  The possibility for an appreciable many-body contribution to the
electron-phonon interaction (EPI) in Fe-pnictides is discussed in the model
where EPI is due to the electronic polarization of As- ions. The EPI-pol
coupling ismuch larger than the one obtained in the LDA band structure
calculations. It contributes significantly to the intra-band s-wave pairing and
an appreciable positive As-isotope effect in the superconducting critical
temperature is expected. In the Fe-breathing mode the linear (in the
Fe-displacements) EPI-pol coupling vanishes, while the non-linear (quadratic)
one is very strong. The part of the EPI-pol coupling, which is due to the
"potential" energy (the Hubbard U) changes, is responsible for the giant
magneto-elastic effects in MFe_{2}As_{2}, M=Ca, Sr, Ba since it gives much
larger contribution to the magnetic pressure than the band structure effects
do. This mechanism is contrary to the LDA prediction where the magneto-elastic
effects are due to the "kinetic" energy effects, i.e. the changes in the
density of states by the magneto-elastic effects. The proposed $EPI-pol is
expected to be operative (and strong) in other Fe-based superconductors with
electronically polarizable ions such as Se, Te, S etc., and in high-temperature
superconductors due to the polarizability of the O-ions.
