  Palladium hydrides display the largest isotope effect anomaly known in
literature. Replacement of hydrogen with the heavier isotopes leads to higher
superconducting temperatures, a behavior inconsistent with harmonic theory.
Solving the self-consistent harmonic approximation by a stochastic approach, we
obtain the anharmonic free energy, the thermal expansion and the
superconducting properties fully ab initio. We find that the phonon spectra are
strongly renormalized by anharmonicity far beyond the perturbative regime.
Superconductivity is phonon mediated, but the harmonic approximation largely
overestimates the superconducting critical temperatures. We explain the inverse
isotope effect, obtaining a -0.38 value for the isotope coefficient in good
agreement with experiments, hydrogen anharmonicity being the main responsible
for the isotope anomaly.
