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In the section on icosahedral
symmetry operations, we showed some of the symmetry operations that
can be performed on an isolated fullerene molecule which will leave the
molecule looking the same.
In our papers on STM imaging of C60 adsorbed on a surface
substrate (see publication
list here), we have assumed that the surface perturbs that atoms nearest
to it in some way that makes the direction perpendicular to the surface
different to the directions in the plane of the surface, and also breaks
the symmetryof the molecule about a horizontal plane. We can imagine this
by drawing the fullerene as an egg-shape (rather than a football [soccerball]).
We are not saying that the molecule will necessarily adopt this shape
(although it could) - just that the effect on the symmetry is the same
as with this shape.
The diagams below illustrate this for a molecule adsorbed with a pentagon
facing upwards.
The fullerene molecule will still look exactly the same
if it is rotated by 2*pi/5 radians (=360/5o) about a vertical
C5 axis.
Five of the atoms around the pentagon facing upwards have been coloured
differently as a guide to the eye. Obviously, in a real fullerene molecule
all of the carbon atoms look the same. The image on the right shows the
result of rotating the image on the left by 2*pi/5 radians (i.e. 1/5 of
a complete revolution). If the atoms had not been coloured differently,
we would not have been able to tell that the rotation had taken place.The
same is not true for 5-fold rotations about axes through the centres
of the other pentagons though.
The fullerene molecule will also look exactly the same
if it is reflected about various planes.
The image on the right shows the effect of reflecting the image on the
left about the vertical plane shown.
However, there are no longer any 3-fold rotations. The
images below show the effect of a rotation of 2*pi/3 radians (=120o)
about the centre of a hexagon.
For the case of an isolated molecule, this was a symmetry axis.
This time, the image on the right is different to that on the
left. We can tell that the rotation had taken place. Therefore
this is no longer a symmetry operation.
In terms of group theory, the point group symmetry we
now have in this case is C5v.
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