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Isomers of polycyclic conjugated hydrocarbons with arbitrary ring sizes: Generation and enumeration
Computers Chem. Vol. 17, No. 3, pp. 291-296, Printed in Great Britain. All rights reserved
0097~8485/93 $6.00 + 0.00 Copyright 0 1993 I’ergamon Press Ltd
1993
ISOMERS OF POLYCYCLIC CONJUGATED HYDROCARBONS WITH ARBITRARY RING SIZES: GENERATION AND ENUMERATION J. BRLJNVOLL, B. N. CYVIN and Division
of Physical
Chemistry,
The University
(Received 20 October
of Trondheim,
S. J. CYVIN* N-7034 Trondheim-NTH,
1592; in revised form 25 January
Norway
IW3)
Abstraet~ompletely condensed polycyclic conjugated hydrocarbons are studied with respect to their numbers of C,H, isomers ( # I) for arbitrary ring sizes. Direct combinatorial methods are applied and baaed on the ring-edge sum (Zq), viz the sum of all edges in the q-membered rings taken individually. Explicit expressions of # iare reported for two rings (r = 2) and three rings (r = 3). The latter case (r = 3) splits into structures without any internal carbon (hi = 0) and those with one internal carbon (yi = 1). Finally, the corresponding problem for r = 4, q = I is solved by means of computer programmmg.
INTRODUCMON enumeration of isomers is a well-established branch of chemistry, especially in the organic chemistry and now also as a part of computational chemistry. The present work deals with the computational aspects of the isomers of a certain class of polycyclic conjugated hydrocarbons (Chu, 1964). If only sixmembered (benzenoid) rings are present the subclass consists of benzenoid hydrocarbons (Gutman & Cyvin, 1989). Enumerations of chemical isomers started more than a hundred years ago (Cayley, 1875) by the classical studies of alkanes, while the systematic enumerations of benzenoid hydrocarbons are of a relatively new date (Balaban & Harary, 1968; Harary & Read, 1970; Balasubramanian et al., 1980; Knop et al., 1983). Nevertheless, a great deal of information on the numbers of benzenoid isomers has been accumulated to date; the reader is referred to a comprehensive review by Brunvoll et al. (1992) and references cited therein. Much less work has been done on the isomers of polycyclic conjugated hydrocarbons when it is allowed for other rings than six-membered. Dias (1988) has emphasized the chemical interest of such studies and focused his attention on the polycyclic C,H, isomers where C,,H, represents a benzenoid formula, but he has only performed rudimentary generations of such isomers. In the present work the polycyclic conjugated C, H, isomers are treated systematically under a rather general viewpoint: the formula C,H, may or may not include benzenoid isomers, and no restrictions are
The
imposed on the ring size. Combinatorial expressions in explicit forms are reported for the numbers of isomers of the simplest relevant structures. These
expressions
were derived
* Author for correspondence.
from very simple principles
by elementary combinatorics. They were also supported and extended by computer programming. DEFINITIONS
AND PRINCIPLES
The polycyclic conjugated hydrocarbons, which are considered, should be completely condensed in the sense that every carbon atom should belong to at least one ring. Only secondary and tertiary carbon atoms are permitted, and the secondary carbons (to which the hydrogen atoms are attached) should all be found at the outer boundary of the hydrocarbon. This specification excludes molecules with holes (Hall, 1988) as, e.g. kekulene (Staah & Diederich, 1983). The class of hydrocarbons as specified above, can be represented by geometrical systems of polygons, which again represent chemical graphs (TrinajstiC, 1992). These polygonal systems consist of simply connected polygons, where any two polygons either share exactly one edge, or they are disjoint. A formula C,H, is associated with a polygonal system, P, as well as the corresponding hydrocarbon, Then the number of carbon atoms (n) corresponds to the total number of vertices in P, while the number of hydrogens (s) corresponds to the number of vertices of degree two. The number of polygons in P, which corresponds to the number of rings, shall be identified by the symbol r. Another important quantity (among the invariants of P) is the number of internal vertices, denoted by fli. An internal vertex of P is, by definition, shared by three polygons. Here we shall speak about an internal carbon atom when it is shared by three rings. The number of C,H, isomers among the polycyclic conjugated hydrocarbons is defined as the number of nonisomorphic polygonal systems compatible with the given formula (C,H,). Two polygonal systems are nonisomorphic when they cannot be brought into 291
J. BRUNVOLL et 01.
292
Fig. 1. The three C, H, (indenyl) isomers. The numerals, here and in the subsequent figures, indicate ring sizes or q values.
each other by a rotation or a reflection, or a combination of these two operations, occasionally accompanied by deformations of the polygons. The symmetry of a polygonal system P shall presently refer to the highest possible symmetry of P, where also the geometrical planarity is preserved. This symmetry conforms often, but not always, with the observed or expected symmetry of the corresponding hydrocarbon. A polycyclic conjugated hydrocarbon may contain q-membered rings for q = 3,4,5,. . . , q(max). If the number of q-membered rings is denoted by r4 (viz. rX, r,, r,, . . .), then r=rj+re+rTS+...+rHmax,.
(1)
The number of rings, independent fixed for the class of hydrocarbons ation by the expression
of their sizes, is under consider-
r =f(n
-s)+
1.
(2)
Hence we can formulate as a first principle: all isomers of poIycyclic conjugated hydrocarbons for a given formula C,H, have the same number of rings (r)Another quantity, the ring-edge sum, is defined as Zq = 3tj + 4r, + Sr, + . * ++ q(max)rqrmx)
(3)
and indicates the sum of ring edges taken individually. For naphthalene, for instance, Xq = 12. The quantity Zq seems not to have been fully exploited in the previous research of polycyclic isomers, although it has proved to be very useful. It is easily found: Zq=n+2r+nj-2=2n-s+nn,.
derived by elementary methods. In these expressions, the (a) “floor” and (b) “ceiling” functions are employed: (a) LXJ is the largest integer not larger than x; (b) rxl is the smallest integer not smaller than x. The problem is rather trivial for r = 1 and 2, but included here for the sake of completeness and the chemical relevance. Already for r = 3 the combinatorics which are involved, exhibit some complexity. The derived expressions in these cases were checked by simple computer programming. A description of the algorithm gives at the same time a recipe for a generation of the isomers in question. A conjugated hydrocarbon with one or two rings (r = 1,2) cannot have any internal carbon, i.e. n, = 0. For r = 3, on the other hand, structures with n, = 0 and ni = 1 are possible. One ring One conjugated q-membered ring is associated with the formula C,H,. According to the present definitions there is exactly one isomer for each q in this trivial case. The smallest hydrocarbons of this category are well known: cyclopropenyl, cyclobutadiene, cyclopentadienyl and of course benzene for q = 3,4, 5 and 6, respectively. Also cyclooctatetraene (q = 8) has been synthesized, as well as several macrocyclic conjugated hydrocarbons among the annulenes: [I4]annulene (Sondheimer & Gaoni, 1960), [18]-, [24]- and [30]annulene (Sondheimer et al., 1962). Two rings In the case of r = 2 (ni = 0), the appropriate formula has the form C,H, _ 2, where n = Eq - 2; cf. equations (2) and (4). For a given Zq = q, + q2, all the isomers are clearly generated by letting q, assume the values q, = 3,4,5, . , L $q 1. Hence the number of isomers # I in this case is
If Zq is even, to one isomer Otherwise the mathematical
(4)
we can formulate the secondprinciple as: all isomers of polycyclic conjugated hydrocarbons for a given formula C, H, and with a given number of internal carbon atoms (n,), have the same ring-edge sum (Tq).
Introductory
remarks
In this section
the complete solutions are reported for the generation and enumeration of the C,H, s polycyclic conjugated hydrocarbon isomers for r = 1, 2 and 3. Explicit combinatorial formulas for the numbers of isomers, # 1, as functions of Zq were
then q, = q2 will occur and correspond belonging to the symmetry group D,. isomers belong to the C,, symmetry. In terms:
1 -r$cql+L$qj
#WA,)=
In consequence,
GENERATION AND ENUMERATION OF ISOMERS WITH THREE RINGS OR LESS
(5)
#Z=L$qJ-2.
# z(c,,)
(‘5)
= r+zqT - 3.
(7)
As an example, there are 4 C,,H, (naphthalene) isomers of polycyclic conjugated hydrocarbons, viz. 67,, q2) = (3,% (4, Q ($71, @,6), in consistency
00 6
D 14
I
6
14
II
16
6
III
Fig. 2. Benzo[a] [14]annulene (I), benzo[d][l4~nnulene and benzo[lg]annulene (III).
(II),
Isomers and ring sizes
293
Table 1. Numbers of isomers of polycyclic conjugated hydrocarbons with three rings and no internal carbon: r = 3, n, = 0 Q
Fig. 3. Model of a polycyclic conjugated hydrocarbon
with
I = 3, n, = 0.
with Dias (1988), who depicted these four structures and identified three of them with known chemical compounds. Here we show the three CPH7 indenyl (q, = 5, qz = 6) isomers cf. Fig. 1. Relevant chemical examples with larger rings are found among benzoannulenes; see Fig. 2. It should be noticed that, in the framework of the present definitions, it is not distinguished between the two chemical isomers of C,sH,, benzo[ 14]annulene, viz. I and II of Fig. 2 (Staab ez al., I979b). But there are eight C,8H,6 isomers of polycyclic conjugated hydrocarbons when it is allowed for all ring sizes. Correspondingly, there are ten &H, (benzo[ 1Slannulene) isomers. The depicted representative (III in Fig. 2) has been synthesized (Staab et al., 1979a). Three rings without internal carbons Consider the case of r = 3, n, = 0. Then the polycyclic conjugated hydrocarbons to be considered have a form as shown in Fig. 3. The formula is C,,Hn--4r where n = Zq - 4. In Zq = i + j + k, i indicates the size (q-value) of the middle ring. We adopt as a convention for the end rings: j 2 k. The isomers of the category under consideration are generated in the following way. For every i = 4,5,6, . . _, Zq - 6, let k = 3,4, 5, . . . , L@q J - 2 andj = Cq - i - k. Let the combination of the three ring sizes be coded (i)(j, k). Then each code represents Li/2] - I isomers, corresponding to the number of distinct ways the two end rings can be annelated to the middle ring. When j = k and i is even there exists one structure of D, symmetry; otherwise when j = k the structure is C,,. Also when j # k and i is even there exists one structure of C,,. In all other cases the structures belong to C,. The following combinatorial expression was deduced for the relevant number of isomers # I = ; L)L-q - 4J(zq
- 7)rfzq
- 31.
Fig. 4. The four C,,H, (biphenylene)
(8)
F0llflula
D2h
G
Gb GH,
:,
0 2
f: I4 I5 I6 17 18
C, H, GH, Go& C,, “7 CnHa C,,H, CM%
02 3 0 4 0 5
62 6 12 12 20 20
:‘o
‘&W, CM’%,
06
G,H,,
0
G HI, ‘%H,, ‘&HI, G, HI, C,, “I, GW, ‘&Hn, GJ% Cx Hz
7 0 8 0 9 0 10 0 II
30 30 42 42 56 56 12 ;:
21 22 23 24 2s 26 27 28 29 30
For the symmetrical #I(&)
= (1 -
# I(c,,)
Total
C*
IO II
I 2
: 2I 5 8 14
: 14 20 30 40 55
:: 40 55 70 91 II2 I40 lb8 204 240 285 330 385
90 I IO I IO
E 112 140 lb8 204 240 285 330 385 440 506
structures:
r;Ql+L&D(f.%
= r;zq
- 41
x r;_zq
-
- 4) 51.
(9) WJ)
The above results are applicable to the C,,H,, anthracene isomers (Zq = 18). Then equations (8) prescribes 55 isomers with ni = 0, which according to (9) and (10) are distributed into the symmetry groups as 5D,,K+ ZOC,,. + 3OC,. The maximum polygon size has q = 12 in the five isomers to be coded as (12)(3,3). If only the polygon sizes q ,< 9 are taken into account one arrives at 35 C,4HI0 isomers without internal carbons in consistency with the depictions of Dias (1988). Another chemically important case is associated with the formula C,,H, for Zq = 16. Here the number of isomers is 30 (40, + IZC,, + 14C,). The four D, structures are depicted in Fig. 4 and include biphenylene, viz. (4)(6,6). Additional examples are, for instance, fluorenyl(5)(6,6) and the two isomers of benzindenyl, viz. (6)(6,5). But also an abundance of examples of chemical interest is found among the macrocyclic conjugated hydrocarbons. Theoretical studies have been made on the five chemical isomers of C,,H,, dibenzo[l4]annulene and three isomers of CZ6H,, dibenzo[ 1Ilannulene (Stollenwerk et nl., 1983) depicted in Fig. 5. These isomers correspond to (14)(6,6) and (18)(6,6), respectively. It should be noticed that the theoretically possible isomers of
isomers with no internal carbons and symmetry
D,,
294
J.
BRUNVOLLet al.
Fig. 5. Five chemical dibenzo[l4]annulenes (top row) and three. dibenzo[l8]annulenes (bottom row). The first and third structure are identical isomers in the framework of our definitions.
polycyclic conjugated hydrocarbons with the codes (14)(6,6) and (18)(6,6) are six and eight, respectively. If all ring sizes are taken into account, then there are 285 C=H,* (Zq = 26) and 506 C,,H,, (Zq = 30) relevant isomers. The numbers of the isomers for Zq d 30 are collected in Table 1. These numbers display different repeating patterns which easily can be verified from the explicit expressions (8)-( 10); e.g. the columns for C, and Total are shifted by two units in Lz. Three
rings
with one internal
carbon
The case of r = 3, n, = 1 is illustrated in Fig. 6. The formula reads C.H._I as in the preceding case, but now n = Zq - 5. Adopt the convention i > j >, k. A simple algorithm for the generation of the pertinent isomers is given in the following. For every ,.,., LfZqJ, let j=k,k+l,k+2 ,..., k=3,4,5 Li (Pq - 3)J and i = Zq -j - k. Introduce the coding (i, j, k). A & structure emerges for i =j = k, while the C,,, structures are associated with i =j # k or i #j = k; otherwise the symmetry is C,. The combinatorial expressions which were deduced for the appropriate numbers of isomers, read:
when Xq is odd. For the symmetrical # I(&,)
= 1 - r&Q 1 +
structures:
Lf.3J
(13)
#z(c,,)=r~~qi+r~~ql-LLf~g~--. (14) Notice that the expression for # Z(C,) + # Z(D,,) is exceedingly simple. In the first place, the above results can be used to complete the enumerations of C,4H,0 (anthracene) and C1*HB (biphenylene) isomers. For &HI0 the appropriate value of Xq is 19; equation (4). Then equations (12)-(14) yield # Z = 14(6C2, + SC,). Out of these structures there are found to be eight with q Q 9, again in consistency with Dias (1988). For C12H, (Zq = 17) the result is # Z = 10(X, + 5C,). One of these isomers is acenaphthene, viz. (6,6,5), which is known chemically. The numbers of the isomers under consideration for Cq $30 are collected in Table 2. Again some repeating patterns are apparent. In this case there is a shift between the columns of C, and Total by three units in Zq. In Fig. 7 an Table 2. Numbersof isomersof polycydic conjugatedhydrocarbons with three rings and one internalcarbon: r = 3, n, = 1
# Z = Li (Cq - 4) J(Cq - 6 - 3La (Zq - 4) J) (11) when Zq
is even, and
#I=f(Zq-7)+L;(Zq-7)J(Cg-9-3L~(Zq-7)J) (12)
Fig. 6.
Model of a polycyclic conjugated
r = 3, II,= 1.
hydrocarbon with
IO
C,k
0
I
0
II12 13 14 15 t8
C, CT”, “2
C, “4 C,“, Cm& C,, H,
0I
2I 3
0I
:
0
4
2 3 4
4 5 7 IO 8
18
‘AH, CnH,
1
:
:
12
:
C,,“,, C,,H,,
:
6
IO 8
16 14
22 21 23
C,,“,, ‘%“I, C,,H,,
01 0
6 7 8
I2 14 16
:‘: 24
24 25 26 27 28 29 30
C,H,, CwH,, Cx HI, C,,“,, C,,“,, Cll”l C,,HZI
0 1 0 :,
7 9 9 9 10 II IO
:; 24 27 30 33 37
:: 33 37 40 44 48
0 0
1
0 I
I
I
:
295
Isomers and ring sizes
Fig. 7. The three isomers of C,H, polycyclic conjugated hydrocarbons (r = 3) with n, =
illustrating example is shown: the three C, H, isomers for r = 3, ni = 1 (Cq = 12), where all the three possible symmetry groups are represented. ISOMERS WITH FOUR RINGS AND ONE INTERNAL CARBON If the above analysis could be extended to all cases with four rings the applicability to chemically interesting hydrocarbons would be increased substantially. This is a relatively complicated task, but it can be accomplished conveniently with computer aid. Four rings may be combined to polycyclic conjugated hydrocarbons with the numbers of internal carbons equal to zero, one and two. Here we give a solution for the case with r = 4 and ni = 1. Explicit expressions for the numbers of isomers were not achieved in this case, but a simple algorithm was devised for computer programming. The title case is illustrated in Fig. 8. The hydrocarbon formula reads C,H, _ 6, where n = Zq - 7. Adopt the convention i 3 j 2 k along with (i, j, k)(l) for the coding. The possible (i, j, k) structures are generated precisely as in the case for r = 3, n, = 1 (see above) wherein Xq is replaced by Zq - I. One obtains sets of combinations for I = 3,4,5, . . , Zq - 11. For every (& j, k) combination the possibilities of distinct annelations of the q = 1 rings, say a, have to be specified. Here one must take into account the symmetry group of the (i, j, k) structure, whether it is C,, C,, or D3h_ Accordingly, it was found: a(C,)=i+j+ka(C,) u(D&
12+r,
= rf (i + j + k - 12 + rj)l = ri (i tj
+ k - 12)l.
(15) (16) (17)
The number of triangles, viz. r,, may assume the values 0, 1 and 2 in the case of C,; r3 = 0 or 2 in the case of C,,; finally, r, vanishes when the symmetry of
.
I
Fig. 8. Model of a polycyclic conjugated hydrocarbon with r=4, n,= 1.
1.
(i, j, k) is D,,,. The total number of isomers is obtained on adding the a values for every (i, j, k)(l) combination. It remains to detect the symmetries of the whole (i, j, k)(l) structures. Only C,, and C, symmetries are possible. The C, symmetry is realized if and only if two of the values in (i, j, k) are equal while the third is odd, but different from 3. The computational results of this analysis are listed in Table 3 for Iq 6 50. From Table 3 we find #Z = 365 (15C,, + 35OC,) for C16H10(Zq = 23). These numbers represent a part of the important pyrene isomers, which were treated erroneously by Dias (1988). He depicted 149 out of these structures for q < 9, while, according to our analysis, this number should be 217. Among the 693 C,,H,, polycyclic conjugated hydrocarbon isomers with n, = 1 (cf. Table 3) four structures are listed in
Table 3. Numbers of isomers of polycyclic conjugated hydrocarbons with four rings and one internal carbon: r = 4, n, = I ‘ra
F0llXUJla
14 15 16 17 18 19 20 21 22 13 24 25
1 1
3 3 6 6 10 10 15 15 21 21
:; 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 : 47 48 49 50
c,..
:: 36 36 45 45 55 55 66 66 78
CaHN
97: 91 105 105 120 120 136 136 I53 I53 I71 I71 190
TOtid
C. 0
2 6 16 32 60 100 160 240 350 490 612 896 II76 1512 1920 2400 2970 3630 4400 5280 6292 1436 8736 10192 11830 I3650 IS680 I7920 204OO 23120 26112 29376 32946 36822 41040 45600
I 3 9 I9 38 66 110 170 255 365 511 693 924 1204 I548 1956 2445 3015 3685 4455 5346 6358 7514 8814 I0283 II921 I3755 IS785 I8040 20520 23256 26248 29529 33099 36993 41211 45790
I. BRUNVOLL et al.
296
c$$q$3& I
III
Fig. 9. Phenanthro[u] [ 14lannulene (I), tribenzo[u,g,m] [ 18lannulene (II), and tribenzo[a,e,i][ 12]annulene (III). Dias (1988) as chemically known. An isomer of &Hz,, phenanthro[l4]annulene (viz. 1 of Fig. 9) has been synthesized (Staab et aI., 1979b); according to Table 3 it has 4455 isomers of polycyclic conjugated hydrocarbons with n, = 1. Similarly for the C,, H,, tribenzo[ lfl]annulenes (Stollenwerk er al., 1983), one of which being depicted in Fig. 9. Here the corresponding number of isomers is 8814. Finally the interesting macrocyclic conjugated hydrocarbon C,,H,8 (Staab er al., 1971) is worth mentioning; see III of Fig. 9. This formula was found to have 3015 isomers of the category under consideration. COMPUTER PROGRAMS Computer programs were designed in order to produce all the numbers of Tables l-3. In the case of Tables 1 and 2 these computations provided a useful check on the explicit formulas, which were derived (see above). The computer programs were not saved because of their great simplicity and because they are easily retrieved from the combinatorial principles, which are described above. Acknowledgement-Financial support to B.N.C. from The Norwegian Research Council for Science and the Humanities is gratefully acknowledged.
REFERENCES Balaban A. T. % Hararv F. (19681 Tetrahedron 24. 2505. Balasubramanian K., Kaufman J. J., Koski W. S. & Balaban A. T. (1980) 3. Comput. Chem. 1, 149. Brunvoll J., dyvin.B. N. &‘Cyvin S. J. (1992) Topic. Curr. C&m. 162, 181. Cayley A. (1875) Ser. Dtsch. Chem. Ges. 8, 172. Clar E. (1964) Pofycyclic Hydrocarbons, Vols I-II. Academic Press, London. Dias J. R. (1988) Handbook of Polycyclic Hydrocarbons, Part E. Elsevier, Amsterdam. Gutman I. & Cyvin S. J. (1989) Introductian to the Theory of Benzenoid Hvdrocarbons. &xineer. Berlin. Hail G. G. (1988j Theoret. Chi& Asta.73, 425. Hararv F. & Read R. C. (19701Proc. Edinburah Math. Sot. Ser.. II 17, I. Knop J. V., Szymanski K., Jeri&wiC 2. & Trinajstii: N. (1983) J. Comput. Chem. 4, 2.7. Sondheimer F. & Gaoni Y. (1960) J. Am. Chem. Sot. 82, 5765.
Sondheimer F., Wolovsky R. & Amiel Y. (1962) J. Am. Chem. Sot. 84, 274.
Staab H. A., Graf F., Doerner K. & Nissen A. (1971) Chem. Bet-. 104, 1159. Staab H. A., Meissner U. E. & Gensler A (1979a) Chem. Ber. 112, 3907. Staab H. A., Meissner U. E., Weinacht W. & Gensler A. (1979b) Chem. Ber. 112, 3895. Staab H. A. % Diederich F. fl983) Chem. Eer. 116. 3487. Stollenwerk A. H., Kanellakopulos B., Vogler H., JuriC A. & TrinajstiC N. (1983) J. Mol. Strucf. (Theochem.) 102,377. TrinajstiC N. (1992) Chemical Graph Theory, 2nd edn. Boca Raton, Fl, CRC Press.
0097~8485/93 $6.00 + 0.00 Copyright 0 1993 I’ergamon Press Ltd
1993
ISOMERS OF POLYCYCLIC CONJUGATED HYDROCARBONS WITH ARBITRARY RING SIZES: GENERATION AND ENUMERATION J. BRLJNVOLL, B. N. CYVIN and Division
of Physical
Chemistry,
The University
(Received 20 October
of Trondheim,
S. J. CYVIN* N-7034 Trondheim-NTH,
1592; in revised form 25 January
Norway
IW3)
Abstraet~ompletely condensed polycyclic conjugated hydrocarbons are studied with respect to their numbers of C,H, isomers ( # I) for arbitrary ring sizes. Direct combinatorial methods are applied and baaed on the ring-edge sum (Zq), viz the sum of all edges in the q-membered rings taken individually. Explicit expressions of # iare reported for two rings (r = 2) and three rings (r = 3). The latter case (r = 3) splits into structures without any internal carbon (hi = 0) and those with one internal carbon (yi = 1). Finally, the corresponding problem for r = 4, q = I is solved by means of computer programmmg.
INTRODUCMON enumeration of isomers is a well-established branch of chemistry, especially in the organic chemistry and now also as a part of computational chemistry. The present work deals with the computational aspects of the isomers of a certain class of polycyclic conjugated hydrocarbons (Chu, 1964). If only sixmembered (benzenoid) rings are present the subclass consists of benzenoid hydrocarbons (Gutman & Cyvin, 1989). Enumerations of chemical isomers started more than a hundred years ago (Cayley, 1875) by the classical studies of alkanes, while the systematic enumerations of benzenoid hydrocarbons are of a relatively new date (Balaban & Harary, 1968; Harary & Read, 1970; Balasubramanian et al., 1980; Knop et al., 1983). Nevertheless, a great deal of information on the numbers of benzenoid isomers has been accumulated to date; the reader is referred to a comprehensive review by Brunvoll et al. (1992) and references cited therein. Much less work has been done on the isomers of polycyclic conjugated hydrocarbons when it is allowed for other rings than six-membered. Dias (1988) has emphasized the chemical interest of such studies and focused his attention on the polycyclic C,H, isomers where C,,H, represents a benzenoid formula, but he has only performed rudimentary generations of such isomers. In the present work the polycyclic conjugated C, H, isomers are treated systematically under a rather general viewpoint: the formula C,H, may or may not include benzenoid isomers, and no restrictions are
The
imposed on the ring size. Combinatorial expressions in explicit forms are reported for the numbers of isomers of the simplest relevant structures. These
expressions
were derived
* Author for correspondence.
from very simple principles
by elementary combinatorics. They were also supported and extended by computer programming. DEFINITIONS
AND PRINCIPLES
The polycyclic conjugated hydrocarbons, which are considered, should be completely condensed in the sense that every carbon atom should belong to at least one ring. Only secondary and tertiary carbon atoms are permitted, and the secondary carbons (to which the hydrogen atoms are attached) should all be found at the outer boundary of the hydrocarbon. This specification excludes molecules with holes (Hall, 1988) as, e.g. kekulene (Staah & Diederich, 1983). The class of hydrocarbons as specified above, can be represented by geometrical systems of polygons, which again represent chemical graphs (TrinajstiC, 1992). These polygonal systems consist of simply connected polygons, where any two polygons either share exactly one edge, or they are disjoint. A formula C,H, is associated with a polygonal system, P, as well as the corresponding hydrocarbon, Then the number of carbon atoms (n) corresponds to the total number of vertices in P, while the number of hydrogens (s) corresponds to the number of vertices of degree two. The number of polygons in P, which corresponds to the number of rings, shall be identified by the symbol r. Another important quantity (among the invariants of P) is the number of internal vertices, denoted by fli. An internal vertex of P is, by definition, shared by three polygons. Here we shall speak about an internal carbon atom when it is shared by three rings. The number of C,H, isomers among the polycyclic conjugated hydrocarbons is defined as the number of nonisomorphic polygonal systems compatible with the given formula (C,H,). Two polygonal systems are nonisomorphic when they cannot be brought into 291
J. BRUNVOLL et 01.
292
Fig. 1. The three C, H, (indenyl) isomers. The numerals, here and in the subsequent figures, indicate ring sizes or q values.
each other by a rotation or a reflection, or a combination of these two operations, occasionally accompanied by deformations of the polygons. The symmetry of a polygonal system P shall presently refer to the highest possible symmetry of P, where also the geometrical planarity is preserved. This symmetry conforms often, but not always, with the observed or expected symmetry of the corresponding hydrocarbon. A polycyclic conjugated hydrocarbon may contain q-membered rings for q = 3,4,5,. . . , q(max). If the number of q-membered rings is denoted by r4 (viz. rX, r,, r,, . . .), then r=rj+re+rTS+...+rHmax,.
(1)
The number of rings, independent fixed for the class of hydrocarbons ation by the expression
of their sizes, is under consider-
r =f(n
-s)+
1.
(2)
Hence we can formulate as a first principle: all isomers of poIycyclic conjugated hydrocarbons for a given formula C,H, have the same number of rings (r)Another quantity, the ring-edge sum, is defined as Zq = 3tj + 4r, + Sr, + . * ++ q(max)rqrmx)
(3)
and indicates the sum of ring edges taken individually. For naphthalene, for instance, Xq = 12. The quantity Zq seems not to have been fully exploited in the previous research of polycyclic isomers, although it has proved to be very useful. It is easily found: Zq=n+2r+nj-2=2n-s+nn,.
derived by elementary methods. In these expressions, the (a) “floor” and (b) “ceiling” functions are employed: (a) LXJ is the largest integer not larger than x; (b) rxl is the smallest integer not smaller than x. The problem is rather trivial for r = 1 and 2, but included here for the sake of completeness and the chemical relevance. Already for r = 3 the combinatorics which are involved, exhibit some complexity. The derived expressions in these cases were checked by simple computer programming. A description of the algorithm gives at the same time a recipe for a generation of the isomers in question. A conjugated hydrocarbon with one or two rings (r = 1,2) cannot have any internal carbon, i.e. n, = 0. For r = 3, on the other hand, structures with n, = 0 and ni = 1 are possible. One ring One conjugated q-membered ring is associated with the formula C,H,. According to the present definitions there is exactly one isomer for each q in this trivial case. The smallest hydrocarbons of this category are well known: cyclopropenyl, cyclobutadiene, cyclopentadienyl and of course benzene for q = 3,4, 5 and 6, respectively. Also cyclooctatetraene (q = 8) has been synthesized, as well as several macrocyclic conjugated hydrocarbons among the annulenes: [I4]annulene (Sondheimer & Gaoni, 1960), [18]-, [24]- and [30]annulene (Sondheimer et al., 1962). Two rings In the case of r = 2 (ni = 0), the appropriate formula has the form C,H, _ 2, where n = Eq - 2; cf. equations (2) and (4). For a given Zq = q, + q2, all the isomers are clearly generated by letting q, assume the values q, = 3,4,5, . , L $q 1. Hence the number of isomers # I in this case is
If Zq is even, to one isomer Otherwise the mathematical
(4)
we can formulate the secondprinciple as: all isomers of polycyclic conjugated hydrocarbons for a given formula C, H, and with a given number of internal carbon atoms (n,), have the same ring-edge sum (Tq).
Introductory
remarks
In this section
the complete solutions are reported for the generation and enumeration of the C,H, s polycyclic conjugated hydrocarbon isomers for r = 1, 2 and 3. Explicit combinatorial formulas for the numbers of isomers, # 1, as functions of Zq were
then q, = q2 will occur and correspond belonging to the symmetry group D,. isomers belong to the C,, symmetry. In terms:
1 -r$cql+L$qj
#WA,)=
In consequence,
GENERATION AND ENUMERATION OF ISOMERS WITH THREE RINGS OR LESS
(5)
#Z=L$qJ-2.
# z(c,,)
(‘5)
= r+zqT - 3.
(7)
As an example, there are 4 C,,H, (naphthalene) isomers of polycyclic conjugated hydrocarbons, viz. 67,, q2) = (3,% (4, Q ($71, @,6), in consistency
00 6
D 14
I
6
14
II
16
6
III
Fig. 2. Benzo[a] [14]annulene (I), benzo[d][l4~nnulene and benzo[lg]annulene (III).
(II),
Isomers and ring sizes
293
Table 1. Numbers of isomers of polycyclic conjugated hydrocarbons with three rings and no internal carbon: r = 3, n, = 0 Q
Fig. 3. Model of a polycyclic conjugated hydrocarbon
with
I = 3, n, = 0.
with Dias (1988), who depicted these four structures and identified three of them with known chemical compounds. Here we show the three CPH7 indenyl (q, = 5, qz = 6) isomers cf. Fig. 1. Relevant chemical examples with larger rings are found among benzoannulenes; see Fig. 2. It should be noticed that, in the framework of the present definitions, it is not distinguished between the two chemical isomers of C,sH,, benzo[ 14]annulene, viz. I and II of Fig. 2 (Staab ez al., I979b). But there are eight C,8H,6 isomers of polycyclic conjugated hydrocarbons when it is allowed for all ring sizes. Correspondingly, there are ten &H, (benzo[ 1Slannulene) isomers. The depicted representative (III in Fig. 2) has been synthesized (Staab et al., 1979a). Three rings without internal carbons Consider the case of r = 3, n, = 0. Then the polycyclic conjugated hydrocarbons to be considered have a form as shown in Fig. 3. The formula is C,,Hn--4r where n = Zq - 4. In Zq = i + j + k, i indicates the size (q-value) of the middle ring. We adopt as a convention for the end rings: j 2 k. The isomers of the category under consideration are generated in the following way. For every i = 4,5,6, . . _, Zq - 6, let k = 3,4, 5, . . . , L@q J - 2 andj = Cq - i - k. Let the combination of the three ring sizes be coded (i)(j, k). Then each code represents Li/2] - I isomers, corresponding to the number of distinct ways the two end rings can be annelated to the middle ring. When j = k and i is even there exists one structure of D, symmetry; otherwise when j = k the structure is C,,. Also when j # k and i is even there exists one structure of C,,. In all other cases the structures belong to C,. The following combinatorial expression was deduced for the relevant number of isomers # I = ; L)L-q - 4J(zq
- 7)rfzq
- 31.
Fig. 4. The four C,,H, (biphenylene)
(8)
F0llflula
D2h
G
Gb GH,
:,
0 2
f: I4 I5 I6 17 18
C, H, GH, Go& C,, “7 CnHa C,,H, CM%
02 3 0 4 0 5
62 6 12 12 20 20
:‘o
‘&W, CM’%,
06
G,H,,
0
G HI, ‘%H,, ‘&HI, G, HI, C,, “I, GW, ‘&Hn, GJ% Cx Hz
7 0 8 0 9 0 10 0 II
30 30 42 42 56 56 12 ;:
21 22 23 24 2s 26 27 28 29 30
For the symmetrical #I(&)
= (1 -
# I(c,,)
Total
C*
IO II
I 2
: 2I 5 8 14
: 14 20 30 40 55
:: 40 55 70 91 II2 I40 lb8 204 240 285 330 385
90 I IO I IO
E 112 140 lb8 204 240 285 330 385 440 506
structures:
r;Ql+L&D(f.%
= r;zq
- 41
x r;_zq
-
- 4) 51.
(9) WJ)
The above results are applicable to the C,,H,, anthracene isomers (Zq = 18). Then equations (8) prescribes 55 isomers with ni = 0, which according to (9) and (10) are distributed into the symmetry groups as 5D,,K+ ZOC,,. + 3OC,. The maximum polygon size has q = 12 in the five isomers to be coded as (12)(3,3). If only the polygon sizes q ,< 9 are taken into account one arrives at 35 C,4HI0 isomers without internal carbons in consistency with the depictions of Dias (1988). Another chemically important case is associated with the formula C,,H, for Zq = 16. Here the number of isomers is 30 (40, + IZC,, + 14C,). The four D, structures are depicted in Fig. 4 and include biphenylene, viz. (4)(6,6). Additional examples are, for instance, fluorenyl(5)(6,6) and the two isomers of benzindenyl, viz. (6)(6,5). But also an abundance of examples of chemical interest is found among the macrocyclic conjugated hydrocarbons. Theoretical studies have been made on the five chemical isomers of C,,H,, dibenzo[l4]annulene and three isomers of CZ6H,, dibenzo[ 1Ilannulene (Stollenwerk et nl., 1983) depicted in Fig. 5. These isomers correspond to (14)(6,6) and (18)(6,6), respectively. It should be noticed that the theoretically possible isomers of
isomers with no internal carbons and symmetry
D,,
294
J.
BRUNVOLLet al.
Fig. 5. Five chemical dibenzo[l4]annulenes (top row) and three. dibenzo[l8]annulenes (bottom row). The first and third structure are identical isomers in the framework of our definitions.
polycyclic conjugated hydrocarbons with the codes (14)(6,6) and (18)(6,6) are six and eight, respectively. If all ring sizes are taken into account, then there are 285 C=H,* (Zq = 26) and 506 C,,H,, (Zq = 30) relevant isomers. The numbers of the isomers for Zq d 30 are collected in Table 1. These numbers display different repeating patterns which easily can be verified from the explicit expressions (8)-( 10); e.g. the columns for C, and Total are shifted by two units in Lz. Three
rings
with one internal
carbon
The case of r = 3, n, = 1 is illustrated in Fig. 6. The formula reads C.H._I as in the preceding case, but now n = Zq - 5. Adopt the convention i > j >, k. A simple algorithm for the generation of the pertinent isomers is given in the following. For every ,.,., LfZqJ, let j=k,k+l,k+2 ,..., k=3,4,5 Li (Pq - 3)J and i = Zq -j - k. Introduce the coding (i, j, k). A & structure emerges for i =j = k, while the C,,, structures are associated with i =j # k or i #j = k; otherwise the symmetry is C,. The combinatorial expressions which were deduced for the appropriate numbers of isomers, read:
when Xq is odd. For the symmetrical # I(&,)
= 1 - r&Q 1 +
structures:
Lf.3J
(13)
#z(c,,)=r~~qi+r~~ql-LLf~g~--. (14) Notice that the expression for # Z(C,) + # Z(D,,) is exceedingly simple. In the first place, the above results can be used to complete the enumerations of C,4H,0 (anthracene) and C1*HB (biphenylene) isomers. For &HI0 the appropriate value of Xq is 19; equation (4). Then equations (12)-(14) yield # Z = 14(6C2, + SC,). Out of these structures there are found to be eight with q Q 9, again in consistency with Dias (1988). For C12H, (Zq = 17) the result is # Z = 10(X, + 5C,). One of these isomers is acenaphthene, viz. (6,6,5), which is known chemically. The numbers of the isomers under consideration for Cq $30 are collected in Table 2. Again some repeating patterns are apparent. In this case there is a shift between the columns of C, and Total by three units in Zq. In Fig. 7 an Table 2. Numbersof isomersof polycydic conjugatedhydrocarbons with three rings and one internalcarbon: r = 3, n, = 1
# Z = Li (Cq - 4) J(Cq - 6 - 3La (Zq - 4) J) (11) when Zq
is even, and
#I=f(Zq-7)+L;(Zq-7)J(Cg-9-3L~(Zq-7)J) (12)
Fig. 6.
Model of a polycyclic conjugated
r = 3, II,= 1.
hydrocarbon with
IO
C,k
0
I
0
II12 13 14 15 t8
C, CT”, “2
C, “4 C,“, Cm& C,, H,
0I
2I 3
0I
:
0
4
2 3 4
4 5 7 IO 8
18
‘AH, CnH,
1
:
:
12
:
C,,“,, C,,H,,
:
6
IO 8
16 14
22 21 23
C,,“,, ‘%“I, C,,H,,
01 0
6 7 8
I2 14 16
:‘: 24
24 25 26 27 28 29 30
C,H,, CwH,, Cx HI, C,,“,, C,,“,, Cll”l C,,HZI
0 1 0 :,
7 9 9 9 10 II IO
:; 24 27 30 33 37
:: 33 37 40 44 48
0 0
1
0 I
I
I
:
295
Isomers and ring sizes
Fig. 7. The three isomers of C,H, polycyclic conjugated hydrocarbons (r = 3) with n, =
illustrating example is shown: the three C, H, isomers for r = 3, ni = 1 (Cq = 12), where all the three possible symmetry groups are represented. ISOMERS WITH FOUR RINGS AND ONE INTERNAL CARBON If the above analysis could be extended to all cases with four rings the applicability to chemically interesting hydrocarbons would be increased substantially. This is a relatively complicated task, but it can be accomplished conveniently with computer aid. Four rings may be combined to polycyclic conjugated hydrocarbons with the numbers of internal carbons equal to zero, one and two. Here we give a solution for the case with r = 4 and ni = 1. Explicit expressions for the numbers of isomers were not achieved in this case, but a simple algorithm was devised for computer programming. The title case is illustrated in Fig. 8. The hydrocarbon formula reads C,H, _ 6, where n = Zq - 7. Adopt the convention i 3 j 2 k along with (i, j, k)(l) for the coding. The possible (i, j, k) structures are generated precisely as in the case for r = 3, n, = 1 (see above) wherein Xq is replaced by Zq - I. One obtains sets of combinations for I = 3,4,5, . . , Zq - 11. For every (& j, k) combination the possibilities of distinct annelations of the q = 1 rings, say a, have to be specified. Here one must take into account the symmetry group of the (i, j, k) structure, whether it is C,, C,, or D3h_ Accordingly, it was found: a(C,)=i+j+ka(C,) u(D&
12+r,
= rf (i + j + k - 12 + rj)l = ri (i tj
+ k - 12)l.
(15) (16) (17)
The number of triangles, viz. r,, may assume the values 0, 1 and 2 in the case of C,; r3 = 0 or 2 in the case of C,,; finally, r, vanishes when the symmetry of
.
I
Fig. 8. Model of a polycyclic conjugated hydrocarbon with r=4, n,= 1.
1.
(i, j, k) is D,,,. The total number of isomers is obtained on adding the a values for every (i, j, k)(l) combination. It remains to detect the symmetries of the whole (i, j, k)(l) structures. Only C,, and C, symmetries are possible. The C, symmetry is realized if and only if two of the values in (i, j, k) are equal while the third is odd, but different from 3. The computational results of this analysis are listed in Table 3 for Iq 6 50. From Table 3 we find #Z = 365 (15C,, + 35OC,) for C16H10(Zq = 23). These numbers represent a part of the important pyrene isomers, which were treated erroneously by Dias (1988). He depicted 149 out of these structures for q < 9, while, according to our analysis, this number should be 217. Among the 693 C,,H,, polycyclic conjugated hydrocarbon isomers with n, = 1 (cf. Table 3) four structures are listed in
Table 3. Numbers of isomers of polycyclic conjugated hydrocarbons with four rings and one internal carbon: r = 4, n, = I ‘ra
F0llXUJla
14 15 16 17 18 19 20 21 22 13 24 25
1 1
3 3 6 6 10 10 15 15 21 21
:; 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 : 47 48 49 50
c,..
:: 36 36 45 45 55 55 66 66 78
CaHN
97: 91 105 105 120 120 136 136 I53 I53 I71 I71 190
TOtid
C. 0
2 6 16 32 60 100 160 240 350 490 612 896 II76 1512 1920 2400 2970 3630 4400 5280 6292 1436 8736 10192 11830 I3650 IS680 I7920 204OO 23120 26112 29376 32946 36822 41040 45600
I 3 9 I9 38 66 110 170 255 365 511 693 924 1204 I548 1956 2445 3015 3685 4455 5346 6358 7514 8814 I0283 II921 I3755 IS785 I8040 20520 23256 26248 29529 33099 36993 41211 45790
I. BRUNVOLL et al.
296
c$$q$3& I
III
Fig. 9. Phenanthro[u] [ 14lannulene (I), tribenzo[u,g,m] [ 18lannulene (II), and tribenzo[a,e,i][ 12]annulene (III). Dias (1988) as chemically known. An isomer of &Hz,, phenanthro[l4]annulene (viz. 1 of Fig. 9) has been synthesized (Staab et aI., 1979b); according to Table 3 it has 4455 isomers of polycyclic conjugated hydrocarbons with n, = 1. Similarly for the C,, H,, tribenzo[ lfl]annulenes (Stollenwerk er al., 1983), one of which being depicted in Fig. 9. Here the corresponding number of isomers is 8814. Finally the interesting macrocyclic conjugated hydrocarbon C,,H,8 (Staab er al., 1971) is worth mentioning; see III of Fig. 9. This formula was found to have 3015 isomers of the category under consideration. COMPUTER PROGRAMS Computer programs were designed in order to produce all the numbers of Tables l-3. In the case of Tables 1 and 2 these computations provided a useful check on the explicit formulas, which were derived (see above). The computer programs were not saved because of their great simplicity and because they are easily retrieved from the combinatorial principles, which are described above. Acknowledgement-Financial support to B.N.C. from The Norwegian Research Council for Science and the Humanities is gratefully acknowledged.
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Sondheimer F., Wolovsky R. & Amiel Y. (1962) J. Am. Chem. Sot. 84, 274.
Staab H. A., Graf F., Doerner K. & Nissen A. (1971) Chem. Bet-. 104, 1159. Staab H. A., Meissner U. E. & Gensler A (1979a) Chem. Ber. 112, 3907. Staab H. A., Meissner U. E., Weinacht W. & Gensler A. (1979b) Chem. Ber. 112, 3895. Staab H. A. % Diederich F. fl983) Chem. Eer. 116. 3487. Stollenwerk A. H., Kanellakopulos B., Vogler H., JuriC A. & TrinajstiC N. (1983) J. Mol. Strucf. (Theochem.) 102,377. TrinajstiC N. (1992) Chemical Graph Theory, 2nd edn. Boca Raton, Fl, CRC Press.