Static dielectric response of icosahedral fullerenes from ${\mathrm{C}}_{60}$ to ${\mathrm{C}}_{2160}$ characterized by an all-electron density functional theory

The static dielectric response of ${\mathrm{C}}_{60}$, ${\mathrm{C}}_{180}$, ${\mathrm{C}}_{240}$, ${\mathrm{C}}_{540}$, ${\mathrm{C}}_{720}$, ${\mathrm{C}}_{960}$, ${\mathrm{C}}_{1500}$, and ${\mathrm{C}}_{2160}$ fullerenes is characterized by an all-electron density functional method. First, the screened polarizabilities of ${\mathrm{C}}_{60}$, ${\mathrm{C}}_{180}$, ${\mathrm{C}}_{240}$, and ${\mathrm{C}}_{540}$ are determined by the finite-field method using Gaussian basis set containing 35 basis functions per atom. In the second set of calculations, the unscreened polarizabilities are calculated for fullerenes ${\mathrm{C}}_{60}–{\mathrm{C}}_{2160}$ from the self-consistent Kohn-Sham orbitals and eigenvalues using the sum-over-states method. The approximate screened polarizabilities, obtained by applying a correction determined within linear response theory, show excellent agreement with the finite-field polarizabilities. The static dipole polarizability per atom in ${\mathrm{C}}_{2160}$ is $\left(4{\mathrm{\AA }}^3\right)$ three times larger than that in ${\mathrm{C}}_{60}$ $\left(1.344{\mathrm{\AA }}^3\right)$. Our results reduce the uncertainty in various theoretical models used previously to describe the dielectric response of fullerenes and show that quantum size effects in polarizability are significantly smaller than previously thought.

Figure 1

(Color online) The fullerenes structures optimized at $\mathrm{ADFT}∕6\text{−}311{\mathrm{G}}^{**}$ level of theory for which polarizabilities are calculated.

Figure 2

(Color online) The ratio of polarizability, ${\alpha }_{\mathit{RPA}}$, and volume, ${\overline{r}}^3$, as a function of fullerene radius $\overline{r}$. The squares represent polarizability obtained by the finite-field method.

Figure 3

The percent increase in polarizability, ${\alpha }_{\mathit{RPA}}$, as a function of percent increase in volume, ${\overline{r}}^3$, for ${\mathrm{C}}_{960}$ fullerene.