A local-density-functional cluster method is used to calculate the structure and vibrational modes of C60. We find C-C lengths in good agreement with observed values. The effect of doping the molecule with three extra electrons is investigated and shown to result in a surprising shortening of the longer bonds. The second derivatives of the energy are evaluated and have enabled, for the first time, all the normal modes of the molecule to be found. We find these to be in fair agreement with the available experimental results.