Ab initio second-order nonlinear optics in solids: Second-harmonic generation spectroscopy from time-dependent density-functional theory

We present in detail the formulation of the ab initio theory we have developed for the calculation of the macroscopic second-order susceptibility χ ⊃(2⊃). We find a general expression for χ ⊃(2⊃) valid for any fields, containing the ab initio relation between the microscopic and 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 χ ⊃(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 adiabatic local-density approximation 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.