  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.
