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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. Using a backtrack algorithm it is possible to
compute a list of all 12500 possibilities to do that. (It is
already known in literature [14] 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 [3] 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 [12] 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).
| 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
Similar methods are applied by Balasubramanian [1][2][3] and by Fujita [11].
harald.fripertinger@kfunigraz.ac.at
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The resonance
structure of the fullerene C60 |
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