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 An 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 + delta), where delta = 1/4W(infinity) is a small constant distance determined from the work function W-infinity 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 I-n 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 significant 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)).