To reveal new structure-property relationships in the nonlinear optical (NLO) properties of fullerenes that are associated with their open-shell character, we investigated the interplay between the diradical character (y(i)) and second hyperpolarizability (longitudinal component, γ(zzzz)) in several fullerenes, including C(20), C(26), C(30), C(36), C(40), C(42), C(48), C(60), and C(70), by using the broken-symmetry density functional theory (DFT; LC-UBLYP (μ=0.33)/6-31G*//UB3LYP/6-31G*). We found that the large differences between the geometry and topology of fullerenes have a significant effect on the diradical character of each fullerene. On the basis of their different diradical character, these fullerenes were categorized into three groups, that is, closed-shell (y(i)=0), intermediate open-shell (0