Impact of scissors-correction schemes on first-principles calculations of second-harmonic generation in ultraviolet nonlinear-optical crystals
In this work, we assess two widely used scissors-correction schemes for first-principles calculations of second-harmonic generation in representative borate and phosphate ultraviolet nonlinear-optical (UV-NLO) crystals, namely scheme-L [Phys.\ Rev.\ Lett.\ \textbf{63}, 1719 (1989)] and scheme-N [Phys.\ Rev.\ B \textbf{72}, 045223 (2005)]. To enable controlled and numerically robust comparisons, we derive a unified static-limit formulation that avoids spurious divergences and is applicable to both schemes, thereby extending earlier static-limit treatments that were effectively restricted to scheme-L. Benchmark calculations show that both schemes largely preserve the spectral line shape while mainly rescaling the overall response. Scheme-N systematically yields 15\%--25\% larger SHG magnitudes than scheme-L, although for some tensor components and experimental datasets scheme-L shows closer agreement with experiment. We further show that Kleinman symmetry is satisfied in the static limit at the level of the formal theory, whereas apparent violations in practical calculations arise mainly from the numerical approximation used to evaluate generalized derivatives.