6990 J. Phys. Chem. 1993,97, 6990-6998 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 CZ470. 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 ( CZ~)~~ 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, c60 (buckminster- fullerene) and the dodecahedranes. Wel8 have enumerated the isomers of polysubstituted c60 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 c60 using P6lya's theorem. FujitaF6aZ7 on 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 0022-3654/93/2097-6990$04.00/0 buckyball (Ca), CZO, Cz4, (228, C30, and c36 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. Never- theless, it must be noted that all odd substitutions will result in radicals such as CaH, CaoHzF, etc. The ESR studies of radicals derived from fullerenes are becoming increasinglyimportant, 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 buckmin- sterfullerene. 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 thealternatingrepresentationisusedfor theenumeration 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 ~~blx~b~...~,~~ is a cycle representation of gEG 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: (2) 1 if g is a proper rotation x(g) = { -1 if g is an improper rotation 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 0 1993 American Chemical Society
Transcript
6990 J. Phys. Chem. 1993,97, 6990-6998
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, c60 (buckminster- fullerene) 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. FujitaF6aZ7 on 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
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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. Never- theless, it must be noted that all odd substitutions will result in radicals such as CaH, 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 buckmin- sterfullerene. 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. King4 has used this particular GCCI in the context of computing chirality polynomials. However, this appears to be the first time that the GCCI of thealternatingrepresentationisusedfor theenumeration 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 gEG 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:
(2) 1 if g is a proper rotation x(g) = {
-1 if g is an improper rotation 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
0 1993 American Chemical Society
Chiral and Positional Isomers of Fullerene Cages The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 6991
- -
Clz(D3) C36(D6h) c 18 (c 3") C 3 0 ( DS h)
C44(D3h) Cso(D5h) c ~ o ( I ~ )
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 (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-
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
6; = -[xIz8 + 8x1x39 + 3x214 - 6x2 - 6x16x,"] (4)
The GCCI of the totally symmetric A, representation which enumerates the number of positional isomers (without accounting for chirality) is given by
1 24
The antisymmetric representation of the C a buckminsterfullerene is A,, and its cycle index is as follows:
The totally symmetric GCCI of the C ~ O cage is given by
pi = - 'O + 4X514 + 5x23' + X110X230 + 4X,'X1,6 + D5h 20"'
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 antisym- metric 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 equiv- alence of the vertices in the parent cage.
The CGF for the tetrahedral C28 cluster with three different substituents is given by
6992 The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 Balasubramanian
Expressions 12 and 13 are obtained by replacing xk with wlk + w2k+ w3k in the GCCIs (4) and (3, respectively. Thecoefficient of ~ 1 ~ 6 ~ 2 ~ 3 , for example, 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 wI26w2w3 in 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 CGF generating functions.
3. Results and Discussion
The complete three-dimensional diagrams for fullerene cages C2rC70 are available in several references.17,23,24 Figure 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 ~ O cages 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 C20Hn or 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 C20HnFm only. 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 c26 . 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 uV or 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. He 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. Histotalisomercountthusdoesnotadduptothevalueof832 592. 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 C2 group, 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. He gives 1 107 124 isomers in the C1 and 42 isomers in the C3 group, and thus the total number of dl-pairs he
Chiral and Positional Isomers of Fullerene Cages
TABLE I: Enumeration of Chiral Isomers for CkH. and CkHsm (k = 20,24,26, and 28)
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 6993
no. of dl-pairs n m no. of dl-pairs n m no. of dl-pairs n m no. of dl-pairs n m
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 ~ F , 4922 dl- pairs 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, 11 8 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 F , C24HF, 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 ~ F , and 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
6994 The Journal of Physical Chemistry, Vol. 97, No. 27, 1993
TABLE II: Chiral Isomers of Cd-IrCd-I,,
Balasubramanian
no. of dl-pairs n no. of dl-pairs n no. of dl-pairs n no. of dl-pairs n
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 saucerz3 has D3d symmetry. We consider the third possibility shown in ref 24 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,
37 dl-pairs for CaH2 (Dsd), 29 dl-pairs for C ~ H Z (Td), 73 dl- pairs 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 ~ O cage with D5h point group symmetry. It is clear from Table I11 that there are no chiral isomers for the monosubstituted buckminster- fullerene. 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 Z . This 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 semiempirical calculations. 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
Chiral and Positional Isomers of Fullerene Cages
TABLE III: Chiral Isomers of C&, (4) and (ab) The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 6995
no. of dl-pairs n no. of dl-pairs n no. of dl-pairs n no. of dl-pairs n
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 S among 91 030 total i~0mers . l~ We note that t h e r e a r e 9 8 5 5 3 8 1 5 8 3 8 7 1 6 4 d l - p a i r s a m o n g 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 buckmin- sterfullerene 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. He 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 Mat- suzawa et ala’s work 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 CI group 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
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 dl- pairs 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 FujitaZ6 in 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 C60H30 matches 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 C20 cage as expected since all the vertices are equivalent for the Cz0 cage. There are 5 positional isomers for C20H3.15 positional isomersfor C2oH3, and soon. Thenumber
6996 The Journal of Physical Chemistry, Vol. 97, No. 27, 1993
TABLE I V Positional Isomers of C d . and CmH.
Balasubramanian
no. of positional isomers n no. of positional isomers n
TABLE V: 23 Positional Isomers of C&z (See Figure 2) - ~~ _ _ _ ~
isomer positions of isomer positions of no. attachment no. attachment
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
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 13C26 fullerene 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 c30 fullerene 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 13C N M R signals for the 13C36 fullerene 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-,
no. of positional isomers n no. of positional isomers n
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 (D5d) 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 13c N 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.50 fullerene cage has Dsh symmetry. It gives rise to 4 monosubstituted and 78 disubstituted positional isomers. There should be four *3C NMR signals for the parent C ~ O fullerene. As seen from Table V there are 1020 and 11 753 tri- and tetrasub- stituted 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
Chiral and Positional Isomers of Fullerene Cages
TABLE VI: Positional Isomers of CkH, (k = 24-50) and C&,X, no. of positional isomers m n no. of positional isomers m n no. of positional isomers m n no. of positional isomers m n
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,
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
25 23 21 19 17 15 13
29 27 25 23 21 19 17 15
37 35 33 31 29 27 25 23 21 19
39 37 35 33 31 29 27 25 23 21
49 47 45 43 41 39 37 35 33 3 1 29 27 25
6998 The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 Balasubramanian
’Oo0 i
I I 1 I I
20 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
32 42 48
36
0.2 y 1 I I I I
20 30 40 50 60 7 0
n Figure 4. Chirality index ( x = m1/np) plotted as a function of n for the isomers of C,H2. Buckminsterfullerene and 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 C1 isomers 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.
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