Simple, Accurate Electrostatics-based Formulas for Calculating Ionization Potentials, Electron Affinities, and Capacitances of Fullerenes

A set of simple analytic formulas is derived via electrostatics-based methods to accurately calculate the values of electron affinities A_n and ionization potentials I_n for n-carbon icosahedral fullerene molecules as a function of their average radii R_n. These formulas reproduce with accuracy the values of A_n, I_n, and their scalings with 1/R_n that were determined previously in detailed, computationally intensive density functional theory calculations. The formula for A_n is derived from an enhanced image-charge model that treats the valence region of the icosahedral system as a quasispherical conductor of radius (R_n + δ), where δ=1/4W_∞ is a small constant distance determined from the work function W_∞ of graphene. Using this model, though, a formula for I_n that includes only electrostatics-like terms does not exhibit accuracy similar to the analogous formula for A_n. To make it accurate, a term must be added to account for the large symmetry-induced quantum energy gap in the valence energy levels (i.e., the HOMO-LUMO gap). An elementary two-state model based upon a quantum rotor succeeds in producing a simple expression that evaluates the energy gap as an explicit function of A_n. Adding this to the electrostatics-like formula for In gives a simple quantum equation that yields accurate values for I_n and expresses them as a function of A_n. Further, the simple equations for A_n and I_n yield much insight into both the physics of electron detachment in the fullerenes and the scaling with R_n of their quantum capacitances C_n = 1/(I_n − A_n).