[en] We present a first-principles study of the natural optical activity of the gyroelectric Pb$_5$Ge$_3$O$_{11}$ crystal, explicitly accounting for spin-orbit coupling (SOC) effects. We derive a new analytical expression for the gyration coefficients within the recent framework of long-wavelength density-functional perturbation theory [Phys. Rev. Lett. \textbf{131}, 086902 (2023)], which significantly improves computational efficiency by reducing the number of required response functions and includes spin-orbit coupling effects. We use this implementation to investigate the evolution of Pb$_5$Ge$_3$O$_{11}$'s optical rotation across the ferroelectric double-well, from the paraelectric $P\bar{6}$ phase to the ferroelectric $P3$ phase. Our results demonstrate that, in addition to the substantial renormalization of the double-well energy, spin-orbit coupling contributions play an equally crucial role in the natural optical activity, largely through purely electronic contributions, while SOC-induced structural relaxation effects are minor.
