  Employing the recently developed self-consistent variational basis generation
scheme, we have investigated the bipolaron-bipolaron interaction within the
purview of Holstein-Hubbard and the extended-Holstein-Hubbard (F2H) model on a
discrete one-dimensional lattice. The density-matrix renormalization group
(DMRG) method has also been used for the Holstein-Hubbard model. We have shown
that there exists no bipolaron-bipolaron attraction in the Holstein-Hubbard
model. In contrast, we have obtained clear-cut bipolaron-bipolaron attraction
in the F2H model. Composite bipolarons are formed above a critical
electron-phonon coupling strength, which can survive the finite Hubbard $U$
effect. We have constructed the phase diagram of F2H polarons and bipolarons,
and discussed the phase separation in terms of the formation of composite
bipolarons.
