How are 13-atom, 55-atom, and 147-atom icosahedral clusters built?

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cluster Figure 1: The four stable geometric structures of the seven-atom cluster of argon, in order of increasing energy: (A) A pentagonal bipyramid. (B) A regular octahedron with one face capped by the seventh atom. (C) A regular tetrahedron with three of its faces capped by other atoms. (D) A trigonal bipyramid with two of its faces capped by other atoms; although this has the highest energy of the four structures, it is very close in energy to the tricapped tetrahedron.

Icosahedral clusters exhibit unusual stability when they contain specific numbers of atoms that form complete, regular icosahedra. These structures are built in successive shells:1

13-atom cluster: This is the fundamental unit, where all but one atom occupy equivalent sites.1
55-atom cluster: This is constructed by taking the 13-atom icosahedral core and adding 12 atoms to its vertices and 30 atoms to the centers of its edges.1
147-atom cluster: This is built upon the 55-atom core by adding 12 atoms to the vertices of the outermost shell, one atom to the center of each of the 20 faces, and two atoms along each of the 30 edges.1

These clusters are typically bound by van der Waals forces or other simple forces dependent on interatomic distance. Their special stability is determined by the individual stabilities of these atomic shells.1