Some counting problems Colourings of the C60H60-molecule The resonance structure of the fullerene C60

The resonance structure of the fullerene C60

Finally let us investigate the resonance structure of the fullerene C60. For doing this we have to place 30 double bonds and 60 single bonds into the truncated icosahedron, such that each vertex is incident with 2 single bonds and 1 double bond. Obviously it is enough to find the positions of all the double bonds. So we have to determine all sets consisting of exactly 30 edges of the fullerene, such that each vertex is incident with exactly one edge. Applying a SYMMETRICA program which uses a backtrack algorithm it is possible to compute a list of all 12500 possibilities to do that. (It is already known in literature [19] that there are 12500 resonance structures of the fullerene C60.) But many of these structures coincide when applying a symmetry operation on the truncated icosahedron. The symmetry groups R or S act as permutation groups on the set of all edges of the fullerene, so they act on the set consisting of all 30-sets of edges as well. Especially they act on the set of all resonance structures. Since the backtrack algorithm above yields a complete set of all structures we can take a constructive approach to compute not only the number of all the essentially different resonance structures, but a representative of each of these structures and the symmetry groups of all these representatives as well. This method is a constructive approach for the determination of the number of classes of Kekulé structures of C60. In [7] it is stated that this number could be computed by using the inclusion exclusion formula. Let me give the mathematical background for the constructive approach. Each 30-set A of edges (and according to the remark above each resonance structure) can be identified with its characteristic function which is a function χA from the set of edges into the set {0,1}, such that χA(e)=1 if and only if e∈ A. Using a labelling of the edges with labels 1,...,90 these functions can be written as tuples (χA(1),...,χA(90)). The set of these 90-tuples is totally ordered by the lexicographic ordering. The permutation representation of the groups R or S on the set of 30-sets of edges can be rewritten as a group action on the set of these characteristic functions induced by a group action on the domain. Choosing as a canonical representative the lexicographic smallest member of its orbit, we can apply standard algorithms to compute a list of all different resonance structures from the list of all 12500 resonance structures. Together with each representative we also get its stabilizer which is its symmetry group. Since all the elements in one orbit have conjugated stabilizers we can associate an orbit with the conjugacy class Ũ of the stabilizer U of any orbit representative and we say that the orbit is of stabilizer type Ũ . Using the computer algebra system GAP [17] it is possible to derive that there are 22 conjugacy classes Ũ of subgroups U≤S. In table all the conjugacy classes Ũ of S are listed by giving the point group symbol of a representative together with the size of the class Ũ (i.e. number of subgroups conjugated to U), the size of U (i.e. the number of elements in the subgroup U) and the number of orbits of resonance structures of C60 of stabilizer type Ũ . Summarising, there are 158 (260) different resonance structures with respect to the symmetry group S (or R respectively). A list of all the conjugacy classes of subgroups of the icosahedral point group Ih together with the table of marks and the Burnside-matrix of Ih can be found in [14].

 

U |U| |Ũ | # U |U| |Ũ | #
C1 1 1 70 C3v 6 10 3
Ci 2 1 0 D2h 8 5 0
C2 2 15 19 C5v 10 6 1
Cs 2 15 36 D5 10 6 0
C3 3 10 7 C5i 10 6 0
D2 4 5 3 T 12 5 1
C2v 4 15 5 D3d 12 10 3
C2h 4 15 3 D5d 20 6 2
C5 5 6 0 Th 24 5 1
D3 6 10 2 I 60 1 0
C3i 6 10 1 Ih 120 1 1
Resonance structures of the C60 fullerene

The SYMMETRICA routine all_orbits_right_from_vector(a,b,c) computes a complete list c of these representatives. The permutation group acting on the domain of the functions is given by a, which is a VECTOR of generators, which must be PERMUTATION-objects. The VECTOR b is a list of all the functions which must be tested to be a canonical representative or not. The list of all canonical representatives will be computed as the VECTOR c. For our problem of determining all the different resonance structures we have to take for a a VECTOR of the generators of the symmetry group acting on the set of edges of the truncated icosahedron. And for b we have to take the VECTOR of all the 12500 resonance structures generated by the backtrack algorithm described above.

Balasubramanian extensively applied Pólya theory for the enumeration of isomers. He published a review on chemical and spectroscopic applications of this theory in [1]. In [2][3][7] he computes the cycle indices of the symmetry group of C60 acting on its sets of vertices, edges or faces, and he demonstrates how to enumerate isomers of the form C60Hn and C60HnDm. Furthermore he computes the numbers of face and edge colourings of C60 and determines the nuclear-spin statistics for C60 and C60H60. When actually computing the numbers of isomers given in his papers he reports that he had to face complexity problems and arithmetic overflows occurred. So he had to implement a double precision arithmetic into his algorithm. When using SYMMETRICA all these problems do not occur, since SYMMETRICA is working with integers of arbitrary length and with rational numbers stored as fractions.

Fujita [15] computes the numbers of colourings of the truncated icosahedron by stabilizer type. He derives the cycle index of S acting on the set of vertices of C60 by summing over so called partial cycle indices for certain subgroups of S. (For more details on partial cycle indices see [16].)


harald.fripertinger "at" uni-graz.at, May 16, 2011

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