Mechanism for linear and nonlinear optical effects in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>β</mml:mi><mml:mo>−</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">BaB</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">O</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>crystals

Electronic structure calculations of $\ensuremath{\beta}\ensuremath{-}{\mathrm{BaB}}_{2}{\mathrm{O}}_{4}$ from first principles are performed based on a plane-wave pseudopotential method, and the linear optical properties are then obtained. The static second-harmonic generation (SHG) coefficients are calculated at the independent-particle level with a formalism originally given by Aversa and Sipe [Phys. Rev. B 52, 14 636 (1995)] and later rearranged by Rashkeev et al. [Phys. Rev. B 57, 3905 (1998)] to explicitly show Kleinman's symmetry. The formalism is improved to be more efficient in reducing the $k$ points necessary for convergence. A real-space atom-cutting method is suggested to analyze the respective contributions of various transitions among ions and ion groups to optical response. The contribution of the cation Ba to SHG effects is found to be not important but non-negligible, while its contribution to birefringence is negligible.