Zero-point motion energy of the protons is evaluated in the quasi-harmonic approximation for both atomic and molecular phases of highly compressed hydrogen. The results are in good agreement with Kagan et al.'s. In the calculation, the phonon frequencies all over the Brillouin zone are obtained using force constants calculated by the first-principles band theoretical treatments. In the atomic phase, the Cs-IV structure, which is one of the low energy structures in the tetragonal diamond family, is found to be stable with real frequencies all over the Brillouin zone, while the β-Sn structure is unstable with imaginary frequencies near zone-boundaries. In the molecular phase, taking the Cmca structure which is one of the candidate structures above ∼200 GPa, we have studied phonons and the stability of the lattice. We have evaluated the effect of the zero-point energy on the pressure of the molecular dissociation, assuming that it occurs between the Cs-IV and the Cmca structures. With inclusion of the zero-point energy the dissociation pressure is reduced by 90-120 GPa from that estimated by the static energy. The equation of state is in good agreement with the extrapolated one of Loubeyre et al.'s.
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