  Recent studies of pairing and charge order in materials such as FeSe,
SrTiO$_3$, and 2H-NbSe$_2$ have suggested that momentum dependence of the
electron-phonon coupling plays an important role in their properties. Initial
attempts to study Hamiltonians which either do not include or else truncate the
range of Coulomb repulsion have noted that the resulting spatial non-locality
of the electron-phonon interaction leads to a dominant tendency to phase
separation. Here we present Quantum Monte Carlo results for such models in
which we incorporate both on-site and intersite electron-electron interactions.
We show that these can stabilize phases in which the density is homogeneous and
determine the associated phase boundaries. As a consequence, the physics of
momentum dependent electron-phonon coupling can be determined outside of the
trivial phase separated regime.
