Empty versus filled polyhedra: 11 vertex bare germanium clusters

The structures and energetics of centered 10-vertex Ge@Ge₁₀(z) (z = -4, -2, 0, +2, +4) clusters have been investigated by density functional theory (DFT) for comparison with the previously studied isomeric empty 11-vertex Ge₁₁(z) clusters. For the cationic species (z = +2, +4) such centered Ge@Ge₁₀(z) structures are shown to be energetically competitive (within ∼1 kcal mol⁻¹) to the lowest energy isomeric empty Ge₁₁(z) structures. These Ge@Ge₁₀(z) structures can be derived from the lowest energy empty 10-vertex Ge₁₀(z-4) structures by inserting a Ge⁴⁺ ion in the center. The outer 10-vertex polyhedron in the lowest energy Ge@Ge₁₀²⁺ dication structure is the most spherical D(4d) bicapped square antiprism, which is also the lowest energy structure of the empty Ge₁₀²⁻ dianion, as expected from the Wade-Mingos skeletal electron counting rules. For the tetracationic Ge₁₁⁴⁺ /Ge@Ge₁₀⁴⁺ system the lowest energy centered Ge@Ge₁₀⁴⁺ structure can be obtained by inserting a Ge⁴⁺ ion in the center of a C(3v) deltahedral empty Ge10 cluster. Centered 10-vertex polyhedral Ge@Ge₁₀(z) structures were also found for the neutral (z = 0) and dianionic (z = -2) systems but at significantly higher energies than the lowest energy isomeric empty Ge₁₁(z) structures.