High-pressure thermo-elastic properties of beryl (Al₄Be₆Si₁₂O₃₆) from ab initio calculations, and observations about the source of thermal expansion

Abstract

Ab initio calculations of thermo-elastic properties of beryl (Al₄Be₆Si₁₂O₃₆) 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-thermodynamics approach, within the limit of the quasi-harmonic 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 180.2 Gpa and 4.0, respectively) and the low values observed for the thermal expansion coefficient. A zone-centre soft mode $P6/mcc \to P\bar{1}$ 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. in Miner Mag 59:629–639, 1995), where the negative contribution to thermal expansion is ascribed to a geometric effect connected to the tilting of rigid polyhedra in framework silicates.

Appendix

Geometry optimizations, static energies and vibrational frequencies at the (static) equilibrium, and at fixed cell volumes, were performed by means of the ab initio CRYSTAL06 code (Dovesi et al. 2006), which implements the Hartree–Fock and Kohn–Sham, Self Consistent Field (SCF) method for the study of periodic systems (Pisani et al. 1988), by using a Gaussian type basis set.

Basis set, Hamiltonian and computational parameters

The basis set employed was already used for the calculation of the vibrational spectrum of beryl at zero pressure (basis set D2 in Prencipe et al. 2006); it consisted of a 6-31G* contraction for Be, a 88-31G* contraction for both Al and Si, and a 8-411G* contraction for O.

The B3LYP (Becke 1993) and the WC1LYP (Wu and Cohen 2006) Hamiltonians have been chosen which contain hybrid Hartree–Fock/Density-Functional exchange terms. B3LYP is one of the most suitable Hamiltonians for the ab initio calculation of the vibrational properties of molecules, as documented by Koch and Holthausen (2000), as well as for solid state calculations, where it has been shown to provide excellent results for geometries and vibrational frequencies, superior to the one obtained by LDA- or GGA- type functionals (Pascale et al. 2004a, 2005a, b; Prencipe et al. 2004, 2009). WC1LYP (Wu and Cohen 2006) is a relatively new Hamiltonian not yet sufficiently tested on solids; in the present case, it appears to perform slightly better than B3LYP as geometry is concerned, and to provide vibrational frequencies nearly identical to those from the B3LYP calculations, consistently with the findings of Demichelis et al. (2009) on pyrope (Mg₃Al₂Si₃O₁₂), forsterite (α-Mg₂SiO₄), α-quartz (α-SiO₂) and corundum (α-Al₂O₃).

The DFT exchange and correlation contributions to the total energy were evaluated by numerical integration, over the cell volume, of the appropriate functionals; a (75, 974)p grid has been used, where the notation (n_r, n_ω)p indicates a pruned grid with n_r radial points and n_ω angular points on the Lebedev surface in the most accurate integration region (see the ANGULAR keyword in the CRYSTAL06 user’s manual, Dovesi et al. 2006). Such a grid corresponds to 77420 integration points in the unit cell at the equilibrium volume. The accuracy of the integration can be measured from the error in the integrated total electron density, which amounts to −3 × 10⁻⁴|e| for a total of 532 electrons. The thresholds controlling the accuracy of the calculation of Coulomb and exchange integrals have been set to 6 (ITOL1 to ITOL4) and 14 (ITOL5; Dovesi et al. 2006). The diagonalization of the Hamiltonian matrix was performed at six independent k vectors in the reciprocal space (Monkhorst net; Monkhrost and Pack 1976) by setting to 3 the shrinking factor IS (Dovesi et al. 2006).

Geometry, Γ point phonon frequencies and elastic constants

Cell parameters and fractional coordinates were optimized by analytical gradient methods, as implemented in CRYSTAL06 (Civalleri et al. 2001; Dovesi et al. 2006). Geometry optimization was considered converged when each component of the gradient (TOLDEG parameter in CRYSTAL06) was smaller than 0.00001 hartree/bohr and displacements (TOLDEX) with respect to the previous step were smaller than 0.00004 bohr. The volume ranges over which geometries were refined, and static energies calculated, were [703-650 Å³] in the B3LYP case (12 different volume values; static equilibrium found at 692.22 Å³), and [691-650 Å³] in the WC1LYP one (9 volume values; static equilibrium at 686.74 Å³). Results (cell volumes, cell parameters, optimized fractional coordinates and static energies) are provided as supplementary material (Tables S1a and S1b, for the B3LYP and WC1LYP cases, respectively).

Vibrational frequencies and normal modes were calculated at different cell volumes, within the limit of the harmonic approximation, by diagonalizing a mass-weighted Hessian matrix, whose elements are the second derivatives of the full potential of the crystal with respect to mass-weighted atomic displacements (see Pascale et al. 2004b for details). The threshold for the convergence of the total energy, in the SCF cycles, was set to 10⁻¹⁰ hartree (TOLDEE parameter in CRYSTAL06). Results are provided as supplementary material (Tables S2a and S2b for the B3LYP and WC1LYP calculations, respectively).

The elastic constants are the 2nd derivative of the energy with respect to the strain components. They were evaluated through a numerical differentiation of the analytical energy gradient with respect to the cell parameters, by imposing a certain amount of strain along the crystallographic direction corresponding to the component of the elastic tensor. Calculations were carried out by using an automatic scheme recently implemented in the CRYSTAL code (Perger et al. 2009). The calculated (B3LYP) values are provided in Table 5, together the experimental data of Yoon and Newnham (1973).

Table 5 Elastic constants (GPa) calculated at the B3LYP level, and experimental data from Yoon and Newnham (1973)

The K _V , K _R [Eq. (12)] values were 179.9 and 179.4 GPa, respectively; the average value $\overline{K}_{\text{VHR}}$ [Eq. (10)] was 179.7 GPa. The shear moduli μ _V , μ _R and $\overline{\mu }_{\text{VHR}}$ [Eq. (11) and (10)] were 79.1, 77.1 and 78.1 GPa, respectively. The calculated average longitudinal and transversal acustic waves (${\bar{V}}_P$ and ${\bar{V}}_s$) were 10.50 and 5.51 km/sec, respectively, which are very close to the experimental data of Yoon and Newnham (1973). Finally, the average longitudinal and transversal frequencies of the acoustic wave at the Brillouin zone boundary [ν _l and ν _s , Eq. (9)] were 245.5 and 128.8 cm⁻¹, respectively.