  The phase diagram of ice Ih, II, and III is studied by a quasi-harmonic
approximation. The results of this approach are compared to phase diagrams
previously derived by thermodynamic integration using path integral and
classical simulations, as well as to experimental data. The studied models are
based on both flexible (q-TIP4P/F) and rigid (TIP4P/2005, TIP4PQ/2005)
descriptions of the water molecule. Many aspects of the simulated phase
diagrams are reasonably reproduced by the quasi-harmonic approximation.
Advantages of this simple approach are that it is free from the statistical
errors inherent to computer simulations, both classical and quantum limits are
easily accessible, and the error of the approximation is expected to decrease
in the zero temperature limit. We find that the calculated phase diagram of ice
Ih, II, and III depends strongly on the hydrogen disorder of ice III, at least
for cell sizes typically used in phase coexistence simulations. Either ice II
(in the classical limit) or ice III (in the quantum one) may become unstable
depending upon the proton disorder in ice III. The comparison of quantum and
classical limits shows that the stabilization of ice II is the most important
quantum effect in the phase diagram. The lower vibrational zero-point energy of
ice II, compared to either ice Ih or III, is the microscopic origin of this
stabilization. The necessity of performing an average of the lattice energy
over the proton disorder of ice III is discussed.
