In response theory, the hypervirial relations provide alternative formulations for the dynamic (frequency-dependent) molecular response properties. In this contribution, though the length formulation is the most commonly used in the calculations, we derive the full velocity formulations of the dynamic electric-dipole polarizability, \(\alpha _{\alpha \beta }(-\omega ;\omega )\) , and the dynamic electric-dipole second-harmonic generation first hyperpolarizability, \(\beta _{\alpha \beta \gamma }(-2\omega ;\omega ,\omega )\) . Both formulations provide identical responses for exact wavefunctions and for variationally optimized wavefunctions in the limit of a complete basis set. To assess how the exact relations between the length and velocity formulations are satisfied for approximate wavefunctions, we performed TD-DFT and TD-HF calculations of the frequency-dependent polarizability and first hyperpolarizability of R-methyloxirane molecule, using increasingly large basis sets. For \(\alpha _{\alpha \beta }(-\omega ;\omega )\) , the results reproduce the expected qualitative and quantitative trends, with convergence toward the same values when the basis-set size increases. However, for \(\beta _{\alpha \beta \gamma }(-2\omega ;\omega ,\omega )\) , the results are more contrasted. Though the convergence with the basis-set size is evident, the first hyperpolarizability difference between the two formulations is frequency dependent, whereas it is not when the relations are satisfied. Moreover, for \(\beta _{\alpha \beta \gamma }(-2\omega ;\omega ,\omega )\) , we analyzed some of the hyper-Rayleigh scattering main observables and it is clear that the full velocity formulation converges slowly compared with the length formulation.
