Static dipole polarizabilities of icosahedral fullerenes from C$_{60}$ to C$_{2160}$ by all-electron density-functional theory

The electronic response of C$_{60}$, C$_{180}$, C$_{240}$, C$_{540}$, C$_{720}$, C$_{960}$, C$_{1500}$, and C$_{2160}$ fullerenes is characterized by determining their static dipole polarizabilities by all-electron density- functional theory. We first determine the dipole polarizabilities of C$_{60}$, C$_{180}$, C$_{240}$, and C$_{540}$ fullerenes by the finite-field method, using 35 basis functions per atom (NRLMOL basis set), and using the PBE-GGA. In the second set of calculations the sum-over-states (SOS) polarizabilities for all fullerenes from C$_{60}$ through C$_{2160}$ are determined by our fully analytic formulation of density functional theory(ADFT). The 6-311G(d,p) basis set is used in the ADFT calculation, which amounts to 38800 basis functions for the largest fullerene in this series, namely C$_ {2160}$. The SOS polarizabilities are roughly 4 times larger than the finite-field polarizabilities. When scaled by a correction factor obtained within linear response theory, the SOS polarizabilities are within 1-3\% of the finite-field polarizabilities. The polarizability per carbon atom increases from 1.34 \AA$^3$ in C$_{60}$ to 4 \AA$^3$ in C$_{2160}$ while the ratio of fullerene polarizability to its volume approaches unity pointing to quenching of quantum size effects by C$_{2160}$. The results show previous tight-binding calculations greatly exaggerate the electronic response of large fullerenes.