Extended Data Figure 9: Comparison of the EOS for hcp Os calculated using different approximations within DFT with the experimental EOS measured in this work.

Shown are GGA calculations carried out by us using the RSPt method (black solid line) and by Sahu et al.⁵⁶ (blue dashed line), as well as LDA calculations carried out by us using the Wien2k method (red solid line), by Cynn et al.⁴ (pink dashed line), and by Sahu et al.⁵⁶ (red dashed line). The experimental EOS obtained in this work is shown with black circles. The different curves agree reasonably well for pressures up to about 50 GPa. In contrast, there is noticeable discrepancy at ultra-high pressure, and there is no theoretical EOS that accurately describes the experiment for pressures around 0.5 TPa. Local and semi-local approximations within DFT are insufficient to describe the P–V relationship of Os under extreme conditions. Part of the reason for the disagreement between theory and experiment might be related to the improper account of the many-electron effects within the theory. In principle, LDA and GGA work better at high pressure; however, errors that might be important at low pressure⁴⁷ could propagate through the EOS to the whole pressure range, owing to the use of all the calculated points in the fitting of the energy versus volume data by, for example, the third-order Birch–Murnaghan EOS used in this work. To justify this statement, consider the calculated EOS parameters, summarized in Extended Data Table 1. The equilibrium volumes calculated by us differ from the experiment (Extended Data Table 1) by less than 1.5%. On the other hand, the overestimation of the calculated bulk moduli (B) and their pressure derivatives (B′) is greater, about 10%. Our EOS parameters are within the range of theoretical parameters available in the literature, which are fitted for P < 100 GPa (Extended Data Table 1). We deal with a highly incompressible metal, for which typical DFT errors in B, and especially in B′, translate into large differences in P–V relationships at ultra-high pressure. Even in this regime, the error in volume at a fixed pressure remains within typical DFT limits of about 2%–3%. However, the pressure calculated at fixed volume can differ by several tens of gigapascals. This difference is due to errors in B and B′ calculated at ambient pressure, coupled to a very high value of B. The use of more advanced theoretical methods could improve the calculated EOS. In ref. 47, a substantial reduction of B in isoelectronic hcp Fe is demonstrated using a LDA+DMFT approach. Here the effect is expected to be smaller, but may still be sufficient to improve the agreement with experiment. Our results demonstrate a need to further develop the electronic structure theory, with the experiment reported here providing a bench-mark for the theory. On the other hand, we consistently used experimental lattice parameters in the discussion of the electronic structure of Os in this study.