  Chevrel ternary superconductors show an intriguing coexistence of molecular
aspects, large electron-phonon and electron-electron correlations, which to
some extent still impedes their quantitative understanding. We present a first
principles study on the prototypical Chevrel compound PbMo$_{6}$S$_{8}$,
including electronic, structural and vibrational properties at zero and high
pressure. We confirm the presence of an extremely strong electron-phonon
coupling, linked to the proximity to a R$\overline{3}$-P$\overline{1}$
structural phase transition, which weakens as the system, upon applied
pressures, is driven away from the phase boundary. A detailed description of
the superconducting state is obtained by means of fully \textit{ab initio}
superconducting density functional theory (SCDFT). SCDFT accounts for the role
of phase instability, electron-phonon coupling with different intra- and
inter-molecular phonon modes, and without any empirical parameter, and
accurately reproduces the experimental critical temperature and gap. This study
provides the conclusive confirmation that Chevrel phases are phonon driven
superconductors mitigated, however, by an uncommonly strong Coulomb repulsion.
The latter is generated by the combined effect of repulsive Mo states at the
Fermi energy and a band gap in close proximity to the Fermi level. This is
crucial to rationalize why Chevrel phases, in spite of their extreme
electron-phonon coupling, have critical temperatures below 15~K. In addition,
we predict the evolution of the superconducting critical temperature as a
function of the external pressure, showing an excellent agreement with
available experimental data.
