Symmetry and isomeric structures of giant fullerenes and polyhedral clusters

Abstract

In order to illustrate the extension of cage cluster of given symmetry to a possible larger giant cluster of the same symmetry type, we have designed methods for drawing Schlegel (or quasi-Schlegel) diagrams that show clearly all (or almost all) the faces, vertices and edges of a three-dimensional cluster in a two-dimensional plane. The symmetries and structural details were used to select unit cells which serve as “abbreviations” for huge (fullerene) clusters, and these unit cells were then used as basis to derive mathematical principles for extending the structures up to infinitely large clusters. Possible isomers and various fullerene clusters (up to C 150 of D 5h symmetry) of different point-group symmetries are presented (the point groups involved for different sizes are I h, D 6d, D 5d, D 3d, D 6h, D 5h, D 3h, D 3, T d, C 3, C 3v, and D 6). Simple illustrations are given to show how such clearly visible detailed structures can be used to derive the possible magic numbers of (van der Waals) clusters and to formulate the mechanism and reaction coordinates of isomeric transformations. For mixed clusters with different sets of atoms (molecules), the common (minimum) subgroup symmetry is illustrated, which can be used to determine the total structures. An example of A 8B 12 (e.g. Ti 8C + 12) is given to show the possibility of T h or D 3d symmetry.