## 3 Results
### Equations of State of bcc-Cr and hcp-Cr  
The \((P=0,\,T=0)\) ground state corresponds to a bcc-Cr density of 7.229 g/cm\({}^{3}\) (a lattice constant of 2.88 A or an atomic volume of 11.944 A\({}^{3}\)), or a molar volume of 7.193 cm\({}^{3}\), in excellent agreement with 7.2 cm\({}^{3}\) from experiment [30]. At this density, the NM bcc-Cr is higher in energy by \(\sim\)20 meV/atom, i.e., the ground state of bcc-Cr is correctly predicted to be AF. However, the calculated magnetic moment per atom (MMA), \(\sim\)1.1 \(\mu_{\rm B}\), is somewhat larger than the experimental 0.62 \(\mu_{\rm B}\)[31], which is typical as regards a discrepancy between theory and experiment for this type of theoretical calculation [31; 32]. MMA depends almost linearly on the bcc-Cr lattice constant, \(a:\mu(a)\approx 1.1\,+\,14\,(a-2.88)\), and turns zero at a density of 7.87 g/cm\({}^{3}\) (or \(a\approx 2.8\) A); the corresponding \(P\) is \(\sim\) 22 GPa. This \(P\) is higher than \(\approx\)10 GPa (or \(a\approx 2.84\) A) where the experimental Neel \(T\) turns zero as a function of \(P\) (or \(a\)) [33] but is consistent with another theoretical work [32] which shows that the calculated MMA is suppressed considerably at a pressure of \(\sim\) 20 GPa. At \(P\stackrel{{>}}{{\sim}}\) 20 GPa, the EOS of AF bcc-Cr merges into that of NM bcc-Cr, so that the two EOSs become virtually identical. Both AF bcc-Cr below 20 GPa and NM bcc-Cr above 20 GPa combined are described by the third-order Birch-Murnaghan EOS (here \(\rho\) stands for density, in g/cm\({}^{3}\), and \(B\) and \(B^{\prime}\) for bulk modulus, in GPa, and its pressure derivative, and the subscript 0 implies \((P=0,\,T=0)\)). Figure 1: Cold enthalpies of three other solid structures of Cr as a function of \(P\) in relation to that of bcc-Cr assumed to be identically zero. \[P(\rho)=\frac{3}{2}\,B_{0}\left(\eta^{7/3}-\eta^{5/3}\right)\left[1+\frac{3}{4}(B_ {0}^{\prime}-4)\left(\eta^{2/3}-1\right)\right], \tag{1}\]  
where \(\eta=\rho/\rho_{0}\) and \(\rho_{0}=7.229,\;\;B_{0}=215.2,\;\;B_{0}^{\prime}=4.1\). This EOS is expected to be reliable to \(\sim\)2.5 TPa as discussed above. In Figure 2, we compare our theoretical results on the cold EOS of bcc-Cr to the most recent experimental data by [PERSON] et al. [14] and to the EOS obtained from the shock Hugoniot data by [PERSON] et al. [34]. It is clearly seen that agreement between theory and the most recent experiment is excellent. We note that the above EOS does not explicitly contain PM-Cr. However, we did calculate the EOS of PM bcc-Cr using the GGA + U scheme with spin-orbit coupling developed for the calculation of the properties of paramagnetic materials [35]. Specifically, we used the Dudarev approach [36] in which only the difference \(U-J\) is meaningful rather than the values of the two parameters separately. We adopted \(U-J=4.5\) eV from [37]. We found that the EOSs of both PM bcc-Cr and NM bcc-Cr virtually coincide, because the difference in the two values of \(P\) at the same density decreases with increasing density, and eventually becomes negligibly small. Furthermore, we calculated two melting points of PM bcc-Cr (the calculation of the melting curve is discussed above). Although the corresponding values of \(P\) and \(T\) are slightly different from those of NM bcc-Cr having the same densities, these two melting points lie on the melting curve of NM bcc-Cr, Equation (3) below. Thus, the melting curves of both PM-Cr and NM-Cr are virtually identical. Since their EOSs are virtually identical, too, for the purpose of the present study there is no difference in the choice of the structure of bcc-Cr as either paramagnetic or nonmagnetic. For the calculation of the melting curve of Cr presented in what follows, the nonmagnetic structure was chosen.