The sphericity of the diverse 10-vertex polyhedra found in bare post-transition metal clusters: germanium clusters with interstitial magnesium atoms as model systems

The diverse polyhedra found experimentally in bare 10-vertex centered post-transition element clusters characterized structurally by X-ray crystallography include the D-4d bicapped square antiprism Zn@In-10(8-) in the intermetallic K8In10Zn as well as the centered lead clusters M@Pb-10(2-) in [K(2,2,2-crypt)](2)[M@Pb-10] (M = Ni, Pd, Pt), the C-3 nu tetracapped trigonal prism M@In-10(10-) in the intermetallic K10In10M (M = Ni, Pd, Pt), the D-5d pentagonal antiprism Pd@Bi-10(4+) in Bi14PdBr16, and the D-5h pentagonal prism in the trianions M@Ge-10(3-) (M = Fe, Co). The structures of the 10-vertex germanium clusters containing an interstitial magnesium atom (atomic radius 1.30 angstrom), namely Mg@Ge-10(z) (z = +2, 0, -2, -4), have been optimized as models to analyze the frontier molecular orbitals of the four experimentally known 10-vertex polyhedra noted above. In this connection, the molecular orbital patterns of the most spherical D-4d bicapped square antiprism cluster Mg@Ge-10 as well as that of an approximately spherical C-3 nu cluster Mg@Ge-10(2+) derived from the tetracapped trigonal prism exhibit the spherical harmonic ordering of the jellium model. In this ordering scheme, the 20-orbital {1s, 1p, 1d, 2s, 1f, 2p} shell is filled before any of the 1g molecular orbitals. However, the molecular orbital patterns of the oblate D-5d pentagonal antiprism cluster Mg@Ge-10(4-) and the D-5h pentagonal prism cluster Mg@Ge-10 no longer exhibit the spherical harmonic ordering of the jellium model. In these clusters, the polar 1f(z(3)) orbital is raised to antibonding levels and the pair of hexagonal prism g molecular orbitals become bonding orbitals.