1 Introduction

Perovskites of the \({\mathrm{ABO}}_3\) family are receiving special attention in recent years due to their peculiar physical and chemical properties, which lead to many possible applications [1–8]. Calcium stannate, CaSnO3, is known as an effective phosphorous host [9–20], for its potential as constituent of multifunctional optoelectronic devices [21], and for its unconventional luminescent behavior [22–24]. At ambient conditions, CaSnO3 exhibits a orthorhombic \({\mathrm{GdFeO}}_3\)-type perovskite structure belonging to the Pbnm space group, which is expected to be stable up to pressures of about 40 GPa and temperatures of about 2000 K [25]. Other phases have been reported (ilmenite-like, hexagonal, rhombohedral, tetragonal and cubic) [26–29], the orthorhombic one remaining the most studied [30–38]. Many fundamental properties of this material, however, still have to be reliably determined, which is the case of its thermal properties (its thermal structural and elastic features, its thermodynamic behavior, etc.). Furthermore, the evolution under increasing pressure of such thermal response properties is still not known, which would be of geochemical interest as CaSnO3 is structurally and chemically closely related to \({\mathrm{CaSiO}}_3\) and \({\mathrm{MgSiO}}_3\) (two common constituents of the Earth mantle) [33, 39–41]. In a recent study, we have investigated the effect of pressure on some vibrational spectroscopic features (Raman and IR spectra) of CaSnO3 by means of ab initio calculations [42].

In the present study, a series of state-of-the-art ab initio techniques is used for the inclusion of temperature and pressure effects on several properties (structural, elastic, thermodynamic) of the orthorhombic phase of the CaSnO3 perovskite. A recently developed fully automated scheme, as implemented in a development version of the Crystal14 program [43], is used for performing quasi-harmonic calculations [44, 45], which allows to go beyond the standard harmonic approximation to the lattice potential and to effectively compute the directional thermal lattice expansion, temperature dependence of the bulk modulus, difference between constant-pressure and constant-volume specific heats, adiabatic bulk modulus, etc. [46–50]. In this respect, it is worth mentioning that a recent experimental study by Redfern et al. [51] suggested CaSnO3 to be highly quasi-harmonic (without the need for an explicit anharmonic description of the system, then). The anisotropic elastic response of the system (in terms of elastic constants and directional seismic wave velocities) is investigated as a function of pressure [52–56]. Furthermore, the full phonon dispersion of the system is computed, which allows us to investigate other thermodynamic properties of the system and phonon density-of-states [57, 58]. The Gibbs free energy of formation of CaSnO3 from the two simple oxides CaO (cubic) and SnO2 (tetragonal) is also evaluated as a function of temperature. The atomic structure of the three crystals is sketched in Fig. 1.

The effect of the exchange-correlation functional of the density functional theory (DFT) on different properties (structural, thermal, elastic) is documented by considering different levels of approximation: from local-density, to generalized-gradient, to hybrids and range-separated hybrids.

Fig. 1
figure 1

(color online) Structure of a cubic CaO (\(Fm\overline{3}m\)), b tetragonal SnO2 (\(P4_2/mnm\)) and c orthorhombic CaSnO3 (Pbnm) [59]

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2 Computational methodology and setup

All calculations are performed using a development version of the Crystal14 program [43]. An all-electron atom-centered Gaussian-type-function (GTF) basis set is adopted, where Ca, Sn and O atoms are described by 8(s)6511(sp)21(d), 9(s)763111(sp)631(d) and 6(s)2111(sp)1(d) contractions of primitive GTFs, respectively [42]. Seven different DFT functionals are adopted: SVWN [60, 61], PBE [62], PBEsol [63], B3LYP [64, 65], PBE0 [66], HSE06 [67] and HISS [68, 69].

The tolerances for the evaluation of Coulomb and exchange infinite series are set to default values for all calculations. The shrinking factor is set to 8, which corresponds to a sampling over 125 independent k-points within the first Brillouin zone in reciprocal space. The convergence on energy of the SCF procedure is set to the tight value of \(10^{-10}\) hartree.

Phonon dispersion calculations are performed with supercells (SC) of different sizes within the PBE level of theory. For CaO, two SCs of 216 and 512 atoms are used, for SnO2, two SCs containing 384 and 750 atoms and for CaSnO3, SCs with 160 and 540 atoms. Quasi-harmonic calculations on CaO and SnO2 have been performed with SCs of 54 and 48 atoms, respectively.

2.1 Quasi-harmonic properties

A fully automated scheme for computing quasi-harmonic properties of crystals has recently been implemented in the Crystal program, which relies on computing and fitting (with a cubic polynomial function) harmonic vibration frequencies at different volumes after having performed volume-constrained geometry optimizations [46–50]. Harmonic phonon frequencies are computed by diagonalizing the dynamical matrix following a “direct space” approach [46, 57, 70, 71]. Constant-volume specific heat, \(C_V(T)\), and entropy, S(T), are estimated through standard statistical thermodynamics from harmonic phonon frequencies as computed at the equilibrium zero-temperature, zero-pressure volume. Quasi-harmonic properties are computed by considering a volume range extending from a \(-3\,\%\) compression to a \(+6\,\%\) expansion with respect to the equilibrium volume; four equidistant volumes are considered in this interval.

The isotropic thermal expansion coefficient, \(\alpha _{V}(T)\), of the system is obtained by minimizing the isothermal Helmholtz free energy

$$F^{\mathrm {QHA}}(T,V) = U^{\mathrm {ZP}}_0(V) + k_{\mathrm{{B}}} T \sum _{\mathbf{k} p} \left[ \ln \left( 1 - e^{-\frac{\hbar \omega _{\mathbf{k} p}(V)}{k_B T } } \right) \right]$$
(1)

with respect to volume at several temperatures, where \(k_{\mathrm{{B}}}\) is Boltzmann’s constant and \(U^{\mathrm {ZP}}_0(V)\) is the zero-temperature internal energy of the crystal which includes the zero-point energy of the system: \(E^{\mathrm {ZP}}_0(V) = \sum _{\mathbf{k} p} {\hbar \omega _{\mathbf{k} p}(V)}/{2}\). The anisotropic thermal expansion can be computed as well [48].

The quasi-harmonic approximation (QHA) allows for combining pressure and temperature effects on structural and elastic properties of materials. By differentiating Eq. (1) with respect to the volume and changing sign, the thermal pressure is obtained:

$$P(V;T) = - \frac{\partial F^{\mathrm {QHA}}(V;T)}{\partial V}.$$
(2)

The description of the isothermal bulk modulus of the system at simultaneous high temperatures and high pressures, \(K_T(P,T)\), can be obtained as an isothermal second derivative of Eq. (1) with respect to the volume and by exploiting relation (2):

$$K_T(P,T) = V(P,T) \left( \frac{\partial ^2F^{\mathrm {QHA}}(V(P,T);T)}{\partial V(P,T)^2} \right) _T.$$
(3)

2.2 Pressure-constrained structure optimization

A fully analytical scheme, based on the stress tensor, is used for optimizing the crystal volume under a given external pressure [72]. The stress tensor \(\varvec{\sigma }\) is a symmetric second-rank tensor that can be computed in terms of analytical energy gradients with respect to lattice parameters:

$$\sigma _{ij} = \frac{1}{V} \frac{\partial E}{\partial \epsilon _{ij}} = \frac{1}{V} \sum _{k = 1}^3 \frac{\partial E}{\partial a^\prime _{ki}} a_{kj},$$
(4)

with \(\varvec{\epsilon }\) second-rank symmetric pure strain tensor and \(i,j,k = x,y,z\). In the expression above, \(a_{ij}\) are elements of a \(3\times 3\) matrix, A, where Cartesian components of the three lattice vectors \({\mathbf{a}}_1\), \({\mathbf{a}}_2\) and \({\mathbf{a}}_3\) are inserted by rows and V is the cell volume. When a distortion is applied to the cell, the lattice parameters transform as \(a_{ij}^\prime = \sum _{k = 1}^3 (\delta _{jk} + \epsilon _{jk})a_{ik}\), where \(\delta _{jk}\) is the Kronecker delta. By adding an external hydrostatic “prestress” \(\sigma _{ij}^{\mathrm {pre}} = P\delta _{ij}\) to \(\sigma _{ij}\) and by inverting Eq. (4), one gets the expression for the constrained gradients

$$\frac{\partial H}{\partial a_{ij}} = \frac{\partial E}{\partial a_{ij}} +PV ({\mathbf{A}}^{-1})_{ji}.$$
(5)

With the inclusion of a hydrostatic pressure, the function to be minimized becomes the enthalpy \(H = E + PV\) [73].

2.3 Elastic tensor calculation

If any finite prestress is absent, second-order elastic constants are simply defined as second energy density derivatives with respect to pairs of infinitesimal Eulerian strains:

$$C_{ijkl} = \frac{1}{V_0} \left( \frac{\partial ^2 E}{\partial \epsilon _{ij}\partial \epsilon _{kl}}\right) _{\epsilon = 0}.$$
(6)

An automated scheme for the calculation of the elastic tensor has been implemented in the Crystal program [52, 74] that has been generalized also to low-dimensionality 1D and 2D systems [75]. Applications of this scheme cover a wide range of materials [76–84].

When a finite prestress \(\varvec{\sigma }^{\mathrm {pre}}\) is applied in the form of a hydrostatic pressure P, within the frame of finite Eulerian strain, the elastic stiffness constants become [85–89]:

$$B_{ijkl} = C_{ijkl} + \frac{P}{2}(2\delta _{ij}\delta _{kl} -\delta _{il}\delta _{jk} -\delta _{ik}\delta _{jl}),$$
(7)

provided that the volume \(V_0\) at which Eq. (6) is evaluated is replaced by the equilibrium volume V(P) at pressure P. A fully automated implementation in the Crystal program of the calculation of the stiffness tensor \({\mathbf{B}}\) (and of \({\mathbf{S}} = {\mathbf{B}}^{-1}\), the compliance tensor) under pressure has recently been presented [53, 54]. A two-index representation of the elastic stiffness tensor is obtained (\(B_{ijkl} \rightarrow B_{vu}\)) by exploiting Voigt’s notation, according to which \(v,u = 1, \dots , 6\, (1 = xx, 2 = yy, 3 = zz, 4 = yz, 5 = xz, 6 = xy)\) [90]. For the elastic constant calculation, four strained configurations are considered for each independent strain, with a dimensionless strain amplitude of 0.0075 (i.e., 0.75 %).

3 Results and discussion

3.1 Quasi-harmonic description

The thermal expansion of CaSnO3 as theoretically determined by means of quasi-harmonic lattice dynamical calculations is reported in Fig. 2. The lower panel of the figure reports the equilibrium cell volume as a function of temperature, up to 1000 K, as obtained with seven different DFT functionals and as compared to available experimental data (empty symbols) [51, 91]. We can clearly see that while different functionals provide rather different descriptions of the absolute value of the equilibrium volume, they all describe a similar trend as it comes to the temperature dependence. As expected, the LDA functional underestimates the cell volume. Two functionals (the generalized-gradient PBE and the global hybrid B3LYP) significantly overestimate the volume, while the other functionals provide a reasonable description of the equilibrium structure. The two experimental datasets reported in the figure show some large discrepancies between each other as regards the lattice expansion. Data by Chen et al. [91], which measure a lower expansion, are found to better agree with present quasi-harmonic simulations than those by Redfern et al. [51], where a much larger thermal expansion is reported.

Fig. 2
figure 2

(color online) Thermal expansion coefficient (upper panel) and temperature-dependent cell volume (lower panel) of CaSnO3 up to 1000 K, as computed with seven different DFT functionals. Empty circles and triangles are experimental data from Redfern et al. [51] and from Chen et al. [91], respectively

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In order to better highlight the different description of thermal expansion given by different functionals, we may refer to the upper panel of Fig. 2, where the volumetric thermal expansion coefficient \(\alpha _V(T) = 1/V(T) [\partial V(T)/\partial T]\) is reported. Although all functionals provide a similar description of the lattice thermal expansion of CaSnO3, some fine differences can be noticed: The HISS functional shows the smallest thermal expansion, while the PBE one the largest (followed by PBEsol and B3LYP, which give an almost identical description of \(\alpha _V\)). The other three adopted functionals provide a rather similar description with each other.

Fig. 3
figure 3

(color online) Thermal expansion coefficient (upper panel), isothermal bulk modulus (middle panel) and lattice parameters (lower panel) of CaSnO3 as a function of temperature, at 5 different pressures from 0 to 20 GPa. In the middle panel, the full circle represents the experimental value for the isothermal bulk modulus at room conditions [33]. All data are obtained with the PBE functional

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The quasi-harmonic approximation represents an effective theoretical tool for describing structural and average elastic properties of solids at simultaneous high pressure and high temperature that is at extreme conditions which are difficult to be explored experimentally. In Fig. 3, we report the thermal evolution, up to 2000 K, of the thermal expansion coefficient, of the isothermal bulk modulus and of the three independent lattice parameters a, b and c, at five different pressures up to 20 GPa, as obtained with the PBE functional. From a computational point of view, the lattice dynamics of the system has been evaluated at 15 volumes from a \(-10\,\%\) compression to a 10 % expansion of the equilibrium zero-pressure, zero-temperature volume. From the upper panel of the figure, we can see that the thermal expansion coefficient behaves quite regularly up to 2000 K (with a tiny deviation from linearity), thus confirming the high quasi-harmonic character of CaSnO3. As pressure increases, the thermal expansion obviously decreases and the coefficient becomes even more regular and linear at high temperatures, as expected given that pressure tends to suppress explicit anharmonic terms in the lattice potential. The evolution of the isothermal bulk modulus as a function of pressure and temperature is quantitatively given in the middle panel of the figure, where the experimental value at room conditions is also given as a full circle [33]. In the lower panel of the figure, the directional, anisotropic, thermal expansion of orthorhombic CaSnO3 is documented. The a and c lattice parameters are found to expand more than the b one by about a factor of 1.2.

Experimentally, a much larger factor of about 4.3 was suggested by Redfern et al. [51], which seems way too large for a system such as CaSnO3, which is not very structurally anisotropic. This large experimental anisotropy of the thermal expansion is reflected in the large volumetric thermal expansion reported in Fig. 2 (see empty circles).

Fig. 4
figure 4

(color online) Heat capacity (left panel) and entropy (right panel) of CaSnO3, CaO and SnO2. All data in \(\hbox {J}\,\hbox {mol}^{-1}\hbox {K}^{-1}\). In the left panels, dashed and dotted lines refer to \(C_V\) and continuous lines to \(C_P\), in the right ones, to \(S_V\) and \(S_P\), respectively. Full circles are experimental data by Gurevich et al. [92] while empty ones by Robie [93]

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3.2 Phonon dispersion and thermodynamics

Thermodynamic properties require a good description of phonon dispersion in order to be accurately described at the ab initio level. In this section, we report converged harmonic (constant-volume specific heat \(C_V\) and entropy \(S_V\)) and quasi-harmonic (constant-pressure specific heat \(C_P\) and entropy \(S_P\)) thermodynamic properties of CaSnO3. Furthermore, thermodynamic functions of both CaO and SnO2 are studied, with the purpose in mind to determine the formation free energy of the following solid-state reaction [28, 94]:

$$CaO _{(s)} + SnO _{2(s)} \rightarrow CaSnO _{3(s)}$$
(8)

These thermodynamic functions are reported in Fig. 4 for the three systems up to 2000 K of temperature. Harmonic properties are obtained with the largest super-cells considered (see Sect. 2 for details), which ensure full convergence as regards the sampling of phonon dispersion within the Brillouin zone. The quasi-harmonic constant-pressure specific heat is obtained by summing on top of the harmonic \(C_V\), the term \(C_{P}(T) - C_{V}(T) = \alpha ^{2}_{V}(T)K_T(T)V(T)T\), as derived from the quasi-harmonic quantities given in Figs. 2 and 3. Experimental data from Gurevich et al. [92] (full circles) and from Robie [93] (empty circles) are also reported, when available. We can see that our calculations provide a satisfactory description of the entropy (a key ingredient to the study of the formation free energy), apart from a slight underestimation at very high temperatures. Specific heat of CaO is also reasonably described, whereas for SnO2 we get a poor description of the constant-pressure specific heat, due to a large underestimation of the thermal expansion (the quasi-harmonic approximation seems to fail rather dramatically for this system). Absolute values of \(C_V\) and \(C_P\) at room temperature (in units of J/mol K) are: 40.77 and 41.18 for CaO; 55.92 and 56.06 for SnO2 and 98.24 and 99.26 for CaSnO3, respectively.

The formation Gibbs’s free energy of the solid-state reaction (8) is given in Table 1 at few different temperatures. The CaSnO3 perovskite is found to be thermodynamically more stable than the two isolated oxides at all temperatures, which is consistent with the experimentally observed “irreversibility” of CaSnO3 once formed from both oxides or molten salts [16, 28, 34, 94–97].

Table 1 Gibbs free energy of formation of CaSnO3 from the CaO and SnO2 oxides, as determined with the PBE functional
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From the full phonon dispersion of a system, the phonon density-of-state (PDOS) can be determined, which is defined by Eq. [58]:

$$g(\omega ) = \frac{1}{V_{\mathrm {BZ}}} \int _{\mathrm {BZ}} \sum _{p=1}^{3N} \delta (\omega _{\mathbf{k}p} - \omega )d{\mathbf{k}} \; ,$$
(9)

where \(V_{\mathrm {BZ}}\) is the volume of the Brillouin zone and the integration is performed over it. From Eq. (9), it is seen that the PDOS is normalized to 3N, being N the number of atoms per cell (\(\int g(\omega ) d\omega = 3N\)). The total PDOS can be partitioned into atomic contributions \(g(\omega ) = \sum _a g_a(\omega ) x_a\) where the sum runs over the atomic species of the system, \(x_a\) is the fraction of atomic species a, and

$$g_a(\omega ) = \frac{1}{n_{\mathbf{k}}} \sum _{p,{\mathbf{k}}} \vert {\mathbf{e}}_{p,{\mathbf{k}};a} \vert ^2 \delta (\omega _{\mathbf{k}p} - \omega )\; ,$$
(10)

where \({\mathbf{e}}_{p,{\mathbf{k}}}\) are the eigenvectors of the dynamical matrices \(W^{\mathbf{k}}\) and the integral in Eq. (9) has been replaced by the sum over the \(n_{\mathbf{k}}\) sampled \({\mathbf{k}}\)-points within the first Brillouin zone. Atomic PDOSs are useful tools for discussing which atoms are mostly involved in phonons of given spectral regions. Figure 5 reports the total PDOS (black continuous line) as well as the atomic PDOSs of Ca, Sn and the two symmetry-independent oxygen atoms, apical \({\mathrm{O}}_{ap}\) and equatorial \(\hbox {O}_{eq}\). The PDOS of CaSnO3 clearly shows three distinct regions: a broad one ranging from 0 up to about \(440\,\hbox {cm}^{-1}\), a second one from about 450 to about \(625\,\hbox {cm}^{-1}\) and a third sharp one at about \(700\,\hbox {cm}^{-1}\). The optical vibrational spectrum above approximatively \(350\,\hbox {cm}^{-1}\) is almost entirely dominated by motions of the oxygen atoms (see yellow and green lines). At lower wave-numbers, all atoms are involved in the lattice vibrations, Ca and Sn atoms becoming dominant in very soft collective phonon modes.

Fig. 5
figure 5

(color online) Total and atomic phonon density-of-states of CaSnO3

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3.3 Elastic properties

Table 2 Single-crystal elastic constants \(C_{vu}\,{\mathrm{CaSnO}}_3\) as computed with seven DFT functionals at 0 K
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This section is devoted to the discussion of the anisotropic elastic response of CaSnO3, at ambient conditions and under pressure. The fourth-rank elastic tensor of the orthorhombic CaSnO3 perovskite has 9 symmetry-independent elastic constants \(C_{vu}\), which are reported in Table 2 as computed with several DFT functionals. All functionals describe similarly the anisotropy of the elastic response, given that the ratios among the elastic constants are similar in all cases. On the contrary, absolute values differ rather significantly. Single-crystal elastic constants of CaSnO3 have not yet been experimentally measured so that the performance of the different functionals can be evaluated just on average elastic properties. Following the Voigt-Reuss-Hill (VRH) averaging scheme [98], average elastic properties of isotropic polycrystalline aggregates can be derived starting from computed elastic constants. The average bulk modulus \(\overline{K}\) obtained in this way is given in the table. Computed elastic properties refer to 0 K and neglect even zero-point motion effects. From the quasi-harmonic calculations discussed before, however, the isothermal bulk modulus \(K_T\) can be computed as a function of temperature. Some experimental techniques for measuring the bulk modulus of a crystal involve elastic waves and are characterized by very short timescales that prevent the system from reaching a thermal equilibrium; in these cases, an adiabatic bulk modulus, \(K_S\), is measured instead. Adiabatic and isothermal bulk moduli do coincide with each other at zero temperature only, \(K_S\) always being larger than \(K_T\) at any finite temperature. The quasi-harmonic approximation also offers a way to compute the adiabatic bulk modulus from the isothermal one, given that:

$$K_S = K_T + \frac{\alpha _V^2VTK_T^2}{C_V} = K_T \times \frac{C_P}{C_V},$$
(11)

where the dependence of all quantities on temperature is just omitted for clarity sake. In Table 2, both the isothermal and adiabatic bulk moduli are reported at room temperature and compared with available experimental data. From that comparison, we see that the HISS functional provides the most rigid description of the system, followed by LDA and both PBE0 and HSE06. On the other hand, PBE, PBEsol and B3LYP provide a softer description of the system, which is seen to be much closer to the experimental elastic rigidity of CaSnO3.

Table 3 Elastic stiffness constants \(B_{vu}\) (in GPa) of CaSnO3 under pressure up to 20 GPa, as computed with the PBE functional
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The effect of an applied hydrostatic pressure on the elastic properties of CaSnO3 is documented in Table 3, where elastic stiffness constants \(B_{vu}\) (computed as described in Sect. 2.3) are reported along with other average elastic properties, such as the average bulk \(\overline{K}\) and shear \(\overline{G}\) moduli obtained within the VRH scheme. From the bulk modulus and the average shear modulus, Young’s modulus \(\overline{E}\) and Poisson’s ratio \(\sigma\) can be defined as well:

$$\overline{E} = \frac{9 \overline{K} \overline{G}}{3 \overline{K} + \overline{G}} \quad {\text{and}} \quad \sigma = \frac{3\overline{K} -2 \overline{G}}{2 (3\overline{K} + \overline{G})} .$$
(12)

Average values of transverse (shear) and longitudinal seismic wave velocities, for an isotropic polycrystalline aggregate, can also be computed from the elastic constants and the density \(\rho\) of the crystal as:

$$\overline{v}_s = \sqrt{\frac{\overline{G}}{\rho }} \quad {\text{and}} \quad \overline{v}_p = \sqrt{\frac{\overline{K} + {4}/{3}\overline{G}}{\rho }} .$$
(13)

As expected, all elastic constants do increase as pressure increases. The diagonal elastic constants \(B_{22}\) and \(B_{33}\) are the ones that increase faster with pressure with slopes of 5.9 and 4.9, respectively, followed by \(B_{12}\) with a slope of 4.1. Shear constants display a small dependence on pressure, \(B_{55}\) being the less affected by pressure with a slope of just 0.5. Transverse seismic wave velocities are not much affected by pressure (with a derivative of 0.006 km/s/GPa), at variance with longitudinal ones (with 0.037 km/s/GPa); these values are in qualitative agreement with experimental determinations by Schneider et al. [99] of 0.012 and 0.052 km/s/GPa, respectively.

Fig. 6
figure 6

(color online) Directional seismic wave velocities of CaSnO3 under pressures up to 20 GPa, as computed with the PBE functional

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The propagation of elastic waves in a crystal is an anisotropic physical phenomenon, which, once the fourth-rank elastic tensor has been determined, can be fully characterized by solving Christoffel’s Eq. [100, 101]. Directional seismic wave velocities (a quasi-longitudinal \(v_p\) one and two quasi-transverse \(v_{s1}\) and \(v_{s2}\) ones) of CaSnO3, as computed at different pressures in the present study, are reported in Fig. 6. It is seen that the directions of minimum propagation velocity of longitudinal waves are between the [101] and [001] ones, whereas along directions from [110] and [010] longitudinal waves propagate with the largest velocity. Apart from an obvious increase in the average propagation velocity, the effect of pressure on the anisotropic propagation of elastic waves in calcium stannate is found to be relatively small, with just a slight increase in the anisotropy of longitudinal waves.

4 Conclusions

State-of-the-art techniques for the ab initio investigation of solids have been applied to predict structural, elastic and thermodynamic properties of the orthorhombic CaSnO3 perovskite at high temperatures and pressures, which are not yet experimentally known. Harmonic and quasi-harmonic calculations, as combined with advanced algorithms for the calculation of anisotropic elastic properties, do indeed constitute a powerful tool for the quantum-chemical description of solids at P and T conditions of real interest in fields such as material science and geochemistry.

The effect on computed properties of the DFT functional is explicitly discussed. Functionals belonging to four families of the DFT (local-density, generalized-gradient, hybrid and range-separated hybrid approximations) are considered. If large differences on absolute values of computed properties are found, most properties are seen to be almost independent of the adopted functional as regards their behavior as a function of temperature or pressure. The generalized-gradient PBE and PBEsol and the global hybrid B3LYP functionals are found to provide the best description of the elastic features of the system.