  We analyze the ground-state properties of strongly-correlated electrons
coupled with phonons by means of a generalized Gutzwiller wavefunction which
includes phononic degrees of freedom. We study in detail the paramagnetic
half-filled Hubbard-Holstein model, where the electron-electron interaction can
lead to a Mott transition, and the electron-phonon coupling to a bipolaronic
transition. We critically discuss the quality of the proposed wavefunction in
describing the various transitions and crossovers that occur as a function of
the parameters. Previous variational attempts are recovered as particular
choices of the wavefunction, while keeping all the variational freedom allows
to access regions of the phase diagram otherwise inaccessible within previous
variational approaches.
