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Computational details. To include electronic correlation effects for the partially- filled Os 5d band beyond the standard DFT framework we used an local density approximation 1 dynamical mean-field theory (LDA1DMFT) approach25,31. This approach is based on a full-potential linear augmented plane-wave 1 local orbitals technique as implemented in the Wien2k code40 in conjunction with the DMFT implementation provided by the TRIQS package31–34. Our LDA1DMFT framework is fully self-consistent in the charge density. The LDA1DMFT calcu- lations were performed within the scalar-relativistic approximation and using a k-mesh with 32 3 32 3 32 points in the full Brillouin zone. The spin–orbit coup- ling was not included because LDA calculations show that it has a negligible effect on the electronic structure in the vicinity of the Fermi level. The DMFT quantum impurity problem was solved using the numerically-exact imaginary- time hybridization-expansion continuous-time quantum Monte Carlo (CT- QMC) method35. A large number of Monte Carlo cycles, more than 512 million, were performed to obtain a well converged DMFT local self-energy. We adopted a stochastic version of the maximum entropy method36 for the analytical continua- tion of the CT-QMC self-energy to the real frequency axis. For the Coulomb interaction strength U and ▇▇▇▇’▇ coupling constant J we used the values U 5 2.8 eV and J 5 0.55 eV that are estimated in ref. 37. The qualitative results of our LDA1DMFT calculations are not very sensitive to the exact values of U and J. We used the ‘around mean-field form’38 for the double counting correction, which is suitable for weakly correlated metallic systems. In calculations of band structure at the level of DFT39 within the LDA or semi- local generalized gradient approximation (GGA), we used two complementary methods, the full potential (linear) augmented plane waves 1 local orbitals method as implemented in the Wien2k code40 and the electronic-structure method41 RSPt. Both are all-electron methods, which do not impose any approx- imations on the shape of the one-electron potential, and they are known to gen- erate very similar results. The former method allows us to directly compare the LDA and LDA1DMFT results. These methods are particularly suited to high- pressure calculations because the basis functions for any energy, including nom- inally deep core states, can be treated as ‘valence’ states. For calculations with the Wien2k code, we used a k-mesh consisting of 32 3 32 3 32 k-points in the full Brillouin zone. The size of the plane-wave basis set is given by the cutoff parameter Kmax. In our calculations, we kept the product between Kmax and the radius of the real-space muffin-tin spheres to Kmax 3 RMT 5 10. At pressures of 0 GPa, 134 GPa, 247 GPa, and 477 GPa, we set RMT 5 2.5 atomic units (a.u.), 2.34 a.u., 2.27 a.u., and 2.16 a.u., respectively. ▇▇▇▇▇▇▇ is quite noticeable. In our LDA calculations, we do not observe any ETTs at the C point as pressure increases, because at 0 GPa the corresponding band is already well above the Fermi level. This result is in contrast to that of ref. 13, where an ETT was found at this point, but in agreement with ref. 14, in which no ETT was reported. This discrepancy is due to the different values for the lattice constant that were used in the LDA calculations. We used experimental room-temperature values of the lattice parameters. Using the LDA lattice constants from ref. 13, we recover the band energy below the Fermi level at the C point. Assuming the GGA lattice constants from ref. 14, we reproduce the results of this work at the C point. Our LDA calculations also predict that the L-point ETT occurs at a much smaller pressure than our LDA1DMFT calculations, around 100 GPa (Extended Data Figs 4 and 5). In ref. 14, this band at the L point was predicted to be just below the Fermi level at 129 GPa; no ETTs are reported in this pressure range. Using the same lattice constants as in ref. 14, we reproduce these results within LDA. We also find that along the L–H line, the band energy is just above the Fermi level, but this part of the Brillouin zone is not shown in ref. 14. Thus, we attribute discrepancies between the LDA- and GGA-based studies to differences in the EOS rather than to the exchange-correlation potential. The discussion above shows that the electronic structure at the C and L points is quite sensitive to volume changes, and that the occurrence, as well as the position of ETTs in the LDA/GGA picture, depends sensitively on the accuracy of the assumed EOS. The accuracy of the calculated EOS in Os depends on the approxi- mation for the electron–electron interactions used in calculations, as is discussed below. In view of this uncertainty, the most reliable description of the electronic structure is obtained using the experimentally measured lattice parameters46. We adopted this strategy, and show all the electronic structure plots at the experi- mental lattice parameters. We did not detect any substantial difference between the electronic structure calculated with the Wien2k code and with RSPt methods. Influence of relativistic effects. Because Os is a heavy element, the importance of the spin–orbit coupling (SOC) should be investigated. Using LDA, we calculated the band structure in both the scalar-relativistic approximation and with the inclusion of SOC using the Wien2k code40; the results are shown in Extended Data Figs 7 and 8. Some of the bands are split as a result of the inclusion of SOC. However, no new features are seen in the immediate vicinity of the Fermi level. In both cases, we find that one ETT has already taken place at the L point of the hcp Brillouin zone at a pressure of 134 GPa (Extended Data Fig. 7). We do not see any new ETTs upon increasing the pressure to 477 GPa (Extended Data Fig. 8). Instead, we see that the agreement between LDA and LDA1DMFT improves at high pressure, as expected, because the importance of correlation effects decreases with increasing pressure. This observed agreement indicates the internal consist- ency of our calculations. Calculated equation of state. The calculations of the EOS and the lattice para- meters using the LDA1DMFT approach are very time consuming, and their numerical accuracy is insufficient to distinguish weak peculiarities of the lattice parameters, such as the c/a ratio47. We therefore focus on the results obtained within the LDA and GGA of the DFT, and compare our results with experiment, as well as with data available in the literature (see Extended Data Fig. 9 and 10 and Extended Data Table 1). RESEARCH LETTER ,ffiffi Relationship between electronic transitions and anomalies in lattice para- meters. Let us first consider an ETT due to the change of the Fermi surface topology. Although we have shown the importance of many-electron effects for hcp Os, they mainly influence band positions at the C and L points, while the metal remains weakly correlated. Thus, we can use the one-electron picture for a qual- itative discussion. For three-dimensional systems the main effect of correlations on the ETT is the change of coefficients at the singularities48. The character of the anomalies due to the ETT is different at low temperatures (lower than typical phonon energies) and at high ones. The initial anomaly is in the DOS at the Fermi energy, which within the one-electron picture is a square root singularity, for example, dN(EF)a zh(z), where z is the distance between the Fermi energy EF and the Van Hove singularity EC (z 5 EF 2 EC), and h(z) is the Heaviside func- tion10. In the case of an appearance of a new hole pocket19 below the critical volume VETT, the change in the DOS is dN(EF) < (VETT 2 V)1/2. The anomaly yields a sharp peculiarity in the third derivative of the thermodynamic potential V, and induces some kinks in the second derivative. However, it does not necessarily lead to a visible peculiarity of the pressure dependence of the lattice parameters at T 5 0 K, in agreement with our calculations (Extended Data Fig. 10). Still, in hcp metals the effect of the ETT on the lattice parameters can be detected experi- mentally at finite temperature, owing to the anisotropy of the thermal expansion coefficients ac and aa along the c and a directions of the crystal lattice, respectively. Indeed, ac and aa can be evaluated from the phonon Fphon and the electron Fel contributions to the free energy of the hcp metal49,50: established in equation (4). Similar to the ETT, the anisotropy might be strength- ened by a modification of the non-local pseudopotential, equations (1) and (2), acting on the valence electrons, owing to substantial reconstruction of inner 5p and 4f states at the transition. In the model given by equation (3), reconstruction of the inner states affects the electron–ion interaction via the ion core polarizability at imaginary frequencies. The derivation of equation (3) is based on the diagram consideration of the model of polarized ions in electron gas that is suggested in ref.