We theoretically study hydrogen-bonded molecular conductors synthesized recently, $kappa$-H$_3$(Cat-EDT-TTF)$_2$ and its diselena analog, $kappa$-H$_3$(Cat-EDT-ST)$_2$, by first-principles density-functional theory calculations. In these crystals, two H(Cat-EDT-TTF/ST) units share a hydrogen atom with a short O--H--O hydrogen bond. The calculated band structure near the Fermi level shows a quasi-two-dimensional character, with a rather large interlayer dispersion due to the absence of insulating layers in contrast with conventional molecular conductors. We discuss effective low-energy models based on H(Cat-EDT-TTF/ST) units and its dimers, respectively, where the microscopic character of the orbitals composing them are analyzed. Furthermore, we find a stable structure which is different from the experimentally determined structure, where the shared hydrogen atom becomes localized to one of the oxygen atoms, in which charge disproportionation between the two types of H(Cat-EDT-TTF) units is associated. The calculated potential energy surface for the H atom is very shallow near the minimum points, therefore the probability of the H atom can be delocalized between the two O atoms.
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