  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.
