  Extreme pressure strongly affects the superconducting properties of
``simple'' elemental metals, like Li, K and Al. Pressure induces
superconductivity in Li (as high as 17 K), while suppressing it in Al. We
report first-principles investigations of the superconducting properties of
dense Li, K and Al based on a recently proposed, parameter-free, method. Our
results show an unprecedented agreement with experiments, assess the predictive
power of the method over a wide range of densities and electron-phonon
couplings, and provide predictions for K, where no experiments exist so far.
More importantly, our results help uncovering the physics of the different
behaviors of Li and Al in terms of phonon softening and Fermi surface nesting
in Li.
