Topology and Energetics of Metal-Encapsulating Si Fullerene-Like Cage Clusters

On the basis of a topological discussion as well as an {\it ab initio} calculation, we show that it is possible to construct a fullerene-like Si cage by doping of a transition metal atom. The cage is a simple 3-polytope which maximizes the number of its inner diagonals close to the metal atom. Our topological argument also reveals how closely the structure of the fullerene-like Si cages studied is related to that of fullerenes themselves.