Simple, Accurate Electrostatics-based Formulas for Calculating Ionization Potentials, Electron Affinities, and Capacitances of FullerenesApril 2017
Topics: Electronics (General), Mathematics
A set of simple analytic formulas is derived via electrostatics-based methods to accurately calculate the values of electron affinities An and ionization potentials In for n-carbon icosahedral fullerene molecules as a function of their average radii Rn. These formulas reproduce with accuracy the values of An, In, and their scalings with 1/Rn 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 (Rn + 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 In that includes only electrostatics-like terms does not exhibit accuracy similar to the analogous formula for An. 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 An. Adding this to the electrostatics-like formula for In gives a simple quantum equation that yields accurate values for In and expresses them as a function of An. Further, the simple equations for An and In yield much insight into both the physics of electron detachment in the fullerenes and the scaling with Rn of their quantum capacitances Cn = 1/(In − An).