Dependence of the π-electron eigenvalue sum on the number of atoms in almost spherical C cages

A free-electron-like model is first shown to lead to the eigenvalue sum of the π electrons in almost spherical C cages, being proportional to N, where N is the number of C atoms. The model is also closely related to a tight-binding model, stemming from the Hückel theory of π electrons. The equilibrium radius $R_e$ of the C cages is assumed to be determined by the “rule” of constant surface area per C atom, and hence $R_e\propto N^{1/2}.$ We then compare these models with a series of Hartree-Fock calculations on the fullerenes ${\mathrm{C}}_{50},$ ${\mathrm{C}}_{60},$ ${\mathrm{C}}_{70},$ and ${\mathrm{C}}_{84},$ from which more accurate energy scaling relations are obtained. Finally, the relation between the total energy of the four C cages at equilibrium and the all-electron eigenvalue sum is confirmed by the Hartree-Fock studies.