High temperature and pressure thermoelasticity of hcp metals from ab initio quasi-harmonic free energy calculations: the beryllium case

Abstract: We present a systematic ab initio study of the temperature and pressure dependent thermoelastic properties of hcp beryllium within the quasi-harmonic approximation (QHA). The accuracy of the Zero Static Internal Stress Approximation (ZSISA) and of the volume-constrained ZSISA that are widely applied in ab initio thermodynamic calculations are quantified. Particularly, the effect of ZSISA for the calculation of $C_{11}$ and $C_{12}$ is compared with a novel numerical approach which minimizes the free energy with respect to the atomic positions at each strain. In beryllium, minor deviations are found within ZSISA, which gives ECs in good agreement with the full free energy minimization (FFEM). A substantial difference is found between QHA and the quasi-static approximation (QSA), with the former closer to experiments. Within QSA, we compare the ECs obtained by interpolating from a set of geometries along the "stress-pressure" isotherm at $0$ K (within V-ZSISA) with a more general interpolation on a two-dimensional grid of crystal parameters which allows the calculation of the ECs along the $0$ kbar isobar. This paper provides a practical approach for the investigation of the thermoelastic properties of hcp metals at extreme conditions.