  Motivated by recent experimental studies of hydrogen-bonded molecular
conductors $\kappa$-$X_3$(Cat-EDT-TTF)$_2$ [$X$=H, D], interplays of protons
and correlated electrons, and their effects on magnetic, dielectric, and
optical properties, are studied theoretically. We introduce a model Hamiltonian
for $\kappa$-$X_3$(Cat-EDT-TTF)$_2$, in which molecular dimers are connected by
hydrogen bonds. Ground-state phase diagram and optical conductivity spectra are
examined by using the mean-field approximation and the exact diagonalization
method in finite-size cluster. Three types of the competing electronic and
protonic phases, charge density wave phase, polar charge-ordered phase, and
antiferromagnetic dimer-Mott insulating phase are found. Observed softening of
the inter-dimer excitation due to the electron-proton coupling implies
reduction of the effective electron-electron repulsion, i.e. "Hubbard $U$", due
to the quantum proton motion. Contrastingly, the intra-dimer charge excitation
is harden due to the proton-electron coupling. Implications of the theoretical
calculations to the recent experimental results in
$\kappa$-$X_3$(Cat-EDT-TTF)$_2$ are discussed.
