  This paper investigates some of the successes and failures of density
functional theory in the study of high-pressure solid hydrogen at low
temperature. We calculate the phase diagram, metallization pressure, phonon
spectrum, and proton zero-point energy using three popular exchange-correlation
functionals: the local density approximation (LDA), the Perdew-Burke-Ernzerhof
(PBE) generalized gradient approximation, and the semi-local
Becke-Lee-Yang-Parr (BLYP) functional. We focus on the solid molecular
P$6_3$/m, C2/c, Cmca-12, and Cmca structures in the pressure range from
$100<P<500$ GPa over which phases I, II and III are observed experimentally. At
the static level of theory, in which proton zero-point energy is ignored, the
LDA, PBE and BLYP functionals give very different structural transition and
metallization pressures, with the BLYP phase diagram in better agreement with
experiment. Nevertheless, all three functionals provide qualitatively the same
information about the band gaps of the four structures and the phase
transitions between them. Going beyond the static level, we find that the
frequencies of the vibron modes observed above 3000 cm$^{-1}$ depend strongly
on the choice of exchange-correlation functional, although the low-frequency
part of the phonon spectrum is little affected. The largest and smallest values
of the proton zero-point energy, obtained using the BLYP and LDA functionals,
respectively, differ by more than 10 meV/proton. Including the proton
zero-point energy calculated from the phonon spectrum within the harmonic
approximation improves the agreement of the BLYP and PBE phase diagrams with
experiment. Taken as a whole, our results demonstrate the inadequacy of
mean-field-like density functional calculations of solid molecular hydrogen in
phases I, II and III and emphasize the need for more sophisticated methods.
