We study the electronic and structural properties of the low-temperature ordered phase of hydrogen-bonded molecular conductors, $kappa$-D$_3$(Cat-EDT-TTF)$_2$ and its selenium-substituted analog $kappa$-D$_3$(Cat-EDT-ST)$_2$, by means of first-principles density functional theory~(DFT) calculations. In these compounds, the charge ordering in the $pi$-electron system is coupled with the ordering of the displacements in the deuteriums forming the hydrogen-bond, equally shared by two oxygens in the high-temperature phase. While the structural optimization within the standard DFT method based on the generalized gradient approximation fails to reproduce the structural stability of the charge-ordered (CO) phase, we show that a hybrid functional of Heyd, Scuseria, and Ernzerhof can reproduce structural characters of the CO phase, owing to the more localized nature of the wave functions. Furthermore, using the ability of the hybrid functional to predict the electronic and structural properties, we find a stable noncentrosymmetric CO phase with another pattern of deuterium ordering.
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