   The resonance structure of the fullerene C_{60} 
The resonance structure of the fullerene C_{60}
Finally let us investigate the resonance structure of the fullerene
C_{60}.
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 C_{60}.)
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 30sets 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 C_{60}.
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 30set A of edges (and according to the remark above
each resonance structure) can be identified with its characteristic function which is a function
c_{A} from the set of edges into the set {0,1} ,
such that c_{A}(e)=1 if and only if eÎA.
Using a labelling of the edges with labels 1,...,90 these functions
can be written as tuples (c_{A}(1),...,c_{A}(90)).
The set of these 90tuples is totally ordered by the lexicographic
ordering.
The permutation representation of the groups R or S
on the set of 30sets 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 [~U] of the
stabilizer U of any orbit representative and we say that the
orbit is of stabilizer type [~U] .
Using the computer algebra system GAP [17]
it is possible to derive
that there are 22 conjugacy classes [~U] of subgroups
U£S.
In table
all the conjugacy classes [~U]
of S are listed by giving the point group symbol of a
representative together with the size of the class [~U] (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 C_{60} of stabilizer type
[~U] .
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
I_{h} together with the table of marks and the
Burnsidematrix of I_{h} can
be found in [14].
U  U  [~U]   # 
U  U  [~U]   # 
C_{1}  1  1  70  C_{3v}  6  10  3 
C_{i}  2  1  0  D_{2h}  8  5  0 
C_{2}  2  15  19  C_{5v}  10  6  1 
C_{s}  2  15  36  D_{5}  10  6  0 
C_{3}  3  10  7  C_{5i}  10  6  0 
D_{2}  4  5  3  T  12  5  1 
C_{2v}  4  15  5  D_{3d}  12  10  3 
C_{2h}  4  15  3  D_{5d}  20  6  2 
C_{5}  5  6  0  T_{h}  24  5  1 
D_{3}  6  10  2  I  60  1  0 
C_{3i}  6  10  1  I_{h}  120  1  1

Resonance structures of the C_{60} 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 C_{60} acting on its sets of vertices, edges or faces, and he
demonstrates how to enumerate isomers of the form
C_{60}H_{n} and C_{60}H_{n}D_{m}.
Furthermore he computes the numbers of face and edge colourings
of C_{60} and determines the nuclearspin statistics for C_{60}
and C_{60}H_{60}.
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 C_{60} by summing over
so called partial cycle indices for certain subgroups
of S. (For more details on partial cycle indices see
[16].)
harald.fripertinger@kfunigraz.ac.at,
last changed: January 23, 2001
   The resonance structure of the fullerene C_{60} 