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The second and third order nonlinear susceptibilities of the ferroelectricoxides LiNbO$_3$, LiTaO$_3$, and KNbO$_3$ are calculated from first principles.Two distinct methodologies are compared, one approach is based on aperturbative approach within the frequency-domain, another on thetime-evolution of the electric polarization. The frequency dependence of thesecond harmonic coefficients of the ferroelectric phase of LiNbO$_3$ calculatedwithin the two approaches is in excellent agreement. This is further validatedby experimental data for LiNbO$_3$ and LiTaO$_3$, measured for an incidentrange of photon energies between 0.78 eV and 1.6 eV. The real-time basedapproach is furthermore employed to estimate the third order nonlinearsusceptibilities of all investigated ferroelectric oxides. We further show thatthe quasiparticle effects, considered by means of a scissors-shift incombination with the the computationally efficient independent particleapproximation, result in a shift all spectral features towards higher energiesand decrease the magnitude of the optical nonlinearities. The energy of themain resonances in the hyperpolarizabilities suggests that the spectra can beunderstood by multi-photon adsorption within the fundamental bandgap for allinvestigated materials.

Nonlinear optical response of ferroelectric oxides: first-principles calculations within the time-domain and the frequency-domain