All atoms, molecules and crystals vibrate.  The C₆₀ molecule has 60 atoms that can each move in three dimensions. This gives 180 co-ordinates, or degrees of freedom as they are known. Rather than considering these co-ordinates separately, it is more usual to write the displacements in terms of normal modes. Three combinations of co-ordinates are translations, in which the whole molecule moves in a given direction without changing its shape. Three are rotations, in which the molecule rotates about its centre without changing shape. This leaves 174 patterns of vibration.

Normal modes can be labelled according to how they transform in group theory.  The easiest modes to visualise are the symmetric a modes. For the C₆₀ molecule, these are a breathing mode and a 'pentagonal pinch' mode shown below.

	The breathing mode of vibration of an icosahedron, in which the whole molecule gets larger and then smaller.
	The 'pentagonal pinch' mode of vibration of an icosahedron, in which the pentagonal faces get larger and then smaller (with the overall size of the molecule staying the same)

However, the remaining modes are much more complicated, with mixtures of radial and transverse motion. Nevertheless, we can draw pictures of all 174 normal modes. It is possible to draw just representations of the atoms and the bonds joining them. However, seeing through the molecule can be distracting. Therefore we have two alternatives showing all of the modes with fixed orientations. The results are also organised in two different ways. Click on the link below for the results:

• http://www.nottingham.ac.uk/~ppzjld/Visualise_vibration/ (http://www.nottingham.ac.uk/~ppzjld/Visualise_vibration/)

An alternative visualisation made by 4th year MSci project students Anthony Wood and Arani Navaneethakrishnan is available here.

To explore the modes further, it is useful to be able to rotate the images and change various parameters. We have produced an interactive demonstration to do this. You should be able to see it below if you have the free cdf player installed, which is downloadable from http://www.wolfram.com/products/player/.

The demonstration shows 88 of the 174 normal modes. The remaining normal modes can be produced by rotating those modes labelled x or yx about particular 2-fold axes of an icosahedron (mid-way along the common side joining two hexagons).

The work described here has been published in the Journal of Chemical Education (electronically in June 2010 and in the printed copy in August 2010):

A Pictorial Visualization of Normal Mode Vibrations of the Fullerene (C60) Molecule in Terms of Vibrations of a Hollow Sphere, J. Chem. Educ.87, 819–822 (2010)