Thermodynamic properties on the homologous temperature scale from direct upsampling: Understanding electron-vibration coupling and thermal vacancies in bcc refractory metals

We have calculated thermodynamic properties of four bcc refractory elements - V, Ta, Mo, and W - up to the melting point with full density-functional-theory accuracy, using the recently developed direct-upsampling method [J. H. Jung et al., npj Comput. Mater. 9, 3 (2023)]. The direct-upsampling methodology takes full account of explicit anharmonic vibrations and electron-vibration coupling very efficiently using machine-learning potentials. We have thus been able to compute highly converged free-energy surfaces for the PBE exchange-correlation functional, from which accurate temperature dependencies of various thermodynamic properties such as the heat capacity, thermal expansion coefficient, and bulk modulus are accessible. For all four elements, the electronic contribution is large, including a strong coupling with the thermal vibrations. The atomic forces in W are even affected by the temperature-consistent Fermi broadening, which alters the free energy by around 3 meV/atom at the melting point. Trends within group V and group VI refractory elements are observed and explained by qualitative differences in the electronic density of states. We also provide an estimate of the Gibbs energies of vacancy formation and the vacancy contribution to the thermodynamics. Lastly and most importantly, our results are analyzed in terms of the homologous temperature scale relative to theoretically predicted melting points (for the PBE functional). The homologous temperature dependencies show a remarkable agreement with experiments and reveal the predictive power of self-consistently determined ab initio thermodynamic properties.