A first-principles study of the birefringence and the frequency dependent second harmonic generation (SHG) coefficients of the ternary pnictide semiconductors with formula ABC$_2$ (A = Zn, Cd; B = Si, Ge; C = As, P) with the chalcopyrite structures was carried out. We show that a simple empirical observation that a smaller value of the gap is correlated with larger value of SHG is qualitatively true. However, simple inverse power scaling laws between gaps and SHG were not found. Instead, the real value of the nonlinear response is a result of a very delicate balance between different intraband and interband terms.
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