  The spatially localized bound states of two electrons in the adiabatic
two-dimensional Holstein-Hubbard model on a square lattice are investigated
both numerically and analytically. The interplay between the electron-phonon
coupling g, which tends to form bipolarons and the repulsive Hubbard
interaction $\upsilon \geq 0$, which tends to break them, generates many
different ground-states. There are four domains in the $g,\upsilon$ phase
diagram delimited by first order transition lines. Except for the domain at
weak electron-phonon coupling (small g) where the electrons remain free, the
electrons form bipolarons which can 1) be mostly located on a single site
(small $\upsilon$, large g); 2) be an anisotropic pair of polarons lying on two
neighboring sites in the magnetic singlet state (large $\upsilon$, large g); or
3) be a "quadrisinglet state" which is the superposition of 4 electronic
singlets with a common central site. This quadrisinglet bipolaron is the most
stable in a small central domain in between the three other phases. The pinning
modes and the Peierls-Nabarro barrier of each of these bipolarons are
calculated and the barrier is found to be strongly depressed in the region of
stability of the quadrisinglet bipolaron.
