Alkali-doped fullerides A3C60 (A = K, Rb, Cs) are surprising materials where conventional phonon-mediated superconductivity and unconventional Mott physics meet, leading to a remarkable phase diagram as a function of volume per C60 molecule. We address these materials with a state-of-the-art calculation, where we construct a realistic low-energy model from first principles without using a priori information other than the crystal structure and solve it with an accurate many-body theory. Remarkably, our scheme comprehensively reproduces the experimental phase diagram including the low-spin Mott-insulating phase next to the superconducting phase. More remarkably, the critical temperatures Tc's calculated from first principles quantitatively reproduce the experimental values. The driving force behind the surprising phase diagram of A3C60 is a subtle competition between Hund's coupling and Jahn-Teller phonons, which leads to an effectively inverted Hund's coupling. Our results establish that the fullerides are the first members of a novel class of molecular superconductors in which the multiorbital electronic correlations and phonons cooperate to reach high Tc s-wave superconductivity.
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