A particle‐on‐a‐sphere model may be used as a crude approximation for the π system of a C60 molecule. The radius is 3.51 Å. The energy eigenfunctions are the well‐known spherical harmonics solutions to the purely angular Schrödinger equation, Yl^m. (Ref: See Section 1 of the paper published by Savina et al. in Chem. Phys. Lett. in 1993.) (a) Explain why the highest occupied molecular orbitals (HOMO) and the lowest unoccupied molecular orbitals (LUMO) correspond to l=5 and l=6, respectively.
(b) E= 1(1 + 1)ħ2 /2mer; where me is the mass of an electron and r, is the radius. Calculate the first allowed transition predicted by this model.
(c) Compare your result in (b) with the experimental data of 3.0-3.8 eV.