We present a first-principles calculation of two second-order optical response functions as well as the dielectric function for GaAs and GaP. Specifically, we evaluate the dielectric function \ensuremath{\epsilon}(\ensuremath{\omega}) and the second-harmonic generation response coefficient ${\mathrm{\ensuremath{\chi}}}^{(2)}$(-2\ensuremath{\omega};\ensuremath{\omega},\ensuremath{\omega}) over a large frequency range. The electronic linear electro-optic susceptibility ${\mathrm{\ensuremath{\chi}}}^{(2)}$(-\ensuremath{\omega};\ensuremath{\omega},0) is also evaluated below the band gap. These results are based on a series of self-consistent LDA calculations using the full-potential linearized augmented plane wave method. Self-energy corrections are included at the level of the ``scissors'' approximation, which corrects for the underestimation of the local density approximation band gap and produces a change in the velocity matrix elements. The analytic expressions for the second-order response functions are free of the unphysically divergent terms at zero frequency that have previously plagued such calculations. Results for ${\mathrm{\ensuremath{\chi}}}^{(2)}$(-\ensuremath{\omega};\ensuremath{\omega},0) are in good agreement with experiment below the band gap and those for ${\mathrm{\ensuremath{\chi}}}^{(2)}$(-2\ensuremath{\omega};\ensuremath{\omega},\ensuremath{\omega}) are compared with experimental data where available. We note that despite the equivalence of both of these second-order response functions at zero frequency, there seems to be some discrepancy between the experimental results for these functions in this regime. \textcopyright{} 1996 The American Physical Society.