Recent progress in the fully nonempirical calculation of the superconducting transition temperature (Tc) is reviewed. Especially, this study focuses on three representative light-element high-Tc superconductors, i.e., elemental Li, sulfur hydrides, and alkali-doped fullerides. Here, it is discussed how crucial it is to develop the beyond Migdal-Eliashberg (ME) methods. For Li, a scheme of superconducting density functional theory for the plasmon mechanism is formulated and it is found that Tc is dramatically enhanced by considering the frequency dependence of the screened Coulomb interaction. For sulfur hydrides, it is essential to go beyond not only the static approximation for the screened Coulomb interaction, but also the constant density-of-states approximation for electrons, the harmonic approximation for phonons, and the Migdal approximation for the electron–phonon vertex, all of which have been employed in the standard ME calculation. It is also shown that the feedback effect in the self-consistent calculation of the self-energy and the zero point motion considerably affect the calculation of Tc. For alkali-doped fullerides, the interplay between electron–phonon coupling and electron correlations becomes more nontrivial. It has been demonstrated that the combination of density functional theory and dynamical mean field theory with the ab initio downfolding scheme for electron–phonon coupled systems works successfully. This study not only reproduces the experimental phase diagram but also obtains a unified view of the high-Tc superconductivity and the Mott–Hubbard transition in the fullerides. The results for these high-Tc superconductors will provide a firm ground for future materials design of new superconductors.
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