  We formulate an ab initio downfolding scheme for electron-phonon coupled
systems. In this scheme, we calculate partially renormalized phonon frequencies
and electron-phonon coupling, which include the screening effects of
high-energy electrons, to construct a realistic Hamiltonian consisting of
low-energy electron and phonon degrees of freedom. We show that our scheme,
which we call constrained density-functional perturbation theory (cDFPT), can
be implemented by slightly modifying the conventional DFPT, which is one of the
standard methods to calculate phonon properties from first principles. Our
scheme can be applied to various phonon-related problems, such as
superconductivity, electron and thermal transport, thermoelectricity,
piezoelectricity, dielectricity and multiferroicity. We believe that the cDFPT
provides a firm basis for the understanding of the role of phonons in strongly
correlated materials. Here, we apply the scheme to the fullerene
superconductors and discuss how the realistic low-energy Hamiltonian is
constructed.
