How unusual is C 60? Magic numbers for carbon clusters

Abstract

Icosahedral carbon clusters with pentagonal and hexagonal faces are Goldberg polyhedra. They have 20( b 2 + bc + c 2) atoms, where b and c are non-negative integers, and obey an electron-counting rule similar to the famous Hückel (4 n + 2) prescription. When b − c is divisible by 3 the cluster has a multiple of 60 atoms and is closed-shell; otherwise it has 60 n + 20 atoms and is open-shell. A geometrical interpretation of this rule is: open-shell Goldberg clusters have atoms on the C 3 axes, closed-shell clusters do not. Closed shells are predicted for C 60, C 180, C 240, C 420, C 540,.... Irrespective of pointgroup symmetry, structures of large clusters may be generated by a leapfrog method from smaller ones. A tetrahedral structure generated in this way is the best candidate for C 120