Enumeration of Kekule Structures for Multiple Zigzag

Enumeration of Kekule Structures for Multiple Zigzag Chains and Related Benzenoid Hydrocarbons S. J. Cyvin. B. N. Cyvin, and J. Brunvoll D ivision o f Physical C hem istry, T he U niversity o f T rondheim , N orw ay

I. G u tm a n Faculty o f Science. U niversity o f K ragujevac, Y ugoslavia* Z. N aturforsch. 42 a, 7 2 2 -7 3 0 (1987); received M arch 24, 1987 The benzenoid classes o f m u ltip le zigzag chains, A («, w ), and incom plete m ultiple zigzag chains. A {n. m . /), are considered. N u m erical num bers o f K ekule structures (K ) are given for n up to 5 and m up to 10. A recurrence relatio n is developed for the K num bers o f A(n.m) with fixed values o f rr. a general fo rm u latio n for a rb itrary m values was achieved. Six new benzenoid classes are studied in connection w ith the ap plication o f the John-Sachs theorem to d o u b le zigzag chains. A new a p p ro ac h to the K enum eration is introduced by nonlinearly d ep en d en t recurrence relations. F inally a new type o f c o m b in a to rial K form ulas in term s o f determ inants and based on the John-S achs theorem , is introduced. T he new en u m eratio n techniques are applied to m ultiple zigzag chains.

Introduction

A(n,m)

The research on Kekule structures in conjugated hydrocarbons has been intensivated during the last years. Only in 1985 and 1986 more than 50 papers have appeared dealing with the en um eratio n of Kekule structures and closely related topics. The references [ 1 - 7 ] provide only a representative sample. A great n u m b e r of chemical applications of Kekule structures is known and have been reviewed several times [8 - 1 0 ]; for an exhaustive list of references on this matter, see also [ 1 1 ]. T he present work is a continuation of the analysis of the num ber of Kekule structures for m ultiple zigzag chains [10. 12]. This class of benzenoid hydrocarbons was recognized as an im portant class from the beginning o f the systematic e num eratio n of Kekule structures for classes o f benzenoids [13, 14]. The multiple zigzag chains were included in several later works [2 , 1 5 - 1 7 ] before our previous systematic studies [ 1 2 ]. Let the multiple zigzag chain be designated [12] by A (/?./>;); see Figure 1. F u rth e rm o re , the n u m b e r

* P.O. Box 60. Y U -34000 K ragujevac. R eprint requests to Prof. S. J. Cyvin. D ivision o f Physical C hem istry. T he U niversity o f T ro n d h eim , N -7034 T rondheim -N T H . N orw ay.

A(n,m,l)

Fig. 1. D efinition o f the notatio n for the class o f m u ltip le zigzag chains and the auxiliary class o f incom plete m ultiple zigzag chains.

of Kekule structures ( K ) o f A (/?. m) will be denoted by Z n(m). In the previous studies [ 1 2 ] we derived by very laborious calculations the recurrence relations for Z n{m) with fixed values o f n up to n = 5. In the present work we report an alternative form o f the recurrence relations for Z„(m). A new expression of Z„(m) for arbitrary n is offered. In this analysis the very useful method o f fragmentation due to R andic [18] was employed. In the subsequent part o f this work the newly developed enum eration techniques [ 1 1 ] based on the John-Sachs theorem [19] were applied to m ultiple zigzag chains in general and double zigzag chains in particular.

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S. J. Cyvin et al. • Enum eration of Kekule Structures for M ultiple Zigzag Chains T able 1 a. N um erical values o f

K {A(n, m. /)} = Z^(am) for n = 1,2, 3.

Z \ n (m)

Zp(m) 1

1 0

0 0

0 1

1 0

2

1

2

3

1

3 5

3

5

6

4

6

11

10

14 31 70 157 353 793 1 782 4 004

25 56 126 283 636 1 429 3211 7 215

14 31 70 157 353 793 1 782 4 004 8 997

1

1 0

0 1

1 2

_

2

1 — 1

1 1

0 1 2

3 4 5

8

13

13

21

6

21

7

34 55 89 144

34 55 89 144 233

8

9 10

/= 0

1 1

3 5

1 I

8

T able 1 b. N um erical values o f

m

Z (3n (m) 2

/= 0

- 3

1 1

K |A(/z. in, / ) }

=

_

- 3 2 I 0

1 2

3 4 5 6

7 8

9 10

=0

Z ^ i m ) for

1

3

4

0

0

0

0

0

0

0

1

1 1

1 2

1

1

I

3

5 15 55 190 671 2 353 8 272 29 056 102 091 358 671

9 29 105 365 1 287 45 1 6 15 873 55 759 195 910 688 286

41 146 511 1 798 6 314 22 187 77 946 273 856 962 142

1

1 1

n ==

2

3

0 0 11 2

0 0

0 1

7 19 56 160 462 1 329 3 828

30 85 246 707 2 037 5 864 16 886 48 620

n = 4 and

1

11 021

31 735 91 376

1I

1i

3 9 26 75 216 622 1 791 5 157 14 849 42 756 123 1 11

4 10

30 85 246 707 2 037 5 864 16 886 48 620 139 997

5.

z