Enumeration of benzenoid series having a constant number of isomers

Volume 176,number 6

CHEMICALPHYSICSLETTERS

I February 1991

Enumeration of benzenoid series having a constant number of isomers Jerry Ray Dias Department of Chemistry, University of Missouri, Kansas City, MO 6411 O-2499, USA

Received I3 August 1990;in final form 8 November 1990

Two classes of constant-isomer benzenoid series have been identified. One class is topologicallyunique and the other forms a pairwise topologicallyequivalent class; the former class possesses isomer numbers occurring once and the latter class has isomer numbers occurring twice. These two classes appear in both strictly peri-condensed and total resonant sextet benzenoids.

1. Introduction It is well known that as the number of carbons in an alkane hydrocarbon increases its number of isomers increases [ 11. Our work has shown that there exist benzenoid series whose number of isomers remains constant as the number of carbons increases [ 2,3]. The benzenoid classes that exhibit these characteristics are strictly peri-condensed and, as shown herein, essentially strain-free total resonant sextets [ 3,4]. These are the very benzenoids that are predicted to be among the most stable ones and the most likely ones to be formed pyrolytically. Thus, these benzenoids are very important and have been the principal focus of our research. Herein, we show that a spectacular congruency exists between these two sets of benzenoids. Not only are the isomer numbers between these two benzenoid sets the same, but a oneto-one matching exists in the symmetries between the corresponding benzenoid members.

2. Conceptual tools Using the formula periodic table for benzenoids (table PAH6) and the excised internal structure concept, many strictly peri-condensed benzenoid series possessing a constant number of isomers have been identified [ 21. Strictly peri-condensed benzenoids have all their internal vertices connected and 0009-2614/91/$03.50

have no cata-condensed appendages. Pyrene (C16HID) has ethene, coronene (CZ4HIr) has benzene, and ovalene (&HL4) has naphthalene as excised internal structures and are strictly peri-condensed benzenoids that are members of the oneisomer series found in table 1, Starting with a set of strictly peri-condensed isomers and successively circumscribing with a 2N, + 6 carbon-atom perimeter and incrementing the number of H atoms by six generates the constant-isomer series given in tables 1 and 2. Table 2 contains the constant-isomer series of the less stable odd carbon radical benzenoids. It should be noted that the isomer numbers in tables 1 and 2 alternate between singlet and doublet occurrence [ 31. Total resonant sextet benzenoid hydrocarbons have formula carbons (N,-) divisible by six and represent the most kinetically stable subset of isomers [ 41. For example, triphenylene is the only total resonant sextet benzenoid of the five benzenoid isomers of the formula of CISHIZ. To generate the constant-isomer series of these total resonant sextet benzenoids, one constructs their excised internal structures from the most condensed total resonant sextets of a given number of carbons by converting them to permethylenyl derivatives which are then circumscribed by a perimeter of carbon-atoms with the total number of carbons increasing by 3N, + 18 for the two steps. As will now be shown, the number of sextet isomers and the topological features of the sextet benzenoid members of these constant-sextet-isomer

@ 1991- Elsevier Science Publishers B.V. (North-Holland)

559

1 February 199 I

CHEMICAL PHYSICS LETTERS

Volume 176, number 6

Table I Constant-isomer series of even strictly peri-condensed benzenoids

Table 2 Constant-isomer series of odd strictly peri-condensed benzenoids

Initial term

Initial term

Number of isomers

Series

(p=l) NC

W6

G& C,&K CZZHG ‘AH,, C.wH,, G,H I8 C,& CxHx C&B CKXHX C124H28

G&30 CwHx C,UH,~ GO6HM Cd38

&ho G&, C,,oH,, GoHw

6p'

NH

bp’t4p 6p?t8pt2 6p?t 12p+4 bp’t 16p+8 6p’t2Opt14

6~ 6pt2 6pt4 6pt6 6pt8 6p+lO

6$+24pt20 bp’t28pt28 6p’t32pt38 6pzt36p+48

6pt 12 6pt 14 6pt 16 6pt 18

6p't4Op+60

6pt20

6pzt44pt74 6p’t48pt88 6p~t52ptlO4 6p1t56pt122 6p’t6Optl40 6p2t64pt160 6p’t68pt 182 6pzt72pt204 6$+76pt228 6pLt80pt254

6pt22 6pt24 6pt26

6pt28 6pt30 6pt32 6pt34 6pt36 6pt38 6pt40

I I I 2(l)“’ 3 (1) 3(l) 7 (2) 12 (4) 12 (4) 27 (12) 38 (19) 38 (19) 86 (47) 128 (71) 128 (71) 428 b’ 616 616 1265 xl800 ~1800”

” No. of radical isomers is givenin parentheses. ‘) Sum of radicaland nonradicalisomers. ‘) Predicted value based on induction.

series matches one-for-one with the properties previously obtained for the strictly peri-condensed constant isomer benzenoids [ 51.

3. Enumeration of total resonant sextet benzenoids Fig. 1 shows the first generation benzenoids for the one-isomer and corresponding (permethylenyl derivatives of the) one-sextet-isomer series given in tables 1-4. Circumscribing benzene by an 18 carbonatom perimeter and incrementing with 6 H gives coronene ( CZ4H12).Circumscribing coronene by a 30 carbon-atom perimeter and incrementing with 6 H gives circum(30)-coronene ((&H,*). Successively repeating this process leads to the relevant one-isomer series. Similarly, starting with the base structures (p=l) of naphthalene (C,,H,), pyrene (G6HIO), phenalenyl (C13H9), benzo[cd]pyrene (ClsHI1), and benzo[bc]coronene (C27H,3), suc560

Number of isomers

Series

(p=l)

Nc

N”

6p2t6pt 1 6p’tlOpt3 6pZ+14p+7 6p2t18p+ll 6pft22pt 17 6p2t26pt25 6p*t30pt33 6pZt34pt43 6p2t38ptSS 6pzt42pt67 6p2t46pt81 6pz+50pt 97 6pzt54pt 113

6pt3 6ptl

1 1 I

6pt9 6ptll

4

6pt5

2

6pt13

4

6p-t15

13

6pt17 6pt 19 6pt21

20

6pt23

14

20 48

6pt25

74

6pt27

174

6pZ+5Xpt131

6pt29

258

6p1tb2pt 6p2t66pt

6pt31

258

6pt33 6pt35 6pt37

550

151 171

6p2t70pt193

6p2t74pt217

796 796"'

a) Predicted value based on induction.

cessively circumscribing them with carbon-atom perimeters and incrementing with 6 H leads to the other one-isomer strictly peri-condensed benzenoid series. Converting benzene to hexamethylenylbenzene (CIZH12), circumscribing with a 30 carbon-atom perimeter, and incrementing with 6 H gives the total resonant sextet benzenoid, hexabenzo [ bc,eJ; hi,kl,no,qr]coronene (&H18). From benzene to hexabenzocoronene the number of carbons was increased by 3iyHt IS and the number of hydrogens was increased by 12. These incrementation relationships are general. Converting hexabenzocoronene to a permethylenyl derivative and circumscribing with a carbon-atom perimeter leads to the third generation (p= 3) total resonant sextet benzenoid member of the one-sextet-isomer series starting with C6H6 in table 3. All the first generation (p= 1) permethylenyl excised internal structures of the one-sextetisomer series are presented in fig. lj and circumscribing them with carbon-atom perimeters leads to the second generation (p=2) total resonant sextet benzenoid members of the one-sextet-isomer series. The algorithm employed for generating the isomer numbers in tables 3 and 4 uses our prior depictions [3,4] and reduces to the same algorithm [ 61 used to

Volume 176, number 6

CHEMICAL PHYSICS LETTERS

1 February 1991

0

ii%k 00

0

0

5c

Fig. I. Excised internal structures for the one-isomer and one-sextet-isomer series. Table 3 Constant-sextet-isomer series of total resonant sextet benzenoids (analogous to the even carbon constant-isomer series in table 1) Initial term

Series

Number of sextet

k=l) NC

NH

18p2-l8p+6 18p2-6p 18p2+6p 18p*+18~ 18p2+30pt6 18p2+42pt 18 18p2+54p+30 18p2+66pt48 18p2+78pt72 18p2+90pt96 18p2+ 102p+ 126 18p2t 114pt 162 18p2+126pt 198 18p2+138p+240 18p2+150pt288 18p2+162pt336 18p2+174pt390 18p2+186pt450 18p2+198pt510 18p2+210pt576 18p2+222pt648

l2p-6 12p-2

isomers

Table 4 Constant-sextet-isomer series of total resonant sextet benzenoids (analogous to the odd carbon constant-isomer series in table 2 ) Initial term

C&h

C,,H,,

W+I~ GLHIII C54H22

GsHz6 CIDIHX, C&J, C&J~ Go& Cz46H46 C,,& C34A4

CB& C&&Z C5NH66

GszH,, C,,,H,, C,,,H,, Cso& G&W,

12p-t2 12p+6 12ptlO 12pfl4 l2pt 18 12pt22 12pt26 12pt30 12pt34 12pt38 12p+42 12p+46 12p+so 12pt54 12pt58 12pt62 12pt66 12pt70 12pt74

1

1 1 3 4 4 9 16 16 39 51 57”’ 133 199 199 428 616 616 1265 1800 1800

a) This value and all values below are predicted by induction.

(P=

Series

Number of sextet isomers

1)

CIBHU C3oH16 Cd20 GbH24

r&Hz8 C, x,H,z C, ~oH,G C,W,H~O CZZSH~~ C270H48

CWH~Z C372H56 C426H60 Cd64 C551H68 Cbl8H72 C690H76 G68H80

NC

h’H

18pZ 18p2t 12~ 18p2t24pt6 18p2t36pt 12 18p2+48p+24 18p2+60pt42 18p2+72pt60 18pz+84pt84 18p2+96pt 114 18p2r 108pt 144 18p2+120pt 180 18p2+132pt222 l8p*t 144pt 264 18p*- 156pt312 18p’t 168pt366 18p2+180pt420 18p’t 192pt480 18p2+204pt546

l2P 12pt4 12pt8 12ptl2 12pt16 12pt20 l2pt24 12pt28 12pt32 12pt36 12p+40 12pt44 12pt48 12pt52 12pt56 12pt60 12pt64 12pt68

1 1

I 2 4 4 13 20 20 48 74 74 a) 174 258 258 550 796 796

a) This value and all values below are predicted by induction.

561

Volume 176.number 6

CHEMICALPHYSlCS LETTERS

‘3h

I February 1991

c2*

‘Zh

Fig. 2. The only three total resonant sextet isomers of &His and ClosH3,,which are the first and second generation members of the three-sextet-isomer seriesbeginning with Cs6Hi8, obtain the isomer numbers

in tables 1 and 2.

4. Topological characteristics Fig. 1 clearly shows that the one-isomer series of the strictly peri-condensed benzenoids starting with benzene (DC,,), naphthalene (DZh), and pyrene (D2,,) have symmetries which correspond to those of the one-sextet-isomer series of the total resonant sextet benzenoids starting with benzene (DBh), biphenyl ( D2,,), and dibenzoli;p,q] tetracene ( DJ, respectively. Similarly, the one-isomer series of the odd carbon strictly peri-condensed benzenoids starting with phenalenyl (Dr,,), benzo[cd]pyrene (C,,), and benzo [ bclcoronene (C,,) have symmetries that match one-to-one with those of the one-sextet-isomer series of the total resonant sextet benzenoids starting with triphenylene ( D3h), tribenzo vg,ij, rstjpentaphene (C,,), and naphtho[cde-1,2,3,4] -circum (30)-hexamethylenylbenzene (C,,), respectively. Comparison of fig. 2 herein with fig. 1 in ref. [ 51 further illustrates the above congruency between the isomer numbers and topological characteristics of the constant-isomer series of strictly per&condensed and total resonant sextet benzenoids. Fig. 2 gives the only three total resonant sextet benzenoid isomers out of 337 benzenoid isomers having the formula of &Hi8562

These three isomers match one-to-one in symmetry with the only three isomers of the formula C22H12. Transforming these three CJ6H ,* resonant sextet isomers to permethylenyl derivatives and circumscribing with a 54 carbon-atom perimeter generates the only three Cio8Hj0 total resonant sextet benzenoids possible (fig. 2) which match one-to-one the only three benzenoid isomers of the formula C5zHIB. 5. Summary Tables 1 and 2 present constant-isomer series of strictly peri-condensed benzenoids having isomer numbers which precisely correspond to the isomer numbers of the constant-isomer series of the total resonant sextet benzenoids given in tables 3 and 4, respectively. Thus, there exists a spectacular isomorphism between these two classes of benzenoids. References [ 11D.J.Klein, J. Chem. Phys. 75 (1981) 5186. [2] J.R. Dias, Can. J. Chem. 62 (1984) 2914; J. Chem. Inf. Comput. Sci. 24 (1984) 124;30 (1990) 61. [3] J.R. Dias, Handbook of polycyclic hydrocarbons, Part A (Elsevier, Amsterdam, 1987). [4] J.R. Dias Thermochim. Acta 122 (1987); J. Mol. Struct. THEOCHEM 185 ( 1989) 57. [5] J.R. Dias, Theoret. Chim. Acta 77 (1990) 143. [6] JR. Dias, 1. Chem. Inf. Comput. Sci. 30 (1990) 251.