Several thermodynamic properties of ice Ih, II, and III are studied by a quasi-harmonic approximation and compared to results of quantum path integral and classical simulations. This approximation allows to obtain thermodynamic information at a fraction of the computational cost of standard simulation methods, and at the same time permits studying quantum effects related to zero point vibrations of the atoms. Specifically we have studied the crystal volume, bulk modulus, kinetic energy, enthalpy and heat capacity of the three ice phases as a function of temperature and pressure. The flexible q-TIP4P/F model of water was employed for this study, although the results concerning the capability of the quasi-harmonic approximation are expected to be valid independently of the employed water model. The quasi-harmonic approximation reproduces with reasonable accuracy the results of quantum and classical simulations showing an improved agreement at low temperatures (T < 100 K). This agreement does not deteriorate as a function of pressure as long as it is not too close to the limit of mechanical stability of the ice phases.

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.

The phase diagram of ice is studied by a quasi-harmonic approximation. The free energy of all experimentally known ice phases has been calculated with the flexible q-TIP4P/F model of water. The only exception is the high pressure ice X, in which the presence of symmetric O-H-O bonds prevents its modeling with this empirical interatomic potential. The simplicity of our approach allows us to study ice phases at state points of the T-P plane that have been omitted in previous simulations using free energy methods based on thermodynamic integration. The effect in the phase diagram of averaging the proton disorder that appears in several ice phases has been studied. It is found particularly relevant for ice III, at least for cell sizes typically used in phase coexistence simulations. New insight into the capability of the employed water model to describe the coexistence of ice phases is presented. We find that the H-ordered ices IX and XIV, as well as the H-disordered ice XII, are particularly stable for this water model. This fact disagrees with experimental data. The unexpected large stability of ice IX is a property related to the TIP4P-character of the water model. Only after omission of these three stable ice phases, the calculated phase diagram becomes in reasonable qualitative agreement to the experimental one in the T-P region corresponding to ices Ih, II, III, V, and VI. The calculation of the phase diagram in the quantum and classical limits shows that the most important quantum effect is the stabilization of ice II due to its lower zero-point energy when compared to that one of ices Ih, III, and V.