Using the spin-unrestricted density functional theory method, we investigate the interplay between the diradical character y and second hyperpolarizabilities gamma (the third-order nonlinear optical (NLO) properties at the molecular scale) of corannulene derivatives with two phenoxyl radicals. This molecule in the singlet state exhibits intermediate y and thus displays a significantly larger gamma value than the triplet state and the closed shell bis-phenol analogue. We also examine the planar molecules involving a coronene moiety in place of the curved corannulene. The intermediate y and large gamma values of the corannulene systems are found to originate not from their curved skeleton but from the equilibrium between benzenoid/quinoid resonance forms due to delocalization of the radical electrons,of the terminal phenoxyl rings. The longitudinal gamma value of the singlet state is found to be comparable to that of S-indaceno[1,2,3-cd;5,6,7-c'd']diphenalene, which is known to be one of the organic Molecules with the largest two-photon absorption cross section in this size of pure hydrocarbons. The present system is thus expected to be a promising candidate for highly efficient open-shell NLO molecules.