Beyond the Wade-Mingos rules in bare 10- and 12-vertex germanium clusters: Transition states for symmetry breaking processes

The lowest energy structures of bare Ge-n(z) clusters (n=10, 12; z= -6, 0, +.2, +4) obtained using density functional theory (DFT) at the hybrid B3LYP level often are relatively low-symmetry polyhedra not readily recognizable by the Wade - Mingos rules. However, such optimized structures may arise from higher symmetry transition states through symmetry breaking processes. Thus the lowest energy. structures for the Ge-1.0(6-) and Ge-12(6-) clusters with hyperelectronic arachno 2n+6 skeletal electron counts are derived from pentagonal and hexagonal prism transition states, respectively, and retain the pentagonal and hexagonal faces of the prisms upon symmetry-breaking optimization. In addition, a variety of capped cube, prism, and antiprism transition states are found for the hypoelectronic Ge-10(4+), Ge-12, and Ge-12(4+) clusters, which go to low-energy low-symmetry optimized structures, typically C, or C-i, upon following the normal modes of the imaginary vibrational frequencies.