LAUSR

The duals of the most spherical closo borane deltahedra having from 6 to 16 vertices form a series of homologous spherical trivalent polyhedra with even numbers of vertices from 8 to 28. This series of homologous polyhedra is found in endohedral clusters of the group 14 atoms such as the endohedral germanium cluster anions [M@Ge10]3− (M = Co, Fe) and [Ru@Ge12]3− The next members of this series have been predicted to be the lowest energy structures of the endohedral silicon clusters Cr@Si14 and M@Si16 (M = Zr, Hf). The largest members of this series correspond to the smallest fullerene polyhedra found in the endohedral fullerenes M@C28 (M = Zr, Hf, Th, U). The duals of the oblate (flattened) ellipsoidal deltahedra found in the dirhenaboranes Cp*2Re2Bn−2Hn−2 (Cp* = η5-Me5C5; 8 ≤ n ≤ 12) are prolate (elongated) trivalent polyhedra as exemplified experimentally by the germanium cluster [Co2@Ge16]4− containing an endohedral Co2 unit.