  We present in detail the formulation of the ab initio theory we have
developed for the calculation of the macroscopic second-order susceptibility
$\chi^{(2)}$. We find a general expression for $\chi^{(2)}$ valid for any
fields, containing the ab initio relation between the \textit{microscopic} and
\textit{macroscopic} formulation of the second-order responses. We consider the
long wavelength limit and we develop our theory in the Time-Dependent
Density-Functional Theory framework. This allows us to include
straightforwardly many-body effects such as crystal local-field and excitonic
effects. We compute the Second-Harmonic Generation spectra for the cubic
semiconductors SiC, AlAs and GaAs and starting from the Independent-Particle
Approximation for $\chi^{(2)}$, we include quasiparticle effects via the
scissors operator, crystal local-field and excitonic effects. In particular, we
consider two different types of kernels: the ALDA and the "long-range" kernel.
We find good agreement with other theoretical calculations and experiments
presented in literature, showing the importance of very accurate description of
the many-body interactions.
