Ab initio calculations of thermo-elastic properties of beryl (Al 4 Be 6 Si 12 O 36 ) have been carried out at the hybrid HF/DFT level, by using the B3LYP and WC1LYP Hamiltonians. Static geometries and vibrational frequencies were calculated at different values of the unit cell volume, to get static pressure and mode-γ Grüneisen's parameters. Zero point and thermal pressures were calculated by following a standard statistical-thermodynamic approach, within the limit of the quasiharmonic approximation, and added to the static pressure at each volume, to get the total pressure (P) as a function of both temperature (T) and cell volume (V). The resulting P(V,T) curves were fitted by appropriate EoS', to get bulk modulus (K 0 ) and its derivative (K'), at different temperatures. The calculation successfully reproduced the available experimental data concerning compressibility at room temperature (the WC1LYP Hamiltonian provided K 0 and K' values of 180.2Gpa and 4.0, respectively), and the low values observed for the thermal expansion coefficient. A zone-centre soft mode 𝑃6 𝑚𝑐𝑐 ⁄ → 𝑃1 ̅ phase transition was predicted to occur at a pressure of about 14 GPa; the reduction of the frequency of the soft vibrational mode, as the pressure is increased, and the similar behaviour of the majority of the low frequency modes, provided an explanation of the thermal behaviour of the crystal, which is consistent with the RUM model (Rigid Unit Model;Dove et al., 1995), where the negative contribution to thermal expansion is ascribed to a geometric effect connected to the tilting of rigid polyhedra in framework silicates.

We calculated the thermodynamic and thermoelastic properties of periclase and ferropericlase, the latter having a stoichiometric composition of (Fe 0.03 Mg 0.97 )O, at pressures and temperatures which are typical of the Earth's lower mantle. The static lattice energies and vibrational frequencies were derived through ab-initio calculations carried out at the hybrid HF/DFT level. The thermodynamic properties were calculated by following a standard statistical-thermodynamics approach, within the limit of the quasi-harmonic approximation. A third order Birch-Murnaghan equation of state fit to the static E(V) data of periclase yielded K 0 = 163.8 GPa, K' = 4.3 and V 0 = 75.09 Å 3 . The fit at 300K and 0.1 MPa on the P(V) data yielded K 0 = 160.1 GPa, K' = 4.2 and V 0 = 75.99 Å 3 . Such results successfully reproduced the best available experimental and previous computational data. The presence of iron with low spin configuration in the structure had the effects (i) to reduce the cell volume, both at the static (74.19 Å 3 ) and at the ambient conditions (75.14 Å 3 ); (ii) to increase the bulk modulus (respectively 172.2 GPa at the static limit, and 167.4 GPa at 298 K and 0.1 MPa) and (iii) to decrease the thermal expansion (2.79*10 -5 K -1 for periclase and 2.60*10 -5 K -1 for ferropericlase at 300 K). Since the discussed parameters were also calculated at high pressure and temperature conditions simultaneously, the reliability of the quasi-harmonic approximation was tested by evaluating the shape of the potential energy curve, at conditions which simulate those of the Earth's lower mantle. Such test confirmed the applicability of this approximation over all the P/T range considered.

In this work, we calculate the thermo-elastic properties of (Mg 1-x Fe x )O ferropericlase, with x in the [0.06, 0.59] range, and the thermodynamic properties of ferropericlase having the specific stoichiometric composition (Mg 0.54 Fe 0.46 )O, at pressures and temperatures, which are those typical of the Earth's lower mantle. We follow an ab-initio quantum-mechanical approach, with the use of the WC1LYP hybrid Hartree-Fock/density functional theory (HF/DFT) functional, within the framework of the quasiharmonic approximation. Iron is assumed to be in the low-spin configuration, as it proved to be the most stable spin arrangement at the thermo-baric conditions of the deepest lower mantle. The choice of the low-spin configuration, and the use of an ab-initio approach, make this work unique as it is the first time that such a technique is applied for the calculation of the vibrational and thermodynamic properties of the low-spin ferropericlase.We observe a linear increase of the bulk modulus and a linear decrease of the cell volume as iron content increases. More precisely, for x = 0.46, at ambient condition K T = 205.57 GPa, K′ T = 4.242, and V T = 72.216 Å 3 ; for x = 0.03, at the same conditions, K T = 167.42 GPa, K′ T = 4.085, and V T = 75.145 Å 3 .Some thermodynamic parameters and the thermal expansion (C V , C P , S, α) for (Mg 0.54 Fe 0.46 )O are calculated both at ambient condition [C V = 36.11 J/(mol⋅K), C P = 36.38 J/(mol⋅K), S = 26.62 J/(mol⋅K), α = 1.97×10 −5 K −1 ], and at simultaneous high-pressure and high-temperature conditions as a function of the geobar and geotherm curves. The data here proposed can be seen as possible bounds to the values of thermoelastic and thermodynamic parameters employed in the construction of geophysical models and the same data could be used to revise the velocity of the seismic waves in the lower mantle.

Through ab initio calculations carried out at the hybrid HF/DFT level, we calculate the static energies and the vibrational frequencies of CrN at different cell volumes, in both cubic and orthorhombic geometries.The paramagnetic cubic phase is studied in different magnetic arrangements, whereas the orthorhombic one is studied in the antiferromagnetic configuration. The resulting static energies are fitted by appropriate equations of state to get the static cell volume (V 0 ), static bulk modulus (K 0 ) and its pressure derivative (K9). We obtain a static K 0 for the cubic phase ranging from 282 GPa to 302.1 GPa, with a value of K9 that varies from 2.87 to 3.63 and a conventional V 0 oscillating from 73.074 Å 3 to 73.277 Å 3 . At ambient conditions the bulk modulus is reduced by about 6 GPa and the cell volume is increased by about 0.45 Å 3 . For the orthorhombic phase we obtain a value of K 0 ranging from 452.5 GPa to 476 GPa, whose K9 varies from 3.09 to 3.38 and its conventional V 0 oscillates from 73.074 Å 3 to 73.277 Å 3 . At 300 K, K 0 is reduced by about 6 GPa and the V 0 is increased by about 0.45 Å 3 . The computational results here obtained for the orthorhombic phase, although consistent within the methodology used, differ from the very few experimental results on the same phase, and such a difference cannot be explained at the moment. They have to be confirmed by experimental measures in the future.