  The non-linear response of infinite periodic solids to homogenous electric
fields and collective atomic displacements is discussed in the framework of
density functional perturbation theory. The approach is based on the 2n + 1
theorem applied to an electric-field-dependent energy functional. We report the
expressions for the calculation of the non-linear optical susceptibilities,
Raman scattering efficiencies and electrooptic coefficients. Different
formulations of third-order energy derivatives are examined and their
convergence with respect to the k-point sampling is discussed. We apply our
method to a few simple cases and compare our results to those obtained with
distinct techniques. Finally, we discuss the effect of a scissors correction on
the EO coefficients and non-linear optical susceptibilities.
