Magic Number of a Spherical Ca Cluster on C$_{60}$

Since the discovery of fullerenes, there have been a lot of interest in investigating the metal-fullerene clusters. Mass spectrum on the metal(M) covering on a C$_{60}$ complex showed a peak at M$_{32}$C$_{60}$. This magic number was theoretically explained using the geometry based on the C-Ca binding. However, such theories could not clearly reveal why the peak at M$_{32}$C$_{60}$, especially for calcium clusters, is so prominent compared to smaller number of metal atoms. Using {\it ab initio} MD simulations, we have found that for Ca covering with less than 32 atoms, Ca atoms tend to be retracted to a cluster rather than to be bound on each face of C$_{60}$ even though the Ca atoms are deposited symmetrically. Such a cluster does not have specific number of atoms and it is bound to C$_{60}$ through van der Waals interaction. However, once Ca forms a spherical shell with 32 atoms, the structure is quite rigid so that it will not be retracted to a cluster. We have also found that the interaction between an individual Ca atom and each C$_{60}$ face gets loosen so that C$_{60}$ can rotate within Ca$_{32}$ sphere. The phonon spectrum has been obtained by spectral analysis and electronic orbitals of Ca$_{32}$C$_{60}$ will also be presented.