  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 the volume per C60
molecule. Here 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. Most remarkably, the critical temperature
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.
