Cylindrical fullerenes: The smallest nanotubes?

Isomeric five- and sixfold axially symmetric fullerene molecules (belonging to the point groups I h , I , D 5h , D 5d , D 5 , D 6h , D 6d , d 6 ) can be constructed and counted by a formula based on triangulations of the sphere. Two integers characterise the hemispherical cap and hence the cylinder radius. All fivefold symmetric caps are hemi-icosahedra. Expansion of C 60 and C 180 along a C 5 , axis accounts for single-shell nanotube diameters of ~0.7 and ~1.2nm, respectively. Rules for electronic closed shells in cylindrical fullerenes are presented. The C 60 cap allows closed-shell π configurations at 70 + 30k = 70, 100, 130,… atoms; the sixfold symmetric C 72 cap allows closed shells at 84 + 36 k = 84, 120, 156,… atoms; larger caps allow leapfrog closed shells at 60 + 30 k = 60, 90, 120,… atoms (in fivefold symmetry) and 72 + 36 k = 72, 108, 144,… atoms (in sixfold symmetry).