  We study polarons in the one-dimensional Bond-Peierls electron-phonon model,
in which phonons on bonds of a lattice modulate the hopping of electrons
between lattice sites, and contrast the results to those known for the
Breathing-Mode Peierls problem. By inspecting the atomic limit, we show that
polaronic dressing and mass enhancement of Bond-Peierls polarons depend on the
momentum dependence of the phonons. For dispersionless phonons, Bond-Peierls
polarons are perfectly localized in the atomic limit, unlike their
Breathing-mode counterparts, because the carrier creates a string of phonon
excitations that can only be annihilated via processes that retrace the carrier
to its original site. However, inclusion of phonon dispersion leads to a
non-divergent polaron mass even in the atomic limit and depending on the form
of the phonon dispersion may lead to a transition to a non-zero-momentum ground
state akin to that found in the Breathing-mode Peierls model.
