Structural optimizations and total energy calculations at 0 K are performed by using the Vienna ab initio simulation package (VASP).^{[50, 51]} The Perdew– Burke– Ernzerhof (PBE) version^[52] of the general gradient approximation (GGA) is used for the electron exchange– correlation functional. Electrons of 5p⁶6s²5d³ for Ta and 5p⁶6s²5d⁵ for Re are treated as valence configurations. The interaction between the valence electrons and the remaining core electrons is described by the frozen core projector-augmented wave (PAW) method.^[53] Energy convergence (1 mRy per atom) tests are performed to determine the cutoff energy and the number of k-points in whole pressure range for both metals, and the reasonable cutoff energy (400 eV for Ta and 450 eV for Re) and rational k-points (18 × 18 × 18 for Ta and 14 × 14 × 10 for Re) are ascertained. Afterwards, the total energy of undeformed unit cell is calculated in a volume range of 0.45 V_0expt– 1.1 V_0expt, where V_0expt is the experimental equilibrium volume at zero pressure. Then the equation of states at 0 K for both metals are determined according to the fourth-order Brich– Murnaghan equation .^[54] From the relation of P = − (∂ E/∂ V)_{T = 0}, the P– V data are obtained directly, which is the basis of the elastic constants calculations at different pressures.

To verify the accuracy of the present calculations, we list the calculated lattice parameters, the values of equilibrium volume V₀, bulk modulus B₀, and their pressure derivative in Table 1, and compare them with the available experimental results.^{[55– 60]}

Table 1.

Table 1.

Table 1. Calculated values of equilibrium lattice parameter (in unit Å ), atomic volume V0 (in unit Å 3), bulk modulus B0 (in unit GPa), and their pressure derivative , together with the experimental results.[55– 60] GGA– PBE denotes the generalized gradient approximation-Perdew Burke Ernzerhof, XRD the x-ray diffraction, DAC the diamond anvil cell. Method a/Å c/Å c/a V0/Å 3 B0/GPa Present work (GGA– PBE) 3.322 – – 18.33 195 3.47 Ta Expt[55] (XRD in DAC) 3.304 – – 18.04 194 3.52 Expt[56] (XRD in DAC) 3.304 – – 18.04 194 3.55 Expt[57] (XRD in DAC) 3.304 – – 18.03 195 3.40 Present work (GGA– PBE) 2.762 4.457 1.614 14.72 361 4.11 Re Expt[58] (XRD in DAC) 2.762 4.459 1.614 14.73 353 4.56 Expt[59] (XRD in DAC) 2.762 4.455 1.613 14.72 372 4.05 Expt[60] (XRD in DAC) 2.761 4.456 1.614 14.72 350 4.00

Table 1. Calculated values of equilibrium lattice parameter (in unit Å ), atomic volume V0 (in unit Å 3), bulk modulus B0 (in unit GPa), and their pressure derivative , together with the experimental results.[55– 60] GGA– PBE denotes the generalized gradient approximation-Perdew Burke Ernzerhof, XRD the x-ray diffraction, DAC the diamond anvil cell.

From Table 1, we can clearly see that the calculated lattice parameters of Ta and Re are in fair agreement (within 0.5%) with the experimental data.^{[45– 50]} And the equilibrium volumes, bulk modulus B₀ and their pressure derivatives for both Ta and Re are also shown to be in excellent agreement with the existing experimental results.^{[55– 60]} The good agreement indicates that the choices of PAW pseudopotential and GGA– PBE approximation are reasonable for the current study and the calculations are reliable.

Having seen that the calculations of the present work are reliable, we now turn to the results of the high-pressure elastic constants.

The relationships between the pressure and elastic constants, obtained respectively when the energy term is considered and ignored, are displayed in Fig. 1. From the comparison shown in Fig. 1, we can clearly see that though C_ij and C̄ _ij have the same upward tendency with pressure increasing, the values of elastic constant C_ij are not equal to the values of C̄ _ij at nonzero pressures and the difference between them grows larger with the increase of pressure. In what follows, we take the calculated results of Ta for example. As the pressure increases, the values of C̄ ₁₁ and C̄ ₁₂ without the correction of the term respectively have errors about 12% ∼ 16% and 26% ∼ 43% in comparison with the corresponding true data C₁₁ and C₁₂. Moreover, the deviation of C̄ ₄₄ is high up to 53%∼ 71% with pressure increasing. This indicates that the energy term indeed will affect the accuracy of the high-pressure elastic constant. Furthermore, this influence will become larger as pressure increases, especially when the value of the pressure is comparable to that of the elastic constant. Therefore, the energy term should be included in expression (1) when the high-pressure elastic constant is calculated. Otherwise, the calculated results can neither explain the corresponding experimental phenomenon exactly, nor predict the high-pressure elastic property behavior accurately.

Fig. 1.

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	Fig. 1. Pressure dependences of the elastic constants of Ta and Re. Solid symbols represent the elastic constants of Cij obtained when the energy term is considered, and the empty symbols denote the elastic constants of C̄ ij obtained when the energy term is ignored.