  We consider the Hubbard-Holstein model in the adiabatic limit to investigate
the effects of electron-electron interactions on the electron-phonon coupling.
To this aim we compute at any momentum and filling the static charge
susceptibility of the Hubbard model within the Gutzwiller approximation and we
find that electron-electron correlations effectively screen the electron
coupling to the lattice. This screening is more effective at large momenta and,
as a consequence, the charge-density wave phase due to the usual Peierls
instability of the Fermi surface momenta is replaced by a phase-separation
instability when the correlations are sizable.
