Enumeration of chiral and positional isomers of substituted fullerene

J. Phys. Chem. 1993,97, 6990-6998

6990

Enumeration of Chiral and Positional Isomers of Substituted Fullerene Cages (C~O-C,~) K. Balasubramanian Department of Chemistry and Biochemistry, Arizona State University, Tempe, Arizona 85287- 1604 Received: March 2, 1993; In Final Form: April 5, 1993

Enumeration of chiral isomers of substituted fullerene cages (CZO-C~O) is considered using the generalized character cycle index (GCCI) of the alternating representation of the point group of the parent cage. It is shown that there are no chiral isomers for the monosubstituted CZO,c24, C28, C30, c36, and CW fullerene cages but there are chiral isomers for other monosubstituted cages. All cages considered here possess chiral isomers for disubstituted cages. It is shown that the number of positional isomers can be obtained using the GCCI of the totally symmetric representation. We also enumerate the 13C NMR signals of all fullerene cages C Z 4 7 0 .

1. Introduction Chirality in chemistry is a topic of long history dating back to Pasteur's discovery.1-7 Most of the earlier works on chirality dealt with traditional organic or inorganic molecules. Several interesting quantities such as the chirality polynomial and measures of geometric chirality have been pr0posed.~9~Fullerenes have been the topic of myriads of studies especially in the last five years. Numerous papers have appeared dealing with the synthesis, characterization, and properties of the buckminsterfullerene and other fullerene cages ranging in size and compo~ition.~-'~ There are studies on small cages ( C Z ~as) well ~ ~ as large cages (Cm, C ~ O , C78, c76, CEO,etc.) although significant work has been done on the buckyball itself (see ref 12 for a comprehensive bibliography). While there are topological and group theoretical studies on fullerene cages,'8J9 it appears that in general the enumeration of chiral isomers of substituted cages for all fullerene cages has not been done. At present there are a few studies on the enumeration of isomers of two of the fullerenes, namely, c 6 0 (buckminsterfullerene) and the dodecahedranes. Wel8 have enumerated the isomers of polysubstituted c 6 0 for several types of substituents. Fujita26927 has enumerated the isomer counts for the derivatives of c60 and C20. HosoyaZ8 has enumerated the isomers of substituted c 6 0 using P6lya's theorem. FujitaF6aZ7on the other hand, uses the subduction of coset representations of the icosahedral point group (1,)with and without chirality and applies the techniques to derivatives obtained from substituents. His enumeration method uses the unit subduced cycle indices and thus gives the isomer counts in several subgroups. We also note that the derivatives of substituted dodecahedranes have been enumerated by 0thers.2~ Since the chemistry of the substituted fullerenes is becoming increasingly important, it would be useful to have a systematic approach to enumerate the chiral isomers of substituted fullerenes. Ab initio computations of substituted fullerenes are also becoming increasingly important. Since two chiral isomers would have the same energy, it would be useful to have both isomer counts and chirality counts so that chiral partners can be eliminated from quantum studies. The objective of this article is the systematic enumeration of chiral and positional isomers of polysubstituted fullerene cages. We show that the generalized character cycle index proposed previously by us,ZO~Z1 when adapted for the antisymmetric (alternating) representation, directly enumerates the number of chiral pairs of isomers. Previous methods of enumerating chiral isomers are somewhat cumbersome especially for fullerenes in that they require isomer counts both in the rotational subgroup and the point group of the parent cage. In the present study both positional isomers and chiral isomers are enumerated in the same group using the generalized character cycle indices. Our study reveals interesting trends. While we find chiral isomers for most of the monosubstituted fullerene cages, the

buckyball (Ca), CZO,Cz4, (228, C30, and c 3 6 are exceptions in that there are no chiral isomers for monosubstitution for these cages. All fullerene cages considered here possess chiral isomers for disubstitution. In all our enumerations we include the isomer counts for both odd and even numbers of substituents. Nevertheless, it must be noted that all odd substitutions will result in radicals such as C a H , CaoHzF,etc. The ESR studies of radicals derived from fullerenes are becoming increasingly important, and thus isomers resulting from an odd number of substituents are equally important. Our analysis of chiral isomer counts as a function of cage size reveals a significant drop in the number of chiral isomers at n = 60, which corresponds to the buckminsterfullerene. We also note that the generalized character cycle index (GCCI) of the totally symmetric representation enumerates the number of positional isomers without regard to chirality. The number of 13C NMR signals of various parent fullerenes is obtained as a byproduct in our enumerated scheme of positional isomers.

2. Methodology The enumeration of chiral isomers which we call dl-pairs is accomplished through a particular case of the generalized character cycle index (GCCI) proposed and used by in the context of molecular spectroscopy. It is interesting to point out that the use of these GCCIs explicitly in the enumeration of chiral isomers has not received much attention before. King4has used this particular GCCI in the context of computing chirality polynomials. However, this appears to be the first time that the GCCI of thealternatingrepresentationisusedfortheenumeration of chiral isomers. The GCCI which corresponds to the character of an irreducible representation r of the point group G of the cage is defined as

where ~ ~ b l x ~ b is~ a. cycle . . ~ ,representation ~ ~ of g E G if it induces a permutation on the vertices of the carbon cage containing bl cycles of length 1, bz cycles of length 2, ..., 6, cycles of length n. IC(is the number of elements in the group G. For the enumeration of chiral isomers we choose x as the character of the antisymmetric representation, defined as follows: x(g) =

rotation { 1 ifif g isis aanproper improper rotation -1

g

(2)

The above definition is valid insofar as the parent fullerene cage itself is not chiral. If the parent cage itself has no improper axis of rotation then by definition it is chiral and thus every substituted isomer is chiral. In this case the ordinary cycle index for which all x ( g ) are unity suffices to enumerate the chiral isomers. This

0022-3654/93/2097-6990$04.00/0 0 1993 American Chemical Society

Chiral and Positional Isomers of Fullerene Cages

The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 6991 ations without these structures. The C28 cage which was recently isolated with a uranium atom trapped inside has a tetrahedral symmetry with Td point group.I6 The A2 representation of the Td group is the antisymmetric representation. Its GCCI is given by 1 = -[xIz8 8x1x39 3x214- 6x2 - 6x16x,"] (4) 24

+

6;

+

The GCCI of the totally symmetric A, representation which enumerates the number of positional isomers (without accounting for chirality) is given by -

C 3 0 ( DS h)

Clz(D3)

C36(D6h)

-

c 1 8 (c3") The antisymmetric representation of the C a buckminsterfullerene is A,, and its cycle index is as follows:

e=

+ 24x," + OX,'^ + 14xZ3O- 24x,,6 -

-[x160 1 120

OX,'^ - 1 ~ X , ~ X ;(6) ~] Likewise, the GCCI of the totally symmetric A, representation of the buckyball is as follows

= 120[xl6' 1

+ 24x512+ 2 0 ~ +~ 16x23' ' ~ + 24X1,6+ 20X6'0 + 15Xl4X,z8] (7)

The antisymmetric representation of the C70 cage with the D ~ J , point group is AI". This GCCI is given by C44(D3h)

Cso(D5h)

c~o(I~)

1 PI" = -[xl7O + 4X,14 + 5x15- ~~~~x~~~ - 4x5 'x 10 20

Figure 1. Structures of fullerene cages considered in this study.

is readily accomplished with the use of P6lya's theorem. (See ref 21 for a review of this topic.) The number of isomers without considering chirality which we refer to as the number of positional isomers is obtained by choosing the GCCI for the totally symmetric representation. That is we choose x(g) as x(g)

= (1 for all g E G

-

Dsh

(3)

For the above choice of the GCCI the enumerated isomer count corresponds to the total number of positional isomers and thus does not consider chirality. In all of the above enumerations we only use the equivalence or nonequivalence of the substitution sites and thus we do not take into account the actual conformations, that is, if the substituent is endo or exo (inside or outside) with respect to the cage. However the enumeration scheme holds perfectly well as long as all the substituents are endo (or all are exo). In the event some of the substituents are endo while the others are exo, more chiral isomers will be generated compared to the numbers enumerated here due to further lowering of symmetry. Also, geometric distortions to the cage induced by substituents which may lower the symmetry further and increase the chiral isomer counts are not considered. Nevertheless all these factors can only influence the chiral isomer counts enumerated by the antisymmetric representation. The number of positional isomers enumerated by the totally symmetric representation is independent of all these factors as this number, which does not include chiral isomers, depends only on the symmetry equivalence of the vertices of the fullerene cage which is properly taken into account by our enumerational scheme. Fullerene cages whose point groups contain improper axes of rotation areachiral by definition. Consequently the antisymmetric character of the parent cage is well-defined. To illustrate, let us consider two of the fullerene cages. Although complete structures of these fullerene cages can be found in refs 17, 23, and 24, we show the structures of fullerenes considered here in Figure 1 since it is difficult to follow the discussions and isomer enumer-

5x14x233] (8) The totally symmetric GCCI of the C ~ cage O is given by

pi =D5h 20"'

'O

+ 4X514 + 5x23' + X110X230 + 4X,'X1,6 + 5x14x;3] (9)

Let D be the set of carbon nuclei in the parent fullerene cage and let R be a set of substituents such as H, F, C1, etc. Let us assign a weight w(r) to each r E R. The following substitution in the GCCI of the antisymmetric representation directly enumerates the number of enantiomeric pairs (dl-pairs). We will call this resulting generating function the chiral generating function (CGF)

where AR is our abbreviation for the character of the antisymmetric representation and the arrow symbol stands for replacing every x k by C,ER(w(r))k. The coefficient of wl6lwZ62...wnb~ in the CGF gives the number of dl-pairs generated by substituting the vertices of the fullerene cage with bl substituents of the type 1 with weight w1, bz substituents of the type 2 with weight w2, .... For example, the first type of substituent may be the hydrogen atom, the second type may be the chlorine atom, etc. The generating function for the number of positional isomers without regard to chirality is given by

where S R stands for the symmetric representation of the group as defined before with all x(g) values set to unity. The isomer count obtained with PGF is exact in that it does not depend on stereochemistry and is dependent only on the symmetry equivalence of the vertices in the parent cage. The CGF for the tetrahedral C28 cluster with three different substituents is given by

6992

1 24 w33)9

CGF = -[(wl w;

+

+ w2 + w3)" + 8(wI + w2 + w3)(wI3+ + 3(w12 + w; + w32)14 - 6(WI4 + w; + w34)7 6(w1 + w2 + w$(wI2 + w; + w ~ ~ ) " (12) ]

+ w2 + w ~ + 8(w1 ) ~ + w2 ~ + w3)(wI3+ + w33)9+ 3(wI2+ w: + wj2)14 + 6(wI4 + W: + + 6(w1 + w2 + w ~ ) ~ +( Ww; ~+~w32)11] (13)

1 PGF = -[(wl 24

w;

Balasubramanian

The Journal of Physical Chemistry, Vol. 97, No. 27, 1993

Expressions 12 and 13 are obtained by replacing xk with wlk + w2k+ w3kin the GCCIs (4) and (3, respectively. Thecoefficient of ~ 1 ~ 6 ~for2 example, ~ 3 , in the CGF enumerates the number of dl-pairs for C28HF or CzsH26FBr. Note that W I could also stand for no substituent and thus the number of dl-pairs of C2sHF is also enumerated by the same term. Likewise the coefficient of wI26w2w3in the PGF (expression 13) enumerates the number of positional isomers for C2sHF or C28H26FBr. The use of the GCCI of the antisymmetric representation directly provides the number of chiral isomers and also saves a significant amount of computation for fullerene cages. The traditional technique for enumerating chiral isomers (see ref 6, for example) involves two computations using Pblya's theorem. First, one computes the isomer count in the rotational subgroup and subsequently the isomer count in the entire point group. The isomer count in the rotational subgroup less the isomer count in the entire point group is taken to be the number of dl-pairs. However, theuseof theGCCI oftheantisymmetricrepresentation reduces the number of required computations to half of the number in the traditional technique. This is a real advantage especially for larger fullerene cages. The computation of the chiral generating function and the coefficients of the various terms in the CGF are quite intensive. Both the number of different types of terms and the coefficients of the various terms grow astronomically for larger fullerene cages. The complexity of the CGF also increases with the number of different substituents. We have developed a computer code22 for obtaining the generating function and automatic collection of coefficients. This code is adapted for computing the CGF with the appropriate choice of character. We uniformly used the quadruple precision arithmetic since the coefficients grow very large for larger fullerene cages. Consequently all digits reported in this study are valid. In a previous study we23 enumerated isomers of substituted C2rC50 fullerene cages in the rotational subgroups of the point groups of the parent cages. Consequently the enumerated isomer counts included both the chiral and positional isomers. In this study we show how to separate the positional isomer counts from the chiral counts through the C G F generating functions. 3. Results and Discussion

The complete three-dimensional diagrams for fullerene cages C2rC70 are available in several references.17,23,24Figure 1 shows the structures of some of the fullerenecages. We shall also indicate the point group of each fullerene cage considered here. All fullerene cages considered here have only pentagons and hexagons. There are 12 pentagons in each cage as required by the Euler rule. A cage which has isolated pentagons is considered to be a more attractive candidate if there are alternative structures. The C20 and c 6 0 cages have z h point groups. The C24 cage has a D6d point group while C36 has a D6h point group. The C28 cage and one form of C40 (see ref 24) have Td point groups. The C30, C50, and C ~ cages O have DSh symmetries. Another form of C40 has a D5d point group. The c26 and c 4 4 cages have D3h groups while the C38 cage has C3, symmetry. The C32 and C42 cages are chiral with D3 symmetry. Hence they are not considered using

the CGFs since the parent cages themselves are chiral and thus every substituted isomer is chiral for these cages. Table I shows the enumeration of chiral isomers for substituted C2(rc28. Note that all the chiral isomer counts in this study are enumerated under the assumption that all substituents are oriented in the same direction (all endo or all exo). In the event that some substituents are endo while others are exo, the symmetry is lowered further and thus there would be more chiral isomers than the numbers we enumerate. Thus the numbers given in these tables for chiral isomers should be considered as lower bounds. In Table I and all other tables we show only the unique terms in the generating functions and some of the possible isomer counts. A complete set of isomer count lists can be obtained from us. We show in Table I the isomer counts for both one type of substituent (e.g., C20Hn) and two types of substituents (e.g., CZOH,F,,,).The numbers shown are interpreted as follows. If n and m add up to k , where k is the number of carbon atoms, the listed isomer count is for C20Hnor CzoH, or C20H,Fm. Since all these numbers can be proven to be equal in Table I we show only the unique numbers. If n and m do not add up to 20 in Table I, the listed numbers are the number of dl-pairs for C20HnFmonly. Hence there must be a t least two different substituents for these cages. It can be seen from Table I that there are no chiral isomers for any of the cages for the case of n = number of carbon atoms and m = 0. This suggests that the parent cage itself is not chiral as expected. The number of chiral isomers for monosubstituted fullerene cages is enumerated by the isomer count for m = 1 and n m = total number of carbon atoms. It is seen from Table I that this number is zero for C20,C24, and c28 but it is nonzero for c 2 6 . The point group symmetries of C20, (224, and c28 are I h , D6dr and Td, respectively. For these cages, through every atom of the cage at least a plane of symmetry passes and hence all carbon atoms of these cages are nonenantiotopic or achirotopic. Consequently, all monosubstituted cages are achiral since every substituent must lie on a symmetry plane. To the contrary, in the C26 cage which has D3h symmetry there are vertices through which three uVor Uh planes do not pass. As a consequence, these vertices form two equivalent classes such that every vertex in the first class has a chiral partner in the second class. This explains the single set of dl-pairs for C26H. FujitaZ7 as well as others29930 has considered the enumeration of the derivatives of dodecahedrane. Fujita has derived the mark and inverse mark tables of the z h point group. He then constructs the unit subduced cycle indices for all the subgroups of the icosahedral point group. There are 22 subgroups (including the trivial and z h groups) for the z h group. Once the unit cycle indices are constructed, the generating functions for the isomer counts are obtained wherein the coefficients and the isomer counts are partitioned into the 22 subgroups. Consequently in his method, one not only enumerates the total count but also the partitioned counts under each subgroup. H e has considered the isomer counts with chiral and achiral derivatives. Fujita finds one dl-pair (in C2 group) for C20H2 (our m = 18, n = 2), 2 dl-pairs (in the C1 group) for C2oHF (our m = 18, n = l), 6 dl-pairs for C20H3(our m = 17, n = 3), and so on. These numbers agree with ours in Table I. However, we note that his chiral isomer count for the C20HgF7 ( m = 8 and n = 7) is 830 218 andthusdisagreeswithourvalueof830 212. Fujitagives830 212 for the C1 group 2380 for the C, group, and 6 for the C3 group. Histotalisomercountthusdoesnotadduptothevalueof832592. Therefore, we conclude that his count under the C3 group of 6 is incorrect and it should be zero. Fujita gives 968 140 isomers in the C1 group, 1017 in the C2group, 18 in the D2 group, and 3 in the D3 group for C20&F6, and thus the total number of chiral pairs is 969 178 exactly agreeing with our count for m = 8 and n = 6. Note that we take only the counts in those groups that do not have improper axes of rotation since these correspond to chiral isomers by definition. Fujita's last row corresponds to our m = 7 and n = 7. H e gives 1 107 124 isomers in the C1 and 42 isomers in the C3group, and thus the total number of dl-pairs he

+


The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 6993

TABLE I: Enumeration of Chiral Isomers for CkH. and CkHsm(k = 20,24,26, and 28) no. of dl-pairs

n

m

0 1 38 310 1022 1510 23 622 4457 12478 233 4796 29262 69070 6373 58497 184244 73286 322888 5 19040 645606 969178

20 18 16 14 12 10 17 15 13 11 16 14 12 10 14 12 10 12 10 8 9 8

0 2 4 6 8 10 2 4 6 8 2 4 6 8 3 5 7 4 6 8 6 6

0 24 1220 19056 129804 441896 803752 49 4922 114776 1040446 4424162 9654244 7412 287324 3643464 19914360 53109448 73228168 1916942 21866037 10622 1447 247860396

26 24 22 20 18 16 14 24 22 20 18 16 14 22 20 18 16 14 12 19 17 15 13

0 2 4 6 8 10 12 1 3 5 7 9 11 2 4 6 8 10 12 4 6 8 10

no. of dl-pairs

c20

n

m

no. of dl-pairs

n

m

no. of dl-pairs

n

m

0 74 1716 14256 54154 103544 20 1740 33504 244752 8 16440 1351 112 17580 302256 1959696 5717600 8109192 504060 4574238 17 157000 29740424 6861696 343 15680 74354840

23 21 19 17 15 13 22 20 18 16 14 12 19 17 15 13 11 17 15 13 11 15 13 11

1 3 5 7 9 11 1 3 5 7 9 11 3 5 7 9 11 4 6 8 10 5 7 9

0 118 3956 48808 28641 3 892167 1556634 24 3318 93606 1033956 5462958 15202215 23390460 40550 1033736 10352430 49193430 121657452 163774800 1723834 24162354

27 25 23 21 19 17 15 26 24 22 20 18 16 14 23 21 19 17 15 13 21 19

1 3 5 7 9 11 13 1 3 5 7 9 11 13 3 5 7 9 11 13 4 6

c 2 4

0 6 113 609 1340 2 151 1892 8305 15270 1249 13412 50080 76600 22438 117162 230360 175812 46 1000 387192 830212 1107166

19 17 15 13 11 18 16 14 12 10 15 13 11 9 13 11 9 11 9 10 8 7

1 3 5 7 9 1 3 5 7 9 3 5 7 9 4 6 8 5 7 5 7 7

0 11 435 5575 30544 81542 112464 240 8780 100692 489744 1143184 2641 84005 857607 3676350 7434938 111878 17 14884 9803280 24783290 2 144435 17158860 55769 170

24 22 20 18 16 14 12 21 19 17 15 13 20 18 16 14 12 18 16 14 12 16 14 12

0 2 4 6 8 10 12 2 4 6 8 10 2 4 6 8 10 3 5 7 9 4 6 8

1 199 5377 54466 259565 642527 865186 626 27248 383086 2341770 7079404 11263602 54547 1149770 9369450 35403095 67591972 383047 7287379 53 108166 1770405 19 2929277 11

25 23 21 19 17 15 13 23 21 19 17 15 13 21 19 17 15 13 20 18 16 14 12

1 3 5 7 9 11 13 2 4 6 8 10 12 3 5 7 9 11 3 5 7 9 11

0 11 807 15466 128714 545089 1264855 1668324 378 20220 344202 2586705 9835452 20271 114 5009 234564 3622124 2459451 5 83637873 152078448 312814 7245986

28 26 24 22 20 18 16 14 25 23 21 19 17 15 24 22 20 18 16 14 22 20

0 2 4 6 8 10 12 14 2 4 6 8 10 12 2 4 6 8 10 12 3 5

c 2 6

C28

enumerates is 1 107 166 exactly agreeing with our m = 7 and n = 7 chiral isomer count in Table I. Therefore, we conclude that with the exception of m = 8 and n = 7 all the chiral isomer counts of Fujita are correct. The parent c 2 4 cage has D6d symmetry. The parent cage is not chiral, nor are there any monosubstituted chiral cages (see Table I). There are 11 dl-pairs for C24H2,74 dl-pairs for 435 dl-pairsfor C24H4,andsoon. Themaximumvalueof 112 464 is reached for C24H12. There are 20 dl-pairs for CUHF ( m = 22, n = 1 in Table I). This is much larger than the analogous number for CzoHF which is 2. This is indicative of higher symmetry for theC2ocagecomparedtoCa. Themaximumvalueof 350 554 560 is reached for the C24H9F8 cage (or C24HaF7). The parent c 2 6 cage has a D3h point group. It is quiteinteresting that c 2 6 is the smallest fullerene cage exhibiting chiral activity for just monosubstitution. As evidenced in Table I there is one enantiomeric pair of isomers for C26H. There are 24 dl-pairs for C26H2, 199 for C26H3, etc. The maximal number for a single type of substituent is 865 186, and it is attained for C26H13. If there are 2 kinds of substituents there are at least 49 dl-pairs (for example CyHF). There are 626 dl-pairs for C Z ~ H ~4922 F , dlpairs for C26H3F, and so on. The maximal count of dl-pairs is reached for C26H9F9, which is 6 329 699 858.

The tetrahedral C28 cage with a uranium atom inside was recently isolated by Smalley and co-workers.I6 This appears to be the smallest fullerene cage to be found in large abundance. It is interesting that there are no monosubstituted chiral tetrahedral C28 cages (Table I). There are 1 1 dl-pairs for C28H2, 118 dl-pairs for C28H3, and so on. The maximum number for a single typeofsubstituentis 1 668 324 (C28H14). Theisomer counts for m n # 28 correspond to substituted C28 cages with at least two different kinds of substituents. The count for m = 26, n = 1 can be interpreted as the number of dl-pairs for either C28HF or C2sH26F. This number is 24. The corresponding numbers for C ~ O H FC24HF, , and C26HF are 2, 20, and 49, respectively. The smaller numbers of dl-pairs for Cz8HF is attributed to the higher symmetry (Td)of the parent cage. There are 378 dl-pairs for C28H2F, 3318 dl-pairs for C ~ B H ~and F , so on. The maximal number of dl-pairs count is reached for CzaH9Fg which is 26 584 734 120. Table I1 shows the results of our enumeration for the chiral isomers of C30Hn-C~0H,. We restricted our enumeration to a single type of substituent since there are large numbers of possibilities and isomers for two different kinds of substituents. As seen from Table 11, there are no chiral isomers for C3oH and C36H. However, for other cages in Table 11, there are chiral

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6994

Balasubramanian

The Journal of Physical Chemistry, Vol. 97, No. 27, 1993

TABLE II: Chiral Isomers of Cd-IrCd-I,, no. of dl-pairs

n

0 18 1326 29432 29 1720 1499958 4320550 7265664

30 28 26 24 22 20 18 16

0 22 2390 80681 1258578 10583664 52135214 158 145692 304446802 378076778

36 34 32 30 28 26 24 22 20 18

0 37 4542 191709 3844176 42319196 279332168 1160324052 3142566960 5668960934 6892268596

38 36 34 32 30 28 26 24 22 20

0 73 11218 587380 14764230 206749945 1757488814 9579481622 34725897166 85793679910 146752553940 175340183644

44 42 40 38 36 34 32 30 28 26 24 22

no. of dl-pairs

c30Hn

n

no. of dl-pairs

0 180 6966 101112 713416 2727372 5981904 7748884

29 27 25 23 21 19 17 15

0 39 2976 75385 876016 5374695 18813434 39282377 500856 12

32 30 28 26 24 22 20 18 16

0 269 15444 346380 3917265 25018812 96252324 231946814 358166820

35 33 31 29 27 25 23 21 19

0 55 6107 229714 4073618 39388824 225608372 805767304 1853286272 2798110550

38 36 34 32 30 28 26 24 22 20

475 32731 93 1209 13668068 115578444 601634012 201 1216876 4436543358 6564028042

39 37 35 33 31 29 27 25 23 21

0 29 3768 159663 3202929 35314286 232772357 966927156 2618790375 47241 13394 5743533590

40 38 36 34 32 30 28 26 24 22 20

2 1061 90032 3190295 59063047 639060619 4326154667 19158996838 57195608812 117401834927 167717082132

43 41 39 37 35 33 31 29 27 25 23

0 57 11427 793577 26837147 513579375 6069848813 4689 1874543 246183478173 902674405978 2356457183270 4437485932442 6017427274702

50 48 46 44 42 40 38 36 34 32 30 28 26

C36Hn

C44Hm D3h

isomers for the monosubstituted fullerene cages. We believe that this is quite interesting. In Table 11, the isomers of the c42 cage are not listed since this has D3 symmetry. Hence it is easy to enumerate the chiral isomers using the rotational subgroup. Likewise, a stable form of C48 containing pentagons and hexagons has 0 3 symmetry and is thus chiral. Therefore all substituted isomers of C42 and (248 would be chiral. The C30 and C50 parent cages considered in Table I1 have D5h symmetries, the c 3 6 cage has D6h symmetry, and the stable form of C38 cage has C3" symmetry. We considered two forms of C a , one with the Td group and the other with the Dsd There are at least three possible structures for CU. The cage with T point group symmetry23 is not considered since it is strained and chiral. The other possibility called the flying saucerz3has D3d symmetry. We consider the third possibility shown in ref 2 4 to have D3h symmetry, and we consider this structure in the current work. Although there are at least two forms for C32 (D3 and D3d), since the parent D3 cage is chiral we do not consider the D3 cage since the rotational subgroup suffices to enumerate chiral isomers. The optical isomers in Table I1 were enumerated using these symmetries of the parent cages. As evidenced in Table I1 there are 18 dl-pairs for C J O H ~39, dl-pairs for C32H2, 22 dl-pairs for C36H2, 55 dl-pairs for C38H2,

no. of dl-pairs

n

C32Hn

2 400 16678 280048 2336055 10749057 28942865 47 137167

31 29 27 25 23 21 19 17

2 677 41590 1050406 13579375 100262827 451215265 1289220934 2398355480 2945357448

37 35 33 31 29 27 25 23 21 19

1 391 27246 775872 11389506 963 13432 501357393 1676006085 3697 106868 5470008544

39 37 35 33 31 29 27 25 23 21

1 940 105436 4990228 125249282 1867592100 17742710948 112540636364 492367122159 1520293804792 3366367170024 5402156068744 6320523 148364

49 47 45 43 41 39 37 35 33 31 29 27 25

C38Hn

Cdn,

n

Td

CSoHn, DSh

37 dl-pairs for CaH2 (Dsd), 29 dl-pairs for C ~ H (Td), Z 73 dlpairs for CUHZ,and 57 dl-pairs for CsoH2. The general trend is that the isomer counts are larger for parent cages with lower symmetries and vice versa. The largest isomer count is attained at n / 2 where n is the number of atoms in the cage. Table I11 shows the results of our enumeration for the icosahedral buckminsterfullerene (C,) and the C ~ cage O with D5h point group symmetry. It is clear from Table I11 that there are no chiral isomers for the monosubstituted buckminsterfullerene. However, there are 14 dl-pairs for CmH2 among 37 possible isomers without taking into account the stereochemistry of endo or exo substitution. It is interesting to note that Matsuzawa et a1.25 have recently carried out semiempirical calculations on all possible nonchiral isomers of CmH2.19 Of course, the energies of dl-pairs would be identical and are thus not differentiated in a quantum mechanical study. We obtain 23 possible isomers without considering chirality for C ~ H ZThis . is exactly the number considered in Table I of Matsuzawa et al.'s work.2s In Table I of their work, they list the symmetries of the 23 disubstituted isomers based on their semiempiricalcalculations. Based on their calculated point group symmetries we infer that they calculate 15 dl-pairs as opposed to 14 found here. Since Matsuzawa et al. considered all hydrogens attached outside the cage, this discrepancy may be surprising. However, the overall


TABLE III: Chiral Isomers of C&, no. of dl-pairs 0 14 4046 417006 21 319974 628272150 1166 1270420 144548950080 1246735824660 7708578047670 34932033844650 117952327562160 300436550914530 582384704656986 864332859441352 985538158387164

n

(4) and

no. of dl-pairs


(ab) n

no. of dl-pairs

n

Ccdn 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30

0 274 45312 3216564 123181170 2855775000 43056957720 443283298844 3226845176280 17040013308780 66537204375780 194877752875260 432628623838260 733373318297492 953746586743696

59 57 55 53 51 49 47 45 43 41 39 37 35 33 31

symmetry of the computed isomer could be lowered due to changes in bond lengths or the spin multiplicities of the electronic states. There are 274 dl-pairs for C&3 among a total of 577 possible isomers.19 Hence the number of isomers without accounting for the chirality of CwH3 is 404. Likewise, there are 4046 dl-pairs for CwH4 among 8236 total possible isomers.l9 There are 45 3 12 dl-pairs for C ~ H among S 91 030 total i~0mers.l~ We note that there a r e 985 538 158 387 164 dl-pairs among 1 971 076 398 255 692 isomers. Hence the number of dl-pairs is approximately half of the total count for larger cages with larger number of substituents. Fujita26 has enumerated the isomers of substituted buckminsterfullerene using the unit subduced cycle indices for the z h group. We note that Table I1 of ref 26 is identical to Table I1 of his previous paper.27 Fujita lists the number of isomers with a single kind of substituent in our notation (that is m n = k ) . Assuming that the numbers in Table IV of ref 26 are in the same order as his previous work,z7 we can compare his chiral isomer counts with ours. He obtains no chiral isomers for monosubstitution in agreement with our result. H e enumerates 8 isomers in the C1 group and 6 in the C2 group. Thus his chiral isomer count of 14 dl-pairs exactly agrees with our number but disagrees too with Matsuzawa et a 1 . l ~chiral ~ ~ isomer count of 15. Therefore this discrepancy needs further consideration. Fujita’s partitioned enumeration scheme facilitates a possible explanation for the above-mentioned discrepancy between Matsuzawa et ala’swork and the combinatorial enumeration schemes. Matsuzawa et al. havelisted the heats of formation for thevarious isomers and electronic states of CmH2. For a given positional isomer, if one considers the most stable state then there are 12 isomers with CI symmetry, 3 with CZ symmetry, 5 with C, symmetry, 2 with CZ, symmetry, and one with C2h symmetry. These differ from the 8 isomers of C1 symmetry, 6 of C2 symmetry, 6 of C, symmetry, 2 of CZ, symmetry, and 1 of C2h symmetry enumerated by Fujita. This specifically suggests that three of the six combinatorially predicted C2 isomers undergo geometry distortions upon addition of hydrogens, which lowers the symmetry further. Likewise one C, isomer undergoes geometry distortion to C1 symmetry. As noted by Matsuzawa et al. double bonds must rearrange to accommodate the hydrogens, in general. The situation is more complicated for open-shell singlet and triplet states in that the symmetry could be lowered further although we note that none of the CZ,and C2h structures have open shell ground states. Next we compare Fujita’s chiral isomer counts26 for a few other cases with ours. Fujita finds 270 isomers in the CIgroup and 4 in the C, group for C&3 and a total of 274 dl-pairs, in agreement with our results. He computes 3946,98, and 2 isomers in the CI, C2, and D2 groups, respectively for CmH4 and a total

+

0 116 45712 6553764 471996700 19835069256 531943783900 9662683506884 124004477078652 1159806682455 156 8094230026697920 42923947561671540 17542830817593 1980 558671690184032532 1398 157 194190753804 2767386999463338892 4351939234067533932 5453499611158328716

no. of dl-pairs

n

c70Hn 70 68 66 64 62 60 58 56 54 52 50 48 46 44 42 40 38 36

2 2682 604 182 59927728 3251586328 108191575248 2313288884808 36074022700912 3938965864053 12 3174207790946712 19271976308254800 89580412383873840 322788087 128538428 910427939583508 144 2024917315584525192 3570821934373755536 5011323965738610292 56093 13884680224028

69 67 65 63 61 59 51 55 53 51 49 47 45 43 41 39 37 35

of 4046 dl-pairs in agreement with our table. He obtains 985 538 119 566 784,38 774 104,45 936,212,120,and8isomers in c1, c2, C3, C5, D3, and D5 subgroups of z h for C60H30. These numbers add up to 985 538 158 387 164 agreeing exactly with our count for the chiral isomers of C6oH30 (see Table 111, the entry for n = 30). Consequently we conclude that two entirely different combinatorial techniques give exactly the same results. The C70 chiral isomer counts are significantly larger due to its somewhat lower symmetry compared to the buckminsterfullerene. There are 2 dl-pairs for C70H, 116 dl-pairs for C70H2, 2682 dlpairs for C70H3, and so on. The maximal value of 5 609 313 884 680 224 028 is attained for the C70H35 isomers. Table IV shows our computed results based on the PGF as obtained from the totally symmetric representation for Cm and C70. The numbers in Table IV thus give the number of positional isomers for substituted c 6 0 and C70 without considering chirality or stereochemistry. From Table IV we infer that there are 23 positional isomers for the disubstituted buckyball. These are shown in Table V with the position labels as in Figure 2 reproduced from the work of Matsuzawa et al.25 It is evident that our positional isomer count of 23 for c60I-I~ agrees with that of ref 25. The positional isomer counts that we obtain for substituted c 6 0 in Table IV for a single type of substituent correspond to the total number of isomers enumerated by FujitaZ6in Table IV. Fujita obtains a total of 1,23,303,4190, and 45 718 isomers of n = 59, 58, 57, 56, and 54, respectively. His largest total isomer count for C60H30matches exactly with our value in Table IV. However, he has enumerated the isomer counts for only a single type of substituent (that is m n = 60), while in a previous work we18 have enumerated the number of isomers for m n # 60. Our isomer count in the previous workls includes both chiral and positional isomers. Table VI shows the results of enumeration of positional isomers for CkH, ( k = 20-50) for several values of n. Note that we show only unique results in the PGF in Table VI, and thus the isomer counts for CkH, are the same as those for CkHk-,,. These results were obtained using the PGF through the GCCI of the totally symmetric representation. The listed isomer counts do not take into account chirality or stereochemistry in that they depend only on the symmetry equivalence of the sites of the parent cage whereas the chiral isomer counts in Tables 1-111 assume that all the substituents are oriented the same way (endo or exo) and thus the actual number of chiral pairs can be slightly larger. The isomer counts in Table VI are exact since they do not depend on the orientation of the substituents (endo or exo). As evidenced in Table V, there is just one positional isomer for the monosubstituted C20cage as expected since all the vertices are equivalent for the Cz0cage. There are 5 positional isomers for C20H3.15 positional isomersfor C2oH3, and soon. Thenumber

+

+

Balasubramanian

6996 The Journal of Physical Chemistry, Vol. 97, No. 27, 1993

TABLE I V Positional Isomers of C d . and CmH. no. of positional isomers n no. of positional isomers

n

1 23 4190 418470 21330558 628330629 11661527055 144549869700 1246738569480 7708584971055 34932048763560 117952355252550 300436595453640 582384767014701 864332935668892 985538239868528

60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30

1 303 457 18 32202 18 123204921 2855893755 43057432740 443284859624 3226849468425 17040023323785 66537224405790 194877787472550 432628675734195 733373386161407 953746664302456

TABLE V: 23 Positional Isomers of C&z ~~

_ _ _ ~

isomer no.

positions of attachment

isomer no.

1 2 3 4 5 6 7 8 9 10 11 12

1 and 2 1 and 3 1 and6 1 and 7 1 and 9 1 and 13 1 and 14 1 and 15 1 and 16 1 and 23 1 and 24 1 and 31

13 14 15 16 17 18 19 20 21 22 23

no. of positional isomers

n


n

C70Hn

C60Hn

59 57 55 53 51 49 47 45 43 41 39 37 35 33 31

1 143 46275 6561107 472064572 19835545518 531946433532 9662695546292 124004522722680 1159806829161168 809423043 1357966 42923948528653920 175428310192640700 558671693876117652 1398157200150692284 2767387007974984602 4351939244847901422 5453499623286009526

70 68 66 64 62 60 58 56 54 52 50 48 46 44 42 40 38 36

5 2792 606125 59949744 3251766528 108 192710688 237329461 8224 36074046545320 393896669700852 3174208038961440 19271976944758980 89580413804047920 322788089901940448 910427944348609504 2024917322816219232 3570821944096374416 501 1323977341628902 5609313896986061888

69 67 65 63 61 59 57 55 53 51 49 47 45 43 41 39 37 35

(See Figure 2) positions of attachment 1 and 32 1 and 33 1 and 34 1 and 35 1 and 41 1 and 49 1 and 50 1 and 52 1 and 56 1 and 51 1 and 60

of monosubstituted isomers also gives the number of I3C N M R signals for l3C2o. The c24 fullerene has D6d symmetry and thus gives rise to more isomers. There are 2 isomers for C26H, 19 isomers for C26Hz, 96 isomers for C26H3,and so on. The number of 13C N M R signals is enumerated as 2 for the 13C26fullerene with D6d symmetry. The c26 fullerene has D3h symmetry. As a result there are more isomers for the substituted c26 fullerene. There are four isomers for C26H (C26Hz5) and thus there are four I T N M R signals for the c26 cage. There are 37 isomers for C26H2 and 237 positional isomers for C26H3. The c28 fullerene, which has been a topic of recent activity, has tetrahedral symmetry. We predict three isomers for C28H and thus three 13C N M R signals for the Wz8 fullerene. There are 24 positional isomers for C28H2, 161 isomers for C28H3, and so on. The c30fullerene cage has D5h symmetry. It yields 3,33, and 226 positional isomers for mono-, di-, and trisubstitution. We predict three 13C N M R signals for the C30 fullerene cage with D5h point group symmetry. The c32 parent fullerene cage itself is chiral since it has 0 3 symmetry. Consequently, every substituted isomer is chiral, and it suffices to enumerate the isomers in the 0 3 group. This was already done. The c36 fullerene cage has D6h point group symmetry. As seen from Table VI there are 3,41,328, and 2608 mono-, di-, tri-, and tetrasubstituted c36 fullerene isomers. It is also readily seen that there are three 13CN M R signals for the 13C36fullerene with D6h symmetry. Since the c38 cage has only c3"symmetry it gives rise to five monosubstituted isomers and thus five I3C N M R signals for I3c36.There are 72 and 733 di- and trisubstituted positional isomers as seen from Table VI. We consider the C40 fullerene cage in two isomeric forms. One of them, which is probably more stable, has Td point group symmetry while the other isomer has D5d symmetry. It is interesting to compare the isomer counts of both the forms. The D5d form of C40 fullerene yields 3, 51, 51 3, and 4692 mono-, di-,

59

60

Figure 2. Schlegal diagram for Cm with numbering of vertices as in ref 17. See Tables IV and V for the enumeration of positional isomers and Table 111 for the chiral isomers.

tri-, and tetrasubstituted positional isomers. It is evident that there are three I3C N M R signals for the 13Ca ( D 5 d ) fullerene. On the other hand, the C a fullerene with T d symmetry gives rise to 3, 41, 435, and 3904 mono-, di-, tri-, and tetrasubstituted positional isomers. Thus although the T d C a fullerene cage has greater symmetry, it yields the same number ( 3 ) of I3C N M R signals as the D5d cage. Hence ordinary 13cN M R cannot differentiate the two forms of C40 fullerene considered here. The parent C42 fullerene cage has only D3 symmetry and it thus chiral. Consequently every substituted isomer is chiral. It thus suffices to enumerate the isomers of this cage in 0 3 group as done before.24 The CU fullerene cage has D3h symmetry. This structure yields 6, 96, 1157, and 11 532 mono-, di-, tri-, and tetrasubstituted positional isomers. We predict six *3C NMR signals for the fullerene. The c.50fullerene cage has Dsh symmetry. It gives rise to 4 monosubstituted and 78 disubstituted positional isomers. There should be four *3CNMR signals for the parent C ~fullerene. O As seen from Table V there are 1020 and 11 753 tri- and tetrasubstituted C50 fullerenes. Figure 3 shows the plot of the number of dl-pairs for the disubstituted cages as a function of cage size for n = 20-70. We choose the dl-pair count for the disubstituted cage since this is a good measure of chiral isomer count and at the same time is a manageably smaller number. Furthermore, the chiral isomer



TABLE VI: Positional Isomers of CkH, (k = 24-50) and C&,X, no. of positional isomers 1 58 1135 34 4597 274 29739 6557 185308 324428 647706


m

n

20 16 12 17 13 16 12 14 10 10 9


0 4 8 2 6 2 6 3 7 6 6 n

1 19 489 5775 31034 82358 113434

22 20 18 16 14 12

1 24 928 16044 130545 549060 1271126

28 26 24 22 20 18 16

1 41 2608 82123 1264902 10603572 52182175 158230972 304568066

36 34 32 30 28 26 24 22 20

1 51 4692 192699 38487 15 42394620 279372560 1160407692 3 142706 190 5669149226

40 38 36 34 32 30 28 26 24 22

1 96 11532 589929 14778229 206806261 1757662595 9579905405 34726731106 85795019295 146754328729

44 42 40 38 36 34 32 30 28 26 24

1 149 1466 176 8501 1337 50696 22802 23 1550 462820 832592



1 5 9 3 7 3 7 4 8 7 7

n

5 371 1648 674 12716 4984 698 12 59085 740 14 521000 97 1840


m

n


m

n

18 14 10 15 11 14 10 12 12 8 8

2 6 10 4 8 4 8 5 4 8 6

15 693 5 1984 15536 13720 77370 118002 176904 388788 1109966

17 13 18 14 10 13 9 11 11 10 7

3 7 1 5 9 5 9 6 5 5 7

n

23 21 19 17 15 13

1 37 1316 19468 130942 444074 806746

26 24 22 20 18 16 14

3 161 4234 49886 289218 897348 1563630

27 25 23 21 19 17 15

1 33 1467 30173 294255 150605 1 4331275 7279821

30 28 26 24 22 20 18 16

3 328 15972 349260 3928045 25048296 963 13476 232045198 358291230

35 33 31 29 27 25 23 21 19

1 72 6289 230906 4078924 39406096 225651217 805850405 1853414452 2798269382

38 36 34 32 30 28 26 24 22 20

3 513 33073 933147 13675820 1156017OO 601688276 2011317652 4436694522 6564212798

39 37 35 33 31 29 27 25 23 21

1 41 3904 160539 3207006 35328158 232808933 967003044 26189 17113 4724284890

40 38 36 34 32 30 28 26 24 22

6 1151 90974 3196527 59092159 639162691 4326433739 19159606672 57 196691518 117403413751 167718985692

43 41 39 37 35 33 31 29 27 25 23

1 78 11753 796643 26857043 5 13675025 6070204847 46892931437 246 186032172 902679503887 2356465675590 4437497822718 6077441329078

50 48 46 44 42 40 38 36 34 32 30 28 26

Cdn

Cdn

19 15 11 16 12 15 11 13 9 9 8

n

Cd%Xm

2 96 1826 14586 54814 104468

C28Hn

C d L D5d

m

counts for several monosubstituted cages are zero and thus would not suffice to differentiate different fullerene cages on the basis of chiral isomer counts. As seen from Figure 3 there are local peaks at n = 24,32,38,42,48,and 70. The largest value among the cages considered here is attained for c 4 8 since the parent c 4 8 cage itself is achiral. There are local minima at n = 20,28, 36,


Cdn

n

4 237 5589 55 186 261212 645185 868294

25 23 21 19 17 15 13

3 226 7287 102468 717299 2735358 599408 1 7762876

29 27 25 23 21 19 17 15

5 733 4207 1 1053014 13589305 100290961 451276781 1289327090 2398502014 2945520468

37 35 33 31 29 27 25 23 21 19

3 435 27588 777810 11397258 96336688 501411657 1676106861 3697258032 5470193300

39 37 35 33 31 29 27 25 23 21

4 1020 106444 4998212 125294088 1867781780 17743340912 112542321196 4923708 16956 1520300533528 3366377436256 5402169267816 6320537495512

49 47 45 43 41 39 37 35 33

CsoHn

C3d-h

CaHn, T d

CsoHn

31

29 27 25

40, 44, and 60. The parent cages for these counts clearly have higher symmetries compared to their neighbors. Thus a local minimum in the chiral isomer count is indicative of high symmetry in the parent cage. The lowest minimum for larger cages is attained at n = 60,which corresponds to the buckminsterfullerene. The dramatically low chiral isomer count correlates well with the

Balasubramanian

6998 The Journal of Physical Chemistry, Vol. 97, No. 27, 1993

i

’Oo0

32

42

48

36

20

I

I

1

I

I

30

40

50

60

70

n Figure 3. Chiral isomer counts of fullerene cages as a function of cage size. The number of chiral isomers for disubstituted cages (C,H2) is plotted on the y-axis. high symmetry of the buckyball. The higher the symmetry of a cage (as measured by the point group operations) the lower the number of nonequivalent sites and thus the lower the chirality count. The other trend which emerges from Figure 3 is that if two cages have isomorphic point groups, the cage with the larger number of atoms would have the larger chiral count. This trivially follows from more combinatorial possibilities for a larger cage of the same symmetry (for example based on chiral counts C ~ O > CSO> c309 c40 > c28, c60> CZO,etc.1. The plot in Figure 3 is symmetry and size dependent rather than being dependent on chirality. We therefore define a normalized isomer count which factors out the size and symmetry dependence and is truly reflective of chirality. We call this index the kth order chirality index and define it as follows:

where nit) is the number of dl-pairs for C,Hk and nr) is the number of positional isomers for C,Hk. The advantage of the kth order chirality index is that it factors out the “size dependence” by dividing the number of chiral pairs by the number of positional isomers. Consequently the largest value of the chirality index, unity, is attained when the parent cage itself is chiral and thus every substituted isomer is chiral. On this basis, a ~ ( 2 chirality ) index was obtained for all fullerenecages considered here. Figure 4 shows the plot of the x ( ~index ) as a function of n. We see local minima at n = 20, 28, 36, 40,and 60. The C20 cluster has the lowest chirality index (0.2) while the buckminsterfullerene has the second lowest second-order chirality index.

4. Conclusion We enumerated the chiral and positional isomers for substituted fullerene cages C2&70. As a byproduct of our enumerational scheme we obtained the number of 13C N M R signals for all of the parent fullerene cages. We used thecomputerized generalized character cycle index (GCCI) technique for our enumeration, using the GCCIs for the totally symmetric and antisymmetric representations. Weshowed that thechiral isomerscan bedirectly obtained using the GCCIs of the antisymmetric representations. We compared our isomer counts with Fujita’s unit subducedcycle indices scheme for two of the fullerenes considered here (C20 and CSO).While all of the numbers agreed, we found that the number of chiral isomers given by Fujita for C20HaF7 is incorrect in that the isomer count given by him for the C3 subgroup of I h should be 0 instead of 6. We also compared our combinatorial enumerations with the MNDO computations of Matsuzawa et

0.2

y

1

I

I

I

I

20

30

40

50

60

70

n

Figure 4. Chirality index ( x = m1/np) plotted as a function of n for the isomers of C,H2. Buckminsterfullereneand Caexhibit the lowest chirality indices while C32 whose parent cage itself is chiral exhibits the highest chirality index of unity.

al. for CmH2, We noted that the MNDO computation yielded four more C1isomers than predicted by combinatorics. This was tentatively attributed to geometrical distortions for three C2 isomers and one C, isomer resulting in CI symmetries.

Acknowledgment. This research was supported in part by the National Science Foundation under Grant CHE92804999. The author would like to thank Professor K. Mislow and the referees for their invaluable comments. References and Notes (1) Mead, C. A. Top. Curr. Chem. 1974, 49,

1. (2) King, R. B. Theor. Chim. Acta 1983, 63, 103. (3) Mead, C. A. Theor. Chim. Acta 1980, 54, 165. (4) King, R. B. J. Maih. Chem. 1987, I , 45; 1987, I, 15. (5) Buda, R.B.; Heyde, T. A,; Mislow, K. Angew. Chem.,Int. Ed. Engl. 1992, 31, 989. (6) Balasubramanian, K. Theor. Chim. Acta 1979, 51, 37. (7) Mezey, P. G., Ed. New Developmenis in Molecular Chirality; Kluwer: Dordrecht, 1991 ( 8 ) Curl, R. F.; Smalley, R.E. Science 1988, 242, 1017. (9) Kroto, H. W. Comput. Math. Appl. 1989, 17, 417. (10) Kroto, H. W.; Allaf, A. W.; Balm, S. P. Chem. Rev. 1991,91, 1213. (11) Duncan, M. A.; Rouvray, D. H. Sci. Am. 1990, 260, 110. (12) Chibante, L. P. F.; Smalley, R.E. Complete Buckminsterfullerene Bibliography, Sept 9, 1992. (13) Kroto, H. W.; Heath, J. R.;OBrian, S.C.; Curl, R. F.; Smalley, R. E. Nature 1985, 318, 162. (14) Haufler, R.E.; Conceicao, J.; Chibante, L. P. F.; Chai, Y.;Byrne, N. E.; Flangan, S.; Haley, M. M.; O’Brian, S. C.; Pan, C.; Billup, W. E.; Ciufolini,M. A.; Hauge, R. H.;Margrave, J. L.; Wilson, L. J.; Curl, R. F.; Smalley, R.E. J. Phys. Chem. 1990,94,8634. (15) KrHtschmer, W.; Lamb, L. D.; Fositropoulos, K.; Huffman, D. R. Nature 1990, 347, 354. (16) Guo, T.; Perner, M. D.; Chai, Y.;Alford, M. I.; Haufler, R. E.; McClure, S. M.; Ohno, T.; Weaver, J. H.; Scuseria, G. E.; Smalley, R. E. Science, in press. (17) Feyereisen, M.; Gutowski, M.; Simons, J.; Almolf, J. J . Chem. Phys. 1992, 94, 347, 2926. (18) Balasubramanian, K. Chem. Phys. Lett. 1991, 183, 292. (19) Balasubramanian, K. Chem. Phys. Lett. 1990, 182, 257. (20) Balasubramanian, K. J . Chem. Phys. 1981, 74, 6824. (21) Balasubramanian, K. Chem. Rev. 1985, 85, 599. (22) Balasubramanian, K. J . Comput. Chem. 1982, 3, 75. (23) Boo, W. 0. J. J . Chem. Educ. 1992.69, 605. (24) Balasubramanian, K. Chem. Phys. Lett. 1993, 202, 399. (25) Matsuzawa, N.; Dixon, D. A,; Fukunaga, T. J. Phys. Chem. 1992, 96, 7594. (26) Fujita, S. Bull. Chem. SOC.Jpn. 1991,64, 3215. (27) Fujita, S. Bull. Chem. Soc. Jpn. 1990, 63, 2579. (28) Hosoya, H. Gendai-Kagaku 1987,201,38; as referenced toin ref 26. (29) Paquette, L. A.; Ternansky, R. J.; Balogh, D. W.; Taylor, W. J. J. Am. Chem. Soc. 1983,105, 5441. (30) Balaban, A. T. Rev. Roum. Chim. 1986, 31, 679. I

Enumeration of chiral and positional isomers of substituted fullerene

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