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Colourings of
the C60H60-molecule |
Colourings of the
C60H60-molecule
The cycle indices for the action on the set of vertices can be
applied to investigate the C60
H60-molecule. Consider a colouring of the 60
vertices C with 2 colours H or Cl. In other words a colouring is a
function f from the set of vertices of the truncated
icosahedron into the set {H,Cl}. Then Pólyas theorem says
that the number of
C60HkCl60-k molecules is
given as the coefficient of xk in the expansion
of the cycle index of the symmetry group R or S
acting on the set of vertices when all the indeterminates
xi are replaced by 1+xi. (In
mathematical terms we are speaking of weighted enumeration when we
define the weight of a colouring as a product weight. Let
w(H)=x and w(Cl)=1, where x is an
indeterminate over the set of rationals. Then the product weight of
the colouring f is defined to be
where the product runs over all vertices v.) In table you can find the numbers of different
molecules C60HkCl60-k both
for the symmetry groups R and S.
| k |
S |
R |
| 0, 60 |
1 |
1 |
| 1, 59 |
1 |
1 |
| 2, 58 |
23 |
37 |
| 3, 57 |
303 |
577 |
| 4, 56 |
4190 |
8236 |
| 5, 55 |
45718 |
91030 |
| 6, 54 |
418470 |
835476 |
| 7, 53 |
3.220218 |
6.436782 |
| 8, 52 |
21.330558 |
42.650532 |
| 9, 51 |
123.204921 |
246.386091 |
| 10, 50 |
628.330629 |
1256.602779 |
Number of
C60HkCl60-k molecules
More or less in the same way the numbers of hetero
fullerenes, these are fullerenes where some of the carbon
atoms are replaced by other atoms, can be computed from the cycle
index of the symmetry group acting on the set of vertices. For
instance replacing some C atoms by B atoms the numbers in table can be interpreted as the numbers of
essentially different C60-kBk
molecules.
harald.fripertinger@kfunigraz.ac.at
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Colourings of
the C60H60-molecule |
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