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 pi bondings with Dewar structure bondings plus free radical pi electrons, positive and negative ions, metal locations, hydrogen additions etc. Cluster interactions include C36H12, C36H18, C24H12, C20H20, C20H10. Free radical pi-electrons are shown inside C-88(D-3), C-82(C-3v), C-74(D-3h), C-60(I-h), C-36(D-3h), C-34(C-3v) along with their endohedral interaction, with single metals Sc, Y, La and Lu having three top electrons s(2)d. Similarly, free-radical pi-electrons are shown inside C-28(T-d) and C-56(T-d) along with their interaction with single metals Ti, Zr, Hf and Th having four-top electrons s(2)d(2) (plus metal U having f(3)d). The extra case of Ti@C28H4 (C-3v or D-2d) is also considered. For dimetal interactions, we consider C-84(T-d) yielding Sc-2@C-84 and Lu-3@C-84 with subgroup symmetry D-2d and C-80(I-h) yielding Lu-2@C-80 with subgroup symmetry D-3d. For trimetal interactions, we consider C-82(C-3v) yielding Sc3C82 and La-3@C-82 with subgroup symmetry C-3v and C-88(D-3) yielding La-3@C-88 with subgroup symmetry C-3. For exohedral interactions, we consider C-30(D-5h) yielding LaC29+ and C-34(C-3v) yielding LaC33+ with subgroup symmetry C-s.