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11 Ring Partition of pi-Electrons for Clar´s Fully Benzenoid Systems (`Claromatic Benzenoids´)

11 Ring Partition of pi-Electrons for Clar´s Fully Benzenoid Systems (`Claromatic Benzenoids´)

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8



Structural Approach to Aromaticity and Local Aromaticity. . .



B



A



A



A



C



2



1



193



3



A



A

D



B

D



A



F



5



4



6



B



A

A



E



C



B

B'



8



7



A

A



A

E



D



D



9



B

B



10



D

D'



11



Fig. 8.24 Several symmetrical claromatic benzenoids with labeled Clar sextet rings



the attempt to do this, the glass tube melted before the compound did! We use the

letter A for an “exposed” benzenoid ring having one common CC bond with the rest

of the molecule, letters B, B0 and C for benzenoid rings having two CC bonds

common to the rest of the molecule, letter D for benzenoid rings having three CC

bonds common to the rest of the molecule, E for benzenoid rings having four

CC bonds common to the rest of the molecule, and F for interior benzenoid rings

having all CC bonds common to the rest of the molecule. In Table 8.5 we have



194



A.T. Balaban and M. Randic´

Table 8.5 Data for fully benzenoid (claromatic) compounds from Fig. 8.24

Comp.

Ring

EC

GT

1

A

48/9 ¼ 5.333

8/9 ¼ 0.889

2

A

107/20 ¼ 5.350

18/20 ¼ 0.900

B

96/20 ¼ 4.800

16/20 ¼ 0.800

3

A

214/40 ¼ 5.350

36/40 ¼ 0.900

B

192/40 ¼ 4.800

32/40 ¼ 0.800

4

A

240/45 ¼ 5.333

40/45 ¼ 0.889

B

216/45 ¼ 4.800

36/45 ¼ 0.800

D

186/45 ¼ 4.133

32/45 ¼ 0.711

5

A

551/104 ¼ 5.298

90/104 ¼ 0.865

D

425/104 ¼ 4.087

72/104 ¼ 0.692

6

A

1200/250 ¼ 4.800

200/250 ¼ 0.800

F

600/250 ¼ 2.400

128/250 ¼ 0.512

7

A

535/100 ¼ 5.350

90/100 ¼ 0.900

B

480/100 ¼ 4.800

80/100 ¼ 0.800

480/100 ¼ 4.800

80/100 ¼ 0.800

B0

E

360/100 ¼ 3.600

64/100 ¼ 0.640

8

A

1059/198 ¼ 5.348

178/198 ¼ 0.899

B

916/198 ¼ 4.808

160/198 ¼ 0.808

C

882/198 ¼ 4.828

162/198 ¼ 0.818

9

A

2770/520 ¼ 5.327

460/520 ¼ 0.885

D

1904/520 ¼ 4.031

360/520 ¼ 0.692

E

1856/520 ¼ 3.285

320/520 ¼ 0.615

10

A

539/101 ¼ 5.337

90/101 ¼ 0.891

B

484/101 ¼ 4.792

80/101 ¼ 0.792

D

419/101 ¼ 4.149

72/101 ¼ 0.713

11

A

1211/227 ¼ 5.335

202/227 ¼ 0.890

B

1088/227 ¼ 4.793

180/227 ¼ 0.793

D

918/227 ¼ 4.044

160/227 ¼ 0.705

D0

938/227 ¼ 4.132

162/227 ¼ 0.714



listed the EC values and the graph theoretical ring indices GT for the symmetry

non-equivalent benzenoid rings qualifying as ‘aromatic or Clar sextets’ of these

claromatic systems.

We see from Table 8.5 that the highest p-content belongs to rings A, which has

over 5.300 p-electrons per ring. Most of the B-rings and C rings have values around

4.800, the smallest value belongs to 5/5. As rings become more “deeply” attached to

the rest of the molecule, the p-ring content decreases to around 4.000 and finally

drops below 4.000 for rings that are in the interior part of the molecule.

In the last column of Table 8.5 we give the graph theoretical local aromaticity

ring index GT [70] based on the count of Kekule´ valence structures in which a ring

has three double and three single CC bonds (that is, it appears as one of the Kekule´

structures of benzene). There is very good correlation (r2 ¼ 0.985, Fig. 8.25)

between the two indices, i. e. the p-electron ring partition (EC) and the count of

‘benzenic’ ring fragments with three double bonds in a larger benzenoid for

individual benzenoid rings. This is very interesting and clearly illustrates that



8



Structural Approach to Aromaticity and Local Aromaticity. . .



Fig. 8.25 Plot of the

correlation between the GT

index and the p-electron

partition (EC value) for the

Clar sextet rings of

benzenoids from Fig. 8.24



195



GT vs EC Clar

0.9



0.8



0.7



0.6



0.5

2



3



4



5



6



Kekule´ valence structures do involve more information than one might superficially

anticipate. It appears that the fate of Kekule´ valence structures followed the

proverbial: “throwing away the baby with dirty linen” – because so much information innate to Kekule´ valence structures has apparently been overlooked and would

have never been recognized, were it not for the rise of Chemical Graph Theory and

the application of Discrete Mathematics to chemical structure problems.

Interesting correlations between the p-electron ring partition (EC) and

Gutman’s energy effect for bipartite conjugated hydrocarbons (including, in addition to benzenoids, biphenylene and related compounds) were recently found

(Balaban et al. 2010c). Gutman’s observation of the ‘PCP Rule’ was extended to

the effect of outer benzenoid rings twice removed from a central ring (the PCP

Rule involved a central five-membered ring). Remarkably, the correlations could

be expressed in simple numerical terms for just three types of outer benzenoid

rings, depending on their condensation: linear, angular, or geminal (Balaban

et al. 2010b).



8.12



r-Sequences and Signatures



On considering all K Kekule´ structures of a benzenoid with n carbon atoms and

n p-electrons, in addition to partitioning these p-electrons as discussed till now, it is

possible to ascribe them in a few other different ways to individual rings leading to

several types of local aromaticity accounts.

For sextet-resonant benzenoids (called “claromatic”) there is a “winner takes all”

solution, in which the Clar sextet rings get six p-electrons each, and the remaining

rings remain “empty”, as seen in the left-hand structure of Fig. 8.26; its p-electron



A.T. Balaban and M. Randic´



196



Clar structure



Partition

5.35

1.90

5.35



30 π−electrons



Signature

5.35



4.80 1.90

5.35



30 π−electrons



s6: 12.0

s5: 13.0

s 4: 2.4

s 3: 0.6

s 2: 1.3

s 1: 0.7



30 π−electrons



Fig. 8.26 Three ways of accounting for the p-electrons in a 7-ring claromatic perifusene. From

left to right: five Clar sextets (“winner takes all”); partitions (note that the Clar sextets correspond

to the highest partitions); and signatures



partition is shown in the middle structure, and one sees that Clar-sextet rings

correspond to the highest EC values (Balaban and Randic´ 2007, 2008a, b; Balaban

et al. 2008; Pompe et al. 2008; Randic´ and Balaban 2006, 2008). Finally, the same 30

p-electrons may be accounted for by what we called the signature of the benzenoid

(Balaban and Randic´ 2008; Randic´ and Balaban 2008), which is explained in the

following.

Let us have a look at Fig. 8.27 showing for each ring of a heptaperifusene, from

left to right, the number qi of Kekule´ structures with i ¼ 6, 5, 4, 3, 2, 1, 0 p-electrons

according to the partition convention discussed earlier. Sums for each row are equal

to the number K of Kekule´ structures, Si qi ¼ K. Then partitions of electrons to

individual rings are given by EC ¼ i  qi/K. Of course, the sum of all partitions for

a given benzenoid is the number n of carbon atoms and p-electrons. There is a close

correspondence between Clar sextet rings and the rings with the highest partitions,

but even for claromatic benzenoids there are no longer extreme values of EC ¼ 6

and EC ¼ 0.

The last row of numbers for columns with various i values, printed in boldface

characters, is the column sum (Ci), and indicates how many times there are i

p-electrons in any of the R rings for all Kekule´ structures. The sum of all these

numbers (also printed in boldface characters under the partition values) is

Si Ci ¼ RK. The integer Ci numbers allow us to group together the p-electrons in

a different way: divide by K to obtain the ri sequence, and multiply by i to obtain the

signature si. Of course, on summing all signatures for a benzenoid we obtain again

the numbers of p-electrons, as shown in Fig. 8.26. For four hexaperifusenes,

Fig. 8.28 presents their ri-sequences, partitions, and Clar structures.

All non-branched catafusenes have C0 ¼ C1 ¼ 0, and s1 ¼ 0. There are some

interesting correlations among r6 and r5, illustrating the grouping of catafusenes into

several classes with common structural features of their dualists, as discussed in

detail in two side-by-side papers (Balaban and Randic´ 2008; Randic´ and Balaban

2008). Whenever a benzenoid ring in catafusenes or perifusenes has at least three



8



Structural Approach to Aromaticity and Local Aromaticity. . .



001(.)1



197



001(.)2

A



B



F

C



G



D

E



ring



6-ring

6

0

0

2

12

8

9

37



5-ring

6

12

12

0

12

0

9

51



4-ring

14

14

14

0

2

8

8

60



3-ring

0

0

0

2

0

9

0

11



2-ring

0

0

0

9

0

1

0

10



1-ring

0

0

0

10

0

0

0

10



0-ring

0

0

0

3

0

0

0

3



r-sequence



r6

1.4231



r5

1.9615



r4

2.3077



r3

0.4231



r2

0.3846



r1

0.3846



r0

0.1154



signature



s6

8.5386



s5

9.8075



s4

9.2308



s3

1.2693



s2

0.7692



s1

0.3846



A

B

C

D

E

F

G

K = 26



partit.

4.6923

4.4615

4.4615

1.7692

5.3846

4.1923

5.0385

182



Fig. 8.27 Partitions, the r-sequence, and the signature (s-sequence) for both isoarithmic heptacatafusenes shown above



surrounding benzenoid rings as in triphenylene, this benzenoid has a nonzero

r0 value. A combination of several ri values may serve as a yardstick for

benzenoids, allowing an ordering or at least a partial ordering for isomeric systems

(Pompe et al. 2008).



8.13



Biphenyl-Type Conjugation



Aromatic benzenoid hydrocarbons are important raw materials for the chemical

industry, allowing the manufacture of plastics, fibers, strong textiles and a large

variety of other products. However, aromatic benzenoid hydrocarbons as such do

not have many uses, especially since some of them (benzene, benzanthracene,

benzopyrene) are proved carcinogens. One of the few uses involves a mixture of

meta- and para-terphenyl which has such a high thermal and radiolytical stability

that it can be used as moderator and heat transfer fluid in “organic-moderator”



A.T. Balaban and M. Randic´



198



r6,r5,r4,r3,r2,r1,r0

8, 18, 24, 22, 6, 0, 0



3.31 3.77 4.92

4.92 3.77 3.31



3.13 3.81 4.94

12, 32, 20, 15, 11, 5, 1



4.75 2.06

5.31



5.41

14, 35, 20, 14, 10, 7, 2



1.71 3.65 5.06

4.71 3.59



4.80

18, 54, 12, 4, 14, 14, 4



5.35 1.90 1.90 5.35

4.80



Fig. 8.28 From left to right: ri sequences, p-electron partitions, and Clar structures of four

hexaperifusenes (top to bottom) with 2, 3, 3, and 4 Clar sextet rings illustrating the fact that the

Clar sextet rings have the highest partitions, and the “empty rings” have the lowest partitions

among all isomeric structures



nuclear reactors. Indeed, biphenyl-type conjugation is also the cause for the high

stability of various polychlorobiphenyls which now (like the freons which are also

very stable) cause environmental problems.

Looking at Clar structures of polycyclic benzenoids, one sees that Clar sextet

rings benefit always from biphenyl-type conjugation: triphenylene, the smallest

claromatic structure, is derived from ortho-terphenyl by intramolecular dehydrogenation. Structures presented in Fig. 8.29 make it evident that meta- and paraterphenyl conjugation contributes to the stability of systems having closely-situated

Clar sextet rings. With two exceptions to be discussed below, all structures in

Fig. 8.29 reveal that the highest EC values correspond to Clar sextets. The first

three rows contain isoarithmic structures, namely a pair of symmetrical catafusenes

1/29 and 2/29, a pair of nonsymmetrical perifusenes 6/29 and 7/29, and a triplet of

symmetrical catafusenes (3/29 to 5/29). The symmetrical (8/29) and



8



Structural Approach to Aromaticity and Local Aromaticity. . .



3.42



4.67



3.42



5.25



3.45

3.45



4.69



4.67



5.25



1



4.69



199



2



3.24



3.24

5.24



5.24



3.45



4.69



3.24



5.24



4



3



5



3.47

5.20



3.47



3.60

3.60



4.67



4.67

3.87 3.20



3.87 3.20

5.20



7



6



3.33 3.33

4.67



3.33



8



2.94

3.44



3.25



3.44



4.63



3.19



5.13



9

Fig. 8.29 Several benzenoids with Clar structures and EC values



nonsymmetrical perifusene (9/29) happen to have a Clar sextet ring with the same

EC value as a ring that does not claim a Clar sextet; such situations are seldom

encountered.



A.T. Balaban and M. Randic´



200



8.14



Concluding Remarks



In concluding this analysis of mathematical properties of individual Kekule´ valence

structures we should not be complacent and self-satisfied that we have come to the

“end of the road” concerning Kekule´ valence structures. The recent extension of

this analysis of Kekule´ valence structures to non-benzenoid structures has already

hinted to some areas for future explorations that may be of interest. For instance,

while in the case of benzenoid hydrocarbons it has been recently recognized that the

inverse problem of Clar structures (that is, the mathematical characterization of

Clar’s structures) can be solved by selecting only the Kekule´ valence structures

with the highest “degree of freedom” (df), this however does not extend to nonbenzenoid systems. For example, in the case of C60 (buckminsterfullerene) only a

few of the Kekule´ valence structures contribute to the Clar structure, namely: those

having the degree of freedom df ¼ 10 (one structure), df ¼ 9 (two structures) and

df ¼ 8 (19 structures out of 32) (Randic´ 2002). Similarly, while the “resonance

graph” of benzenoid hydrocarbons is always “connected” (i.e. all Kekule´ structures

form a single connected graph), this is not the case for nonbenzenoids. Thus already

in the case of biphenylene, one structure from the five Kekule´ structures has no

single conjugated circuit R1, and this cannot be part of the resonance graph. In our

view, these differences in mathematical properties of benzenoids and nonbenzenoids deserve close attention. It seems promising to focus attention not to

all Kekule´ valence structures but to a subset of “important” Kekule´ valence

structures. One such subset is the set of Kekule´ valence structures contributing to

the Clar structure – and we are looking into this matter. Another subset is even

larger: the set of Kekule´ valence structures that constitute the “connected” resonance graph.

Klein and coworkers (Flocke et al. 1998; Wu et al. 2003) have shown that in the

case of C60, out of 12,500 Kekule´ valence structures, less than half make the

dominant contribution to molecular resonance energy (RE), namely about 99.9%

of RE, and these all form a connected resonance subgraph. It appears of considerable interest to find out when one must restrict attention to these subsets of Kekule´

valence structures, how they reflect upon the p-electron ring partition and on the

graph theoretical benzenoid ring characterization. In other words, it will be

interesting to examine how well some qualified subsets of Kekule´ valence

structures can characterize benzenoid and non-benzenoid hydrocarbons, possibly

including fullerenes. This is a task that appears worthy of attention. That this has

not been hitherto considered may be due primarily to the preconceived idea that

such ‘refinement’ of analysis of Kekule´ valence structures will not make any

difference in the case of benzenoid hydrocarbons, although it may be significant.

However, one may expect that in the case of non-benzenoids we may see more

visible aspects of the modified view on Kekule´ valence structures. That this may be

expected has already been reflected in the consideration of linearly and angularly

fused higher homologues of biphenylene – the angular members of which, if one

takes into account all Kekule´ valence structures would be less and less stable as the



8



Structural Approach to Aromaticity and Local Aromaticity. . .



201



number of fused biphenylene units increases – but Vollhardt has been able to

synthesize the “unstable” systems” (Dosche et al. 2002; Han et al. 2002; Miljanic´

and Vollhardt 2006) thus showing that the model based on all-Kekule´ valence

structures is deficient. However, when one restricts attention to Kekule´ structures

with higher degree of freedom one can see that there is no substantial difference in

the stability of linear and angular higher biphenylenes (Randic´ 2003; Milicˇevic´

et al. 2004; Trinajstic´ et al. 1991).

Although the present discussion was centered on benzenoids, p-electron partitions

of heterocyclic compounds have also been studied, adding a new dimension to the

findings presented in the preceding pages (Balaban et al. 2007, 2008). Finally, it was

interesting to investigate p-electron partitions of polyhedral carbon aggregates

(Balaban et al. 2008c). One can conclude that “the more we discover, the more

diverse are the directions into the unknown, awaiting exploration (Balaban, 2011)”.



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11 Ring Partition of pi-Electrons for Clar´s Fully Benzenoid Systems (`Claromatic Benzenoids´)

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