Valence points and symmetry views of the structures of carbon clusters, fullerenes and metallofullerenes

Planar Schlegel diagrams of fullerene cages for the reaction mechanisms between open clusters and between metals and fullerenes are used to determine the qualitative symmetry properties. The detailed Schlegel diagrams help to illustrate the subgroup symmetries of the cages that contain localized π bondings with Dewar structure bondings plus free radical π electrons, positive and negative ions, metal locations, hydrogen additions etc. Cluster interactions include C36H12, C36H18, C24H12, C20H20, C20H10. Free radical π-electrons are shown inside C88(D3), C82(C3v), C74(D3h), C60(Ih), C36(D3h), C34(C3v) along with their endohedral interaction, with single metals Sc, Y, La and Lu having three top electrons s2d. Similarly, free-radical π-electrons are shown inside C28(Td) and C56(Td) along with their interaction with single metals Ti, Zr, Hf and Th having four-top electrons s2d2 (plus metal U having f3d). The extra case of [email protected]28H4 (C3v or D2d) is also considered. For dimetal interactions, we consider C84(Td) yielding Sc2@C84 and Lu2@C84 with subgroup symmetry D2d and C80(Ih) yielding Lu2@C80 with subgroup symmetry D3d. For trimetal interactions, we consider C82(C3v) yielding Sc3C82 and La3@C82 with subgroup symmetry C3v and C88(D3) yielding La3@C88 with subgroup symmetry C3. For exohedral interactions, we consider C30(D5h) yielding LaC29+ and C34(C3v) yielding LaC33+ with subgroup symmetry Cs.