Symmetry of fullerenes

Volume 21 I, number 2,3

CHEMICAL PHYSICS LETTERS

13 August 1993

Symmetry of fullerenes D. BabiCa, D.J. Klein b and C.H. Sah ’ ’ Institute “Ruder BoskoviC P.O.Box 1016,410Ol Zagreb, Croatia b Department ofMarineSciences, Texas A&M University at Galveston, Gafveston. TX 77553-1675, USA ’ Department OfMathematics, State UniversityofNew York at Stony Brook, Stony Brook, NY I1 794-3651, USA

Received 30 April 1992;in final form 21 June 1993

All fullerene isomers with up to 70 atoms have been generated and their symmetry analyzed. A safe and efficient algorithm for determination of aymmetricaIly equivalent vertices, edges and faces in the corresponding molecular graph is outlined. The automorphism group determined in this way represents the maximal symmetry which the pertinent molecule can have. All analyzed isomers are classified by their symmetry, and the results are tabulated.

1. Introduction Fullerenes are spherically shaped molecules made up of only carbon atoms. Besides the two best known and most accessible: Cbo,buckminsterfullerene, and CrO,falmerene [11, researchers have isolated a number of others: CT6,CT*,CB2,Cs4, C9,,and Cgq [ 2-41. Related assemblies obtained in the laboratories so far are tubular structures (buckytubes) [5-61 and concentric fullerene molecules [ 7-81. The structure of fullerenes relies on a simple topology of carbon bonding [ 91. Each atom is sigma bonded to the three others, sharing with them three valence electrons. The remaining fourth valence electron is included into a delocalized pi-system spread over the entire molecular surface. Stability arguments suggest limitation to only five- and six-membered rings in the structure [ 9 1. An application of the Euler formula reveals that there must be exactly 12 pentagons and in - 10 hexagons (as reviewed in ref. [ 91); n denotes the number of carbon atoms in a molecule. For a fixed n, there are many different structures satisfying the above structural conditions. We call these structures fullerene isomers. A large number of them are chiral and occur in two enantiomeric forms, but each enan’tiomeric pair will be counted here as one isomer. A knowledge of all isomers is important because they provide a collection of new molecules whose investigation has only started. 0009-2614/93/$06.00

In the present Letter we deal with the symmetry properties of fullerene isomers. Symmetry properties provide a natural tool for classification of isomers whose number increases very quickly with n. Also symmetry plays an important role in an identification of isolated isomers [ 4,101, and in predicting their various properties [ 11-l 5 1. A number of papers deal with the symmetry of fullerene isomers and related species [ 16-221. In a recent one by Fowler et al. [20], possible symmetries of fullerenes were determined and all pertinent point groups have been identified and listed. The symmetry classification of all CsOisomers has also been given. The main result in the present Letter is the same classification for all isomers, C,, n < 70. Isomers have been generated and classified by their symmetry, and also according to cardinalities of their vertex, edge and face orbits. An algorithm for the determination of symmetry elements of fullerene isomers and the pertinent point groups has been described in ref. [ 151I The method relies on the geometric representation derived from the Stone’s tensor surface harmonic bonding theory [ 23 1. It seems to be quite efficient and reliable, but the proof that the geometric representation fully preserves the symmetry of the graph has not been given yet. Another method which determines symmetry equivalent atoms in fullerenes was used in ref. [ 24 1. It is based on certain topological characteristics of individual atoms and was successfully applied to a

0 1993 Elsevier Science Publishers B.V. All rights reserved.

235

Volume211,number2,3

CHEMICAL PHYSICS LETTERS

few isomers. However, its applicability in general remains an open question. By symmetry we mean the automorphism-group symmetry of a graph. The symmetry of a graph, also called a topological symmetry, accounts only for the bond relations between atoms, and does not fully determine molecular geometry. The symmetry of the graph does not need to be the same as (i.e. isomorphic to) the molecular point-group symmetry. However, it does represent the maximal symmetry which the geometrical realization of a given topological structure may possess. That this maximal symmetry is realizable is ensured by the theorem of Mani [ 251. The theorem states that for every planar triconnected graph there exists a polyhedron with the point symmetry group isomorphic to the symmetry group of the graph. Whether such a molecule can also exist as a stable species, depends mainly on its electronic structure.

2. Generation and coding of fullerenes The existence theorem of Mani has been strengthened by a recent result of Thurston [ 26 ] that leads to a theoretical enumeration of all fullerene structures. Thurston’s result is a generalization of the method of Coxeter [27] which is also called the “projection method” in ref. [ 2 I 1. Unfortunately, Thurston’s theorem leads to the enumeration of equivalence classes of lattice points in a suitable lattice sitting in a complex vector space of complex dimension 10 with prescribed normsquare (equal to the number of vertices in the fullerene graph). The norm arises from a Hermitian inner product of signature (1,9), i.e. a Lorentzian lattice. So far, there has been no attempt to write an algorithm based on Thurston’s work, The principal difficulty is the absence of a procedure that would produce a molecular graph from a lattice point. However, Thurston’s result does produce a crude asymptotic estimate that the number of theoretical fullerenes with n carbon atoms is at least of the order of n g. For a given symmetry group, it is often possible to give a more explicit enumeration of the fullerenes [ 28 1. However, in the generic case, the symmetry group is trivial and Thurston’s lower estimate gives a measure of the degree of difficulty in any classification scheme. 236

13August1993

Thus, efficient and safe algorithmic generation of fullerene isomers is as yet an unsolved problem. Several algorithms were developed [ 29-3 1 ] but only one of them [ 301 is known to be complete. In fact, the simplest and the most used, spiral ring algorithm [ 291, was recently proven to be incomplete [ 32 ] by effectively using the classification of fullerenes with tetrahedral symmetries [ 28 1. However, it is highly probable that all these algorithms are complete for n under about 100, and this is just the range of their practical application. The counterexample for the spiral ring algorithm has 380 atoms [ 321. Some particular types of isomers, e.g., those having no adjacent pentagons in the structure, can be generated by hand, for n approaching even 90 [ 331. However, in order to generate all isomers one must devise a convenient computer program, mainly because of the rapidly growing numbers of isomers. Here we have used an algorithm described in ref. [ 3 I 1. It is a modification of the spiral ring algorithm [ 291 in which its reliability was improved with similar efficiency. The program works by successively adding new rings to a previously built fragment until a complete fullerene structure is realized. In each stage the fragment boundary contains some divalent vertices. An arc on the boundary between a pair of successive divalent vertices, together with their pendant edges, bounds a place where a new ring will be added, see fig. 1. If the conditions permit us to do

Fig. 1. A typical stage in the procedure for generation of fullerenes. The boundary is represented by bold line. Note the divalent vertices on the boundary with pendant bonds. Regions between them are shaded to denote positions where new rings would be joined.

Volume 211, number 2,3

CHEMICAL PHYSICS LETTERS

so, we added a pentagon in one case and a hexagon the second time. After addition of a ring, the boundary is updated and the process can be repeated. On a given boundary there is more than one place where a new ring could be added (see fig. 1), and only one of them must be chosen. The theorem in ref. [ 341 proves that every existing fullerene can be built in the way described. However, the choice of the position for joining new rings may be critical. A safe algorithm should try, separately, each possible position, and this is the approach taken in Galveston [ 301. In the algorithm we used, it was the position between any of the most distant pairs of divalent vertices, as it appears to be a safer choice than others; the counterexample for the spiral ring algorithm [32] can be successfully created by this approach. The overall counts of isomers agree with that obtained via the complete algorithm. All the algorithms work with some redundancy, i.e. the same isomers are produced more than once. In order to recognize those already generated, some unique code is needed. All atoms in fullerenes are of the same kind, and the same holds for bonds by themselves. That is why graphs are particularly convenient for representation of fullerene structures. The problem of the unique code appears then as the famous graph isomorphism problem. The recent solution [ 351 can be applied, but here we used the more convenient approach. Fullerenes belong to a special class of graphs for which the isomorphism problem has been solved long ago [ 36-381. The unique code is based on an Eulerian cycle in planar three-connected digraphs, derived from the given graph by converting unoriented edges into pairs of oppositely directed edges. An Eulerian cycle is a path starting and ending at the same vertex, and passing through each edge of the graph exactly once. Fullerene graphs as they are do not contain an Eulerian cycle. However, after all edges are converted into pairs of oppositely directed edges, fullerene graphs meet the necessary and sufficient criterion for possessing an Eulerian cycle; each vertex contains equal numbers of inward and outward edges. Eulerian cycles traverse each edge once in each direction. For coding purposes the graph is (imagined to be) drawn in a plane so that no two edges are crossing. In the program the graph was stored by specifying a

13 August 1993

list of incident edges for each vertex. It is important to list them in order, either counterclock- or clockwise. This was implemented already in the course of generating the isomers. An Eulerian cycle is started from any vertex by any of the three edges. When a vertex v is reached, the choice of the next edge depends upon several conditions. If the vertex v is visited for the first time, it gets an integer label (one greater than the last used label), and the path is continued to the rightmost edge, with respect to the incoming edge. If the vertex has already been visited, the path continues to the incoming edge in the opposite direction if it was not already traversed in this direction. Otherwise one chooses the rightmost free edge. Fig. 2 shows an Eulerian cycle on a fullerene graph obtained by following the above rules. These

Eulerian code, C: 1.2.3.4.5.6.1.6.7.8.9.1.9.10.11.2.11.12.13.3.13.14.15.4.15.16.17. 5.17.18.7.18.19.20.8.20.21.10.21.22.12.22.23.14.23.24.16.24.l9. 24.23.22.21.20.19.18.17.16.15.14.13.12.11.10.9.8.7.6.5.4.3.2.1 C’: 1.9.8.7.6.5.1.5.4.3.2.1.2.17.12.9.12.11.10.8.10.16.15.7.15.14.13, 6.13.18.4.18.21.22.3.22.23.17.23.24.11.24.19.16.19.20.14.20.21. 20.19.24.23.22.21.18.13.14.15.16.10.11.12.17.2.3.4.5.6.7.8.9.1

Fig. 2. An Eulerian cycle traced in a small fullerene C,, with Dbd symmetry. Original labels are placed directly on vertices, while those derived in the course of traversing the graph are near the path. The C’ below the Eulerian code is obtained by substituting original labels into the code.

237

Volume211,number 2,3

CHEMICALPHYSICSLETTERS

rules guarantee that each edge will be traversed once in both directions. Such a realization of Eulerian cycles is identical to the familiar strategy for getting out of a labyrinth; edges correspond to passages, and vertices to intersections. The Eulerian cycle is easily coded by writing down vertex labels in the order they were visited. The code obtained in this way we call the Eulerian code of a graph. Only one graph corresponds to each code, and the reconstruction is straightforward. However, different Eulerian codes are possible for the same graph, depending on the starting vertex and edge and the orientation of a planar representation. For the unique one we deliberately chose, lexicographically, the smallest. This algorithm requires cn(n’) in time [ 361. In the literature one can find more efficient algorithms [37,38], but we used the one described as it appeared to be most efficient for the present purpose.

3. Perception of symmetry in fullerene graphs Symmetry operations on a graph are called graph automorphisms. They affect only the labels of vertices by permuting them so that the adjacency matrix of the graph remains unchanged. The graph symmetry is completely determined by all the automorphisms it has, i.e. by specifying all the permutations which leave the adjacency matrix intact. In fullerene graphs these permutations can be inferred from their Eulerian codes. Fullerene graphs may be coded in 6n different ways since the Eulerian cycle can be started at any of the n vertices, by any of the three edges, and in either of the two planar orientations. After the starting vertex, the edge and orientation are chosen, the above specified rules completely determine the cycle and the resulting code. If the codes obtained by two different cycles are the same, there is an underlying automorphism. The proof follows. Assume that the code is derived from a particular (planar) representation of the graph, with vertices labeled in some arbitrary way. These labels we denote as the original labels. While performing an Eulerian cycle, new labels are produced and used for coding. Denote the two different Eulerian cycles producing the same code by Cl and C2. If the labels derived during the code generation are substituted with 238

13August 1993

the original ones, one obtains new, distinct, codes Cl’ and C2’. To interconvert Cl’ and C2’, one should perform a certain permutation on Cl’. If this permutation would have been applied to original labels on the graph representation, instead of Cl’ one would obtain C2’. Since both Cl’ and C2’ correspond to exactly the same graph representation (with the same adjacency matrix), the permutation represents a graph automorphism. On the other hand, for each automorphism there is a pair Cl’ and C2’ related by the appropriate permutation. Just take Cl’ which starts at some of the permuted vertices and apply the permutation. This produces C2’, which corresponds to another Eulerian cycle with the same code, Therefore, to analyze fullerene graph symmetry it suffices to consider the set of its Eulerian codes. Even more, it is sufficient to consider only a maximal subset of equal codes. The permutations which interconvert codes in the subset form a group, and the cardinality of the subset equals the order of the group. As all the same holds for every maximal subset of equal codes, the order of the group must be a divisor of 6n, the total number of codes. To determine the automorphism permutations, labels in Eulerian codes are substituted by the original labels. One of the codes is taken as a reference, and by comparison with all others, the underlying permutations are deduced. There are various notations for permutation groups in the mathematical literature, but in order to keep the common relation to the geometry, we adhered to the usual physico-chemical notation for symmetry groups, and represented the automorphism groups by the appropriate isomorphic point groups. After the permutations of vertices were determined, the corresponding permutations of edges and faces were also, respectively, derived. Vertex, edge and face orbits were then determined by tracing the action of permutations. Each permutation has been identified as one of the symmetry elements characteristic for point groups. It was done by considering the structure of vertex, edge and face orbits. After that, the point group was easily determined. The procedure will be presented in full detail elsewhere [ 391.

-

-~

T* lh

CI Ci C. Ct C, CZ” C 3” CZb C,, J& D3 DS J&h J&I l&l DE., J&4 J&d J&d J&d S4 S, T

1

20

-

1

24

-

1

26

I

1

28

1

2

30

1

1

I 1

2

32

J

2 3

34

.-

1 2

1

2

1

--

1

1

2 1

5

7

2 2 4

38

36

1

2

1

3

1

2

- -

1

4

6 11

7 14

1

23

42

8

40

1

1

3

2

6 2

3

I 22

42

44

-

19 22 2 4

69

46

2

2

I

5

1

3

16 52

117

48

1 2

2

25 37 2 6 1

195

50

-.

1

5

2

9

26 78 3 3 2 I

307

52

-

1

1

8

38 62

470

54

-

1

1 2

1

10 6

2

-

700 1 49 135 3 13

56

-

58 98 4 6 2

1037

58

-._

1

2

19 3 1 1

9 1 4

67 189

1508

60

.--

2

4

1

80 142 4 16 1

2135

62

1

17

2990 2 118 316 8 4 4 4

64

.-

2

18 1

112 211

4134

66

.~

2

I

2

7 1 28 10

122 411 5 21

5714

68

-._

2

186 300 8 14 5

7634

70

Table 1 Classification of fullerene isomers according to symmetry. Numbers at the top of the columns denote the number of atoms in carbon cluster. A number in the table is the number of isomers of the size indicated by the column with point group indicated by the row. Point groups which contain only optically active isomers are marked by a star

-

Volume 21 I, number 2,3

CHEMICAL PHYSICS LETTERS

4. Results

Table 1 gives the review of point groups of fullerenes with up to 70 carbon atoms. For each vertex number n, isomers are classified according to their symmetry, and the associated isomer counts are given in table 1. The supplementary material also contains the structure of vertex, edge and face orbits, and can be obtained from D. BabiC upon request. The content of table 1 is in full agreement with the related results at n= 60 given in ref. [ ZO]. The only exception regards the smallest isomer with the point group Ci, but this is apparently due to a trivial oversight [ 401. Optically active molecules are characterized by point groups with neither a plane of symmetry nor a rotoreflexive axis. As table 1 shows, these molecules prevail. It is also apparent that for each n > 36 there is at least one fullerene structure without any symmetry, other than trivial symmetry. In fact, as n grows, the fraction of isomers with no symmetry becomes ever closer to 1. An interesting fact is a disparity between the number of isomers with C, or C, point groups, on one side, and the number of isomers with Ci group, on the other. The Ci group is possible only in isomers with 4m carbon atoms (since the number of hexagons must be even), but otherwise it puts similar requirements on the arrangement of atoms as CZ and C, groups. However, isomers with Ci symmetry seem to be relatively rare. Although the upper size limit of the isomers considered only touches the area of experimentally obtained isomers, one could expect that the efforts of synthetic chemists [ 4 1,421 will also directed toward smaller fullerenes. A more comprehensive review of the symmetry properties of fullerene isomers will appear as a separate monographt [43].

Acknowledgement

Support from the Ministry of Science, Technology and Informatics of the Republic of Croatia through grant l-07-159 is gratefully acknowledged. CHS is

240

13 August 1993

supported by a grant from the Paul and Gabriela Rosenbaum Foundation.

References [ I ] R. Taylor, J. Chem. Sec. Perkin Trans. II ( 1992) 3. [2] F. Diederich, R. Ettl, Y. Rubin, R.L. Whetten, R. Beck, M. Alvarez, S. Anz, D. Shensharma, F. WudI, K.C. Khemani and A. Koch, Science 252 (1991) 548. [ 31 F. Diederich, R.L. Whetten, C. Thilgen, R. Ettl, I. Chao and M. M. Alvarez, Science 254 (1991) 1768. [4] K. Kikuchi, N. Nakahara, T. Wakabayashi, S. Suzuki, H. Shiromaru, Y. Miyake, K. Saito, I. Ikemoto, M. Kainosho and Y. Achiba, Nature 357 ( 1992) 142. [S] T.W. Ebessen and P.M. Ajayan, Nature 358 (1992) 220. [6] M. Endo and H.W. Kroto, J. Phys. Chem. 96 (1992) 6941. [ 71 D. Ugarte, Nature 359 (1992) 707. [8] H.W. Kroto, Nature 359 (1992) 670. [9] T.G. Schmalz, W.A. Seitz, D.J. KleinandG.E. Hite, J. Am. Chem.Soc.llO(1988) 1113. [IO] R. Taylor, G.J. Langley,T.J.S. Dennis, H.W. Krot0andD.R. M. Walton, J. Chem. Sot. Chem. Commun. (1992) 1043. [ 1I ] P.W. Fowler, J. Chem. Sot. Perkin Trans. 2 (1992) 145. [ 121 P.W. Fowler and J.I. Steer, J. Chem. Sot. Chem. Commun. (1987) 1403. [ 131K. Balasubramanian, Chem. Phys. Letters 197 (1992) 55. [ 141K Balasubramanian, J. Chem. Inf. Comput. Sci. 32 ( 1992) 47. [ 151D.E. Manolopoulos and P.W. Fowler, J. Chem. Phys. 96 ( 1992) 7603. [ 161 P.W. Fowler and C.M. Quinn, Theoret. Chim. Acta 70 (1986) 333. [ 171P.W. Fowler and D.B. Redmond, Theoret. Chim. Acta 83 (1992) 367. [ 181A. Tang, Q. Li, C. Liu and J. Li, Chem. Phys. Letters 201 (1993) 465. [ 191W.O.J. Boo, J. Chem. Educ. 69 (1992) 605. [20] P.W. Fowler, D.E. Manoclopoulos, D.B. Redmond and R.P. Ryan, Chem. Phys. Letters 202 ( 1993) 371. [21] M.S. Dresselhaus, G. Dresselhaus and R. Saito, Phys. Rev. B 45 (1992) 6234. [ 22 ] D.J. Klein, W.A. Seitz and T.G. Schmalz, J. Phys. Chem. 97 (1993) 1231. [ 231 A.J. Stone, Inorg. Chem. 20 ( 1981) 563. [24] 0. Ori and M. D’Mello, Chem. Phys. Letters 97 ( 1992) 49. [25] P. Mani, Math. Ann. 192 (1971) 279. [26] W. Thurston, Shapes of polyhedra, res. report GCG 7, Geometry Center, Univ. of Minn., (1991). [27] H.S.M. Coxeter, Virus macromolecules and geodesic domes, in: A spectrum of mathematics, ed. J.C. Butcher (Auckland Univ. Press, Oxford Univ. Press, 1971) pp. 98-107. [28] P.W. Fowler, J.R. Cremona and J.I. Steer, Theoret. Chim. Acta 73( 1988) 1. 1291D.E. Manolopoulos, J.C. May and S.E. Down, Chem. Phys. Letters 181 (1991) 105.

Volume 2 11, number 2,3

CHEMICAL PHYSICS LETTERS

[ 301 X. Liu, D.J. Klein, T.G. Schmalz and W.A. Seitz, J. Comput. Chcm. I2 (1991) 1252. [ 3 1 ] D. BabiCand N. Trinajstic, Comput. Chem., in press. [32] D.E. Manolopulos and P.W. Fowler, Chem. Phys. Letters 204 (1993) I. [33] C.H. Sah, Croat. Chem. Acta, in press. [341 H. Brucgesser and P. Mani, Math. Stand. 29 ( 1971) 197. [35] X. Liuand D.J. Klein, J. Comput. Chem. 12 (1991) 1243. [36] L. Weinberg, IEEETrans. Circuit Theory CT-13 (1966) 142. [37] J.E. Hopcroft and R.E. Tajan, J. Comput. Sys. Sci. 7 (1973) 323.

13 August 1993

[ 381 J.E. Hopcroft and JR. Wong, Proc. 6th Annual ACM Symp. on the Theory of Computing, Assoc. Comput. Math., New York 1974, pp. 172-l 84. [ 391 D. Babic and C.H. Sah, in preparation. [40] D.E. Manolopoulos, private communication. [4 1] F. Diederich and Y. Rubin, Angew. Chem. Intern. Ed. Eng. 31 (1992) 1101. [42] 5. Anthony, C.B. Rnobler and F. Diederich, Angew. Chem. Int. Ed. Eng. 32 (1993) 406. [43] P.W. Fowler and D.E. Manolopoulos, An Atlas of fullerenes, (Oxford Univ. Press, Oxford), in preparation.

241