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[en] Highlights: • Second-order geometrical derivative with fractional occupation number is implemented. • IR and Raman spectra of graphene nanoribbons are simulated. • The analytic derivative is faster than the numerical derivative. Quantum chemical simulation of infrared (IR) and Raman spectra for molecules with open-shell, radical, or multiradical electronic structure represents a major challenge. We report analytic second-order geometrical derivatives of the Mermin free energy for the second-order self-consistent-charge density-functional tight-binding (DFTB2) method with fractional occupation numbers (FONs). This new method is applied to the evaluation of NO radical stretching modes in various open-shell molecules and to the prediction of the evolution of IR and Raman spectra of graphene nanoribbons with increasing molecular size.