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

Abstract

Density functional theory (DFT) at the hybrid B3LYP level has been applied to Ge₁₂^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 Ge₁₂²⁻ although it remains a low energy structure for Ge₁₂²⁻ lying only 5.6 kcal mol⁻¹ above a bicapped arachno structure conforming to the Wade–Mingos rules. The three lowest energy structures for Ge₁₂⁴⁻ 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 Ge₁₂ are seen to be distorted versions of the icosahedron and the bicapped 10-vertex arachno lowest energy structures for Ge₁₂²⁻. The low energy structures for the even more hypoelectronic Ge₁₂²⁺ and Ge₁₂⁴⁺ are even more unusual including a hexacapped octahedron, a tetracapped square antiprism, and a double cube for Ge₁₂²⁺ and a C_2v structure with a central unique degree 6 vertex for Ge₁₂⁴⁺.