  Using the momentum average approximation we study the importance of adding
higher-than-linear terms in the electron-phonon coupling on the properties of
single polarons described by a generalized Holstein model. For medium and
strong linear coupling, even small quadratic electron-phonon coupling terms are
found to lead to very significant quantitative changes in the properties of the
polaron, which cannot be captured by a linear Holstein Hamiltonian with
renormalized parameters. We argue that the bi-polaron phase diagram is equally
sensitive to addition of quadratic coupling terms if the linear coupling is
large. These results suggest that the linear approximation is likely to be
inappropriate to model systems with strong electron-phonon coupling, at least
for low carrier concentrations.
