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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 10,
2016
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The resonance
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