Ab initio calculation of second harmonic generation in solids

An ab initio formalism for the calculation of macroscopic second order responses within the time dependent density functional theory (TDDFT) framework is presented. The linear Dyson-like equation of TDDFT is generalized to second order and the resulting microscopic response is related to the macroscopic non-linear susceptibility. The macroscopic relations are obtained considering Maxwell's equations and the non-linear relation between the microscopic and macroscopic field. Some approximations have to be made to account for many body effects. The impact of the several levels of approximation on the second harmonic spectrum is illustrated for the example of bulk AlAs in the zincblende structure.