  Density-functional theory (DFT) has been widely used to study water and ice
for at least 20 years. However, the reliability of different DFT
exchange-correlation (xc) functionals for water remains a matter of
considerable debate. This is particularly true in light of the recent
development of DFT based methods that account for van der Waals (vdW)
dispersion forces. Here, we report a detailed study with several xc functionals
(semi-local, hybrid, and vdW inclusive approaches) on ice Ih and six proton
ordered phases of ice. Consistent with our previous study [Phys. Rev. Lett.
107, 185701 (2011)] which showed that vdW forces become increasingly important
at high pressures, we find here that all vdW inclusive methods considered
improve the relative energies and transition pressures of the high-pressure ice
phases compared to those obtained with semi-local or hybrid xc functionals.
However, we also find that significant discrepancies between experiment and the
vdW inclusive approaches remain in the cohesive properties of the various
phases, causing certain phases to be absent from the phase diagram. Therefore,
room for improvement in the description of water at ambient and high pressures
remains and we suggest that because of the stern test the high pressure ice
phases pose they should be used in future benchmark studies of simulation
methods for water.
