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 (yi) and second hyperpolarizability (longitudinal component, γzzzz) in several fullerenes, including C20, C26, C30, C36, C40, C42, C48, C60, and C70, 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 (yi=0), intermediate open-shell (0<yi<1), and almost pure open-shell compounds (yi≅1), which originated from their diverse topological features, as explained by odd-electron-density and spin-density diagrams. For example, we found that closed-shell fullerenes include C20, C60, and C70, whereas fullerenes C26 and C36 and C30, C40, C42, and C48 are pure and intermediate open-shell compounds, respectively. Interestingly, the γzzzz enhancement ratios between C30/C36 and C40/C60 are 4.42 and 11.75, respectively, regardless of the smaller π-conjugation size in C30 and C40 than in C36 and C60. Larger γzzzz values were obtained for other fullerenes that had intermediate diradical character, in accordance with our previous valence configuration interaction (VCI) results for the two-site diradical model. The γzzzz density analysis shows that the large positive contributions originate from the large γzzzz density distributions on the right- and left-extended edges of the fullerenes, between which significant spin polarizations (related to their intermediate diradical character) appear within the spin-unrestricted DFT level of theory.
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