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

Abstract

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@Pb10 2− in [K(2,2,2-crypt)]2[M@Pb10] (M = Ni, Pd, Pt), the C3v tetracapped trigonal prism M@In 10 10− in the intermetallic K10In10M (M = Ni, Pd, Pt), the D 5d pentagonal antiprism Pd@Bi10 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 A), namely Mg@Ge 10 (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@Ge10 as well as that of an approximately spherical C 3v 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@Ge10 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. The sphericities of the four distinct 10-vertex polyhedra found experimentally in centered bare 10-vertex post-transition metal clusters have been evaluated by comparing the relative energies of the cluster bonding molecular orbitals in model Mg@Ge 10 (z = −4 to +2) species with the spherical harmonic ordering predicted by the jellium model. Using this as a criterion, the D 4d bicapped square antiprismatic and C 3v tetracapped trigonal prismatic structures can be regarded as essentially spherical. However, the molecular orbital patterns of the oblate D 5d pentagonal antiprismatic and D 5h pentagonal prismatic clusters deviate significantly from spherical harmonic ordering.