Density functional theory study of twelve-atom germanium clusters: conflict between the Wade-Mingos rules and optimum vertex degrees

Density functional theory (DFT) at the hybrid B3LYP level has been applied to Ge-12(z) bare germanium clusters (z = -6, -4, -2, 0, +2, +4, +6) starting from 11 initial configurations. The Wade-Mingos rules are seen to have limited value in rationalizing the results since they frequently require vertex degrees higher than the optimum vertex degree of 4 for germanium. Thus the expected I-h regular icosahedron is no longer the global minimum for Ge-12(2-) although it remains a low energy structure for Ge-12(2-) lying only 5.6 kcal mol(-1) above a bicapped arachno structure conforming to the Wade-Mingos rules. The three lowest energy structures for Ge-12(4-) within 11 kcal mol(-1) are a prolate (elongated) polyhedron with six quadrilateral faces and eight triangular faces, the dual of the bisdisphenoid with four trapezoidal and four pentagonal faces, and a polyhedron with two quadrilateral and 16 triangular faces related but not identical to the polyhedron found in the known tetracarbon carboranes R4C4B8H8. The lowest energy structures for the neutral Ge-12 are seen to be distorted versions of the icosahedron and the bicapped 10-vertex arachno lowest energy structures for Ge-12(2-). The low energy structures for the even more hypoelectronic Ge-12(2+) and Ge-12(4+) are even more unusual including a hexacapped octahedron, a tetracapped square antiprism, and a double cube for Ge-12(2+) and a C-2 nu structure with a central unique degree 6 vertex for Ge-12(4+).