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 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 R(4)C(4)B(8)H(8). 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(2v) structure with a central unique degree 6 vertex for Ge(12)(4+).

Zintl‐Ionen, Käfigverbindungen und intermetalloide Cluster der Elemente der 14. und 15. Gruppe

“…Die höchsten besetzten MOs der [E n ] 2− ‐Cluster weisen dagegen lone‐pair‐Charakter auf. Das HOMO von [B 12 H 12 ] 2− ist der oberflächenbindende g u ‐Satz,99, 307 und das HOMO der [E 12 ] 2− ‐Anionen ist der h g ‐Satz (E=Si,307 Ge,308 Sn99, 309) oder der t 1u ‐Satz (E=Pb100, 300, 310) mit lone‐pair‐Charakter.…”

“…Strukturoptimierungen für die [E 12 ] 2− ‐Anionen mit E=Si,306, 307, 311, 312 Ge,306, 308, 312 Sn99, 306, 312, 313 und Pb,100, 300, 306, 313 zeigen erwartungsgemäß, dass die ikosaedrische Struktur ( I h ‐Symmetrie) für alle [E 12 ] 2− ‐Cluster einem Grundzustand entspricht. Dieser stellt aber nur für [Sn 12 ] 2− und [Pb 12 ] 2− eindeutig das globale Minimum dar.…”

Beyond the Icosahedron: A Density Functional Theory Study of 14‐Atom Germanium Clusters

“…However, if the otherwise external lone pairs of the underlying Al 8 cube in Al 8 (µ 8 -Si)(µ 4 -AlCp*) 6 are also counted, then the omnicapped cube in this cluster becomes a 24 + (8)(2) = 40-electron system, which is a favored number using the jellium model. [20,21] The research discussed in this paper extends our DFT studies to clusters with more than 12 vertices in order to study effects of electron count on supraicosahedral cluster geometry. The 14-vertex systems were chosen in preference to the 13-vertex systems because of their higher symmetry and greater variety outside polyhedral borane and carborane chemistry.…”

“…The DFT methods used in this work are well established having previously been used by our group to study bare germanium clusters having five to twelve germanium atoms. [22][23][24][25][26][27] …”

Biradical character in the ground state of [Mn@Si12]+: a DFT and CASPT2 study

“…One explanation is that the Cr or W atom is simply too large to fit inside an icosahedral Si 12 cage, and so the prismatic structure, with its larger volume, is preferred. 14,15 Jellium-type models can also be applied in this context, and configurations of 1s 2 1p 6 1d 10 (18 electrons) and 1s 2 1p 6 1d 10 2s 2 (20 electrons) have been invoked to account for the relatively high dissociation energies of Cr@Si 12 and isostructural Fe@Si 12 . [16][17][18] In their initial report, 3 Hiura et al suggested that the stability of W@Si 12 could be rationalised in terms of an 18-electron rule: donation of 12 electrons from the Si 12 cage to the metal raises the electron count to 18, leaving three electrons per Si center to form the three Si-Si bonds.…”