The phase diagram of ice Ih, II, and III: a quasi-harmonic study

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.