Ab initio scaling of the second hyperpolarizabilities of carbon cages

A sum over molecular orbitals scheme in an ab initio $\text{6-31}$ G^* basis set is used to calculate the static second hyperpolarizabilities, γ, of the eight carbon cages now available in macroscopic quantities: ${\mathrm{C}}_{\mathrm{60}},$ ${\mathrm{C}}_{\mathrm{70}},$ one isomer of ${\mathrm{C}}_{\mathrm{76}},$ three isomers of ${\mathrm{C}}_{\mathrm{78}},$ and two isomers of ${\mathrm{C}}_{\mathrm{84}}.$ For ${\mathrm{C}}_{\mathrm{60}}$ and ${\mathrm{C}}_{\mathrm{70}},$ there is good agreement between calculations and available experimental data. The range of nuclearities is extended to cover the most stable isomers of ${\mathrm{C}}_{\mathrm{40}}$ and ${\mathrm{C}}_{\mathrm{100}}.$ It is found that γ increases as a nonlinear function of both the number of atoms and the number of partial double bonds (hexagon–hexagon edges). Two proposed scaling laws lead to very different values of γ for systems with large numbers of carbon atoms but, even for the less steeply rising function, comparison with experimental data shows that, weight for weight, fullerenes should have a larger response than polyenic systems.