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We formulate an ab initio downfolding scheme for electron-phonon coupledsystems. In this scheme, we calculate partially renormalized phonon frequenciesand electron-phonon coupling, which include the screening effects ofhigh-energy electrons, to construct a realistic Hamiltonian consisting oflow-energy electron and phonon degrees of freedom. We show that our scheme,which we call constrained density-functional perturbation theory (cDFPT), canbe implemented by slightly modifying the conventional DFPT, which is one of thestandard methods to calculate phonon properties from first principles. Ourscheme can be applied to various phonon-related problems, such assuperconductivity, electron and thermal transport, thermoelectricity,piezoelectricity, dielectricity and multiferroicity. We believe that the cDFPTprovides a firm basis for the understanding of the role of phonons in stronglycorrelated materials. Here, we apply the scheme to the fullerenesuperconductors and discuss how the realistic low-energy Hamiltonian isconstructed.

Ab initio downfolding for electron-phonon coupled systems: constrained density-functional perturbation theory (cDFPT)