t À 1, respectively.

The cata-condensed systems with three rings of sizes 4–8 presented in Fig. 8.20

reveal two interesting aspects: (i) all of them possess integer EC values, namely for

terminal rings EC ¼ R – 1, and for rings condensed to two other rings EC ¼ R – 2;

Another interesting observation, derived from the preceding one, is that in some cases

presented in Fig. 8.20, rings differing in size have the same EC value. It is not difficult



8



Structural Approach to Aromaticity and Local Aromaticity. . .



187



Table 8.2 Nonalternant Kekule´noids with two, three, or four 5-, 6-, and 7-membered rings.

A hyphen in the column ‘internal vertices’ means ‘not applicable’. A long line in the column

‘compound’ indicates that no examples are illustrated in this review

Catafused nonalternant systems

Perifused nonalternant systems

Ring

Sum of Internal

size

size

vertices Compound

5,5

10

0

1/20

5,7

12

0

2/20

7,7

14

0

3/20

5,5,5

15



None

5,5,6

16

0

10–12/20

5,5,7

17



None

5,6,6

17



None

5,6,7

18

0

13–17/20

5,7,7

19



None

6,6,7

19



None

6,7,7

20

0



5,5,5,5 20

0



7,7,7

21



None

5,5,5,6 21



None

5,5,5,7 22

0



5,5,6,6 22

0

1,2,8, etc./21

5,5,6,7 23



None

5,6,6,6 23



None

5,5,7,7 24

0



5,6,6,7 24

0

3-7,10, etc./21

5,6,7,7 25



None

6,6,6,7 25



None

5,7,7,7 26

0



6,6,7,7 26

0



6,7,7,7 27



None

7,7,7,7 28

0



a

All isomers have integer partitions

b

Aromatic systems



Isomer

count

1a

1a,b

1a

0

3a

0

0

5a,b

0

0

4a

2a

0

0

7a

13

0

0

18a,b

25b

0

0

14a

20

0

8a



Internal

vertices







1



1

1



1

1



2

1

1

2

2

1

1

2

2

1

1

2

2

1

2



Compound

None

None

None

1/22

None

5/22

4/22

None

2/22

6/22

None

7/22

3/22





8–10/22







11,12/22

13-16, etc./22













Isomer

count

0

0

0

1

0

1

1

0

1

1

0

1a

1

2

2a

3

11

4

3a

4

14

5

2a

3

6

1a



to see that this happens only when a terminal r-membered ring with odd ring size r is

connected to a middle (r + 1)-membered ring, namely for systems having two rings of

different sizes with the same EC value: 5, 9–11, 14–17, 22–25, all of Fig. 8.20.

Moreover, in some cases three cata-condensed rings share the same size, as was the

case with 10/20 and 11/20. A similar situation with four rings can be observed for

compounds 11/21 and 7/22, but in the last case it involves a peri-condensed system.

It is not difficult to see that any number of cata-condensed rings can share one EC

value if the ring sizes are r – 1, r, r,. . ., r, r – 1, where r is odd. Finally, one may observe

that several tetracyclic systems such as pyracylene (8/22) and acepleiadylene (11/22)

have integer partitioning of p-electrons to their four rings.



A.T. Balaban and M. Randic´



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6



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4



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3



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4



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21



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3



7

18



6



4

6



4



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3



6

6



4



27



28



6



23



22



7



26



13



5



5



6



7



4



6



12



4



6



5



5



6



7



25



5



17



4



20



5



4



5



4

19



4



5



4



6



4

7



11



16



6



6



4



4



15



2

6



4



4



4



4



10



4



14



6



4



9



4



3



4



6



3



3



4



5



4

3



5



3



4



3



4



6



2



3



2



8



4



24



4



4

6



29



4



6

4

30



Fig. 8.20 Non-alternant bicyclic and tricyclic cata-condensed conjugated hydrocarbons with their

EC values



8.10



Comparison with Other Methods for Estimating Local

Aromaticity of Rings in Polycyclic Benzenoids



We continue this review with a discussion of “local aromaticities” estimated by

EC values in comparison with other methods. Among the many published

approaches to this problem, pioneered by Oskar Polansky (Polansky and Derflinger

1967; Monev et al. 1981) which were presented in Part 6 of our series, entitled

“Partitioning of p-electrons in rings of polycyclic conjugated hydrocarbons.



Structural Approach to Aromaticity and Local Aromaticity. . .



8



5.2



5.2



4.75



2.8



3.25



2.8



3.25

4.75



5.2



1



4.75

2.8



3.25



4.8



5.25



5.2



4.75



3



4



2



6



4

3



5



5



8



4.75



3.6



4.75



4.50



5.2



5



4.75



6



7



4

3



4



4



4



5



9



4.75



5.2



4.50



6.00



6



4.00



4.0



6

3



4



189



4

4



6



4

10



11



4

12



Fig. 8.21 Top row: non-alternant tetracyclic cata-condensed nonbranched conjugated hydrocarbons with their EC values. Bottom row: non-alternant tetracyclic cata-condensed branched

conjugated hydrocarbons with integer EC values



Part 6. Comparison with other methods for estimating the local aromaticity of rings

in polycyclic benzenoids” (Balaban and Randic´ 2005) we shall confine our discussion in this review only to a few other methods: Randic´’s graph-theoretical method

(Randic´ 1975), Schleyer’s nuclear-independent chemical shifts over the ring center

˚ ngstroms from the molecular plane, NICS(y), where y ¼ 0 or 1 A

˚

at a distance of y A

(Chen et al. 2005; Schleyer et al. 1996, 2001) and the harmonic oscillator model of

aromaticity (HOMA) values introduced by Kruszewski et al. (Kruszewski 1971,

1980; Kruszewski and Krygowski 1972; Krygowski and Cyran´ski 1996, 1998, 2001;

Krygowski et al. 1995, 1996; Cyran´ski et al. 2000; Cyran´ski and Krygowski 1996,

1998). A fair correlation exists between EC values and HOMA values, and a

somewhat lower correlation with NICS(1) values (readers should consult Part 6

for details). Here we will present in detail only an unpublished comparison with one

index (indicated in Table 8.3 as GT) for the graph-theoretical local aromaticity of

benzenoid rings in polycyclic aromatic hydrocarbons published in 1975 by Randic´

(1975). This index is easily calculated as the ratio between the number of Kekule´

structures in which a particular benzenoid ring has three conjugated double bonds

and the total number of Kekule´ structures of that polycyclic aromatic hydrocarbon.

For 24 cata- and perifusenes with 3 to 7 benzenoid rings, whose rings are labeled

with capital letters starting with marginal rings belonging to acenic portions, Table 8.3

presents local EC and GT indices. It can be seen that isoarithmic systems such

as zigzag catafusenes and helicenes (fibonacenes with the same number R of rings)



A.T. Balaban and M. Randic´



190



5.33



3.33



3.33 3.33

3.33



5.33



5.33



1



5.33



2



3

3



4



3



4



3



3.4



5.4



13



9



5.4



2.0

5.4



3



4.0



4.67



5



5



4.75 2.75



4



4.50



10



5.25

3.25



14



4.75



15



4.67



6



4.50

2.75 3.75



4

3



5.00



11



12



5.4



4.75

5.2



4.67



5.33



3.00



4



8



5.2



4.67



4.67



3.33 3.33



4



4



3



7



5.33



3



3

3



2.67



3.4

4.0



5.2



16



Fig. 8.22 Non-alternant tricyclic and tetracyclic peri-condensed conjugated hydrocarbons with

their EC values



have identical EC values and very similar GT values. The points in the plot GT versus

EC (Fig. 8.23) reveal three linear correlations. The rightmost trendline corresponds to

marginal rings (labeled A) in acenes or fibonacenes; its equation is:

GT ¼ 0:664 EC À 2:652; and the correlation coefficient is r 2 ¼ 0:9999:

The middle trendline with the highest slope corresponds to rings next to marginal rings in acenes; its equation is:

GT ¼ 0:986 EC À 3:936; and the correlation coefficient is r 2 ¼ 0:9998:

The leftmost trendline with the lowest slope groups together corresponds to kink

rings in catafusenes; its equation is:

GT ¼ 0:377 EC À 0:955; and the correlation coefficient is r 2 ¼ 0:997:

In a very recent paper (Balaban and Mallion 2011) correlations between

three local aromaticity indices for benzenoids, namely EC values, topological

ring currents (Mallion 2008), and six-center delocalization-indices obtained by



8



Structural Approach to Aromaticity and Local Aromaticity. . .



Table 8.3 Local aromaticities of benzenoids (EC and GT values)

Benzenoid

Ring

EC

GT

Benzenoid

Anthracene

A

4.750

0.500

Triphenylene

3/6

B

4.500

0.500

1/12

Naphthacene

A

4.600

0.400

Coronene

(tetracene), 4/6

B

4.400

0.400

7/15

Pentacene

5/6

Hexacene

6/6

Heptacene

8/6



Phenanthrene

1/7

Chrysene

2/7

Picene

3/7, 5/11



A

B

C

A

B

C

A

B

C

D



4.500

4.330

4.330

4.420

4.280

4.280

4.375

4.250

4.250

4.250



0.333

0.333

0.333

0.286

0.286

0.286

0.250

0.250

0.250

0.250



A

B

A

B

A

B

C



5.200

3.600

5.125

3.875

5.154

3.769

4.154



0.800

0.400

0.750

0.500

0.769

0.462

0.615



Benz[a]anthracene

1/11

Dibenz[a,j]anthracene

2/11

Dibenz[a,h]anthracene

3/11

Fulminene

4/7

Zig-zag

[7]fibonacene

5/7 and

heptahelicene



191



Ring

A

B

A

B



EC

4.920

4.620

4.860

4.570



GT

0.615

0.615

0.571

0.571



A

B

C

D

A

B

C

A

B

C



3.290

5.290

5.250

3.420

4.670

5.250

3.420

4.670

4.900

4.600



0.286

0.857

0.833

0.333

0.667

0.833

0.333

0.667

0.600

0.600



A

B

C

A

B

C

D



5.143

3.810

4.048

5.147

3.794

4.088

3.941



0.762

0.476

0.571

0.765

0.471

0.588

0.529



Fig. 8.23 Plot of the correlation between GT values and EC values for the benzenoids

from Table 8.3



A.T. Balaban and M. Randic´



192



Table 8.4 Local aromaticity indices of compound 10/15 (EC, GT and HOMA)

Ring

EC

GT

HOMA

A

5.39

0.929

0.726

B

1.69

0.143

0.670

C

3.70

0.486

0.701

D

5.03

0.668

0.725

E

3.46

0.514

0.694

F

3.46

0.400

0.700



CLAR

1.000

0.000

0.375

0.625

0.500

0.250



A

B

E



C



D



F



10/15



quantum-chemical computations (Mandado et al. 2006) showed similar linear

equations depending on the topology of the ring.

For one and the same benzenoid with 11 rings of six different types, namely

tribenzo[a,h,rst]phenanthra[1,2,10-cde]pentaphene (10/15) with 11 benzenoid

rings (Krygowski et al. 1995; Oonishi et al. 1992) the data in Table 8.4 indicate

fair correlations between EC and various local aromaticity descriptors, albeit with

only six points. Thus for EC and GT, r2 ¼ 0.926); for EC and HOMA, r2 ¼ 0.986;

for EC and the “algebraic Clar structure” index (Randic´ 2011), r2 ¼ 0.864.



8.11



Ring Partition of p-Electrons for Clar’s Fully Benzenoid

Systems (‘Claromatic Benzenoids’)



Clar’s intuitive approach in the characterization of local properties of benzenoids is

necessarily qualitative in nature (Clar 1972). In Clar’s approach individual benzenoid rings in polycyclic systems can be classified into three classes, namely as

‘aromatic sextets’, ‘migrating sextets’, and ‘empty rings’. Fully benzenoid

(claromatic) systems are those that have only ‘aromatic sextets’ or ‘empty rings’,

the former of course being more interesting as the sites of possible chemical

activity. It seems therefore of interest to examine closely individual rings in

“fully benzenoid” systems to see what kind of variation there is between different

classes of rings.

In Fig. 8.24 we show 11 smaller fully benzenoid (claromatic) systems that

include triphenylene as the smallest fully benzenoid molecule (1/23), and

hexabenzocoronene (6/23); the latter was described by Clar as an unusually stable

compound which “resisted” the measurement of its melting point because in



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7 Benzenoids with More Than One Geometric Kekulé Structure Corresponding to the Same Algebraic Kekulé Structure

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