  Using quantum Monte Carlo techniques, we study the effects of electronic
correlations on the effective electron-phonon (el-ph) coupling in a
two-dimensional one-band Hubbard model. We consider a momentum-independent bare
ionic el-ph coupling. In the weak- and intermediate-correlation regimes, we
find that the on-site Coulomb interaction $U$ acts to effectively suppress the
ionic el-ph coupling at all electron- and phonon- momenta. In this regime, our
numerical simulations are in good agreement with the results of perturbation
theory to order $U^2$. However, entering the strong-correlation regime, we find
that the forward scattering process stops decreasing and begins to
substantially increase as a function of $U$, leading to an effective el-ph
coupling which is peaked in the forward direction. Whereas at weak and
intermediate Coulomb interactions, screening is the dominant correlation effect
suppressing the el-ph coupling, at larger $U$ values irreducible vertex
corrections become more important and give rise to this increase. These vertex
corrections depend crucially on the renormalized electronic structure of the
strongly correlated system.
