  We investigate the ground state properties of the bipolaron coupled to
quantum dispersive optical phonons in the one-dimensional Holstein-Hubbard
model. We concentrate on the interplay between the phonon dispersion and the
Coulomb repulsion and their mutual effect on the bipolaron effective mass, the
binding energy, and the phase diagram. Most surprisingly, the sign of the
curvature of the optical phonon dispersion plays a decisive role on the
bipolaron binding energy in the presence of the Coulomb repulsion $U$. In
particular, when the sign of the phonon dispersion curvature matches the sign
of the electron dispersion curvature, the bipolaron remains bound in the strong
coupling limit even when $U\to \infty$ and the binding emanates from the
exchange of phonons between two electrons residing on adjacent sites. At
moderate electron-phonon coupling a light bipolaron exists up to large values
of $U$. Finally, an intuitive explanation of the role of the phonon dispersion
on the bipolaron binding energy is derived using the strong coupling limit
where the binding emanates from the exchange of phonons between two electrons
residing on adjacent sites which leads to enhanced stability of bipolarons at
elevated Coulomb repulsion.
