  The phonon spectrum of the high-pressure simple cubic phase of calcium, in
the harmonic approx- imation, shows imaginary branches that make it
mechanically unstable. In this letter, the phonon spectrum is recalculated
using density-functional theory (DFT) ab initio methods fully including
anharmonic effects up to fourth order at 50 GPa. Considering that perturbation
theory cannot be employed with imaginary harmonic frequencies, a variational
procedure based on the Gibbs- Bogoliubov inequality is used to estimate the
renormalized phonon frequencies. The results show that strong quantum
anharmonic effects make the imaginary phonons become positive even at zero
temperature so that the simple cubic phase becomes mechanically stable, as
experiments suggest. Moreover, our calculations find a superconducting Tc in
agreement with experiments and predict an anomalous behavior of the specific
heat.
