 |
 |
 |
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
The substitution xi↦ 1+xi into the
cycle index is implemented in SYMMETRICA as well. It is called
polya_sub(a,b,c), where a is the cycle
index, b is the number of indeterminates xi
that should be replaced by 1+xi and c is
the result of the expansion of the cycle index.
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
different C60-kBk molecules.
harald.fripertinger "at" uni-graz.at, May 10,
2016
|
|
|
 |
 |
 |
Colourings of
the C60H60-molecule |
 |
 |