# High-Temperature Thermodynamics of Uranium from Ab Initio Modeling
## 2 Ab Initio Computational Methodology
The generalized gradient approximation (GGA) is known to be exceptional for actinides [82] and we apply it here for the electron exchange and correlation functional. Uranium metal is nonmagnetic at ambient conditions in the ground-state orthorhombic \(\alpha\) phase, but it is very close to a magnetic instability that depends sensitively on the volume and crystal structure [83]. Because we are studying uranium at high temperatures, there is a significant thermal expansion and a transition to the bcc \(\gamma\) phase that set the stage for magnetism [83]. Our model therefore includes magnetic interactions and we found that a ferromagnetic alignment of the magnetic moments is favored energetically over the most trivial antiferromagnetic configuration. The trivial antiferromagnetic system is accomplished by repeating the cell once in the \(z\)-axis direction with spins aligning in opposite directions. For this reason, and the fact that the spin moments disorder at elevated temperatures, we assume ferromagnetic order in our ab initio model for \(\gamma\)-U. To obtain as accurate total magnetic moments as possible, we apply orbital polarization (OP) for better orbital moments. The DFT + OP approach has been very effective for plutonium metal [84], and even though orbital polarization is almost negligible for \(\alpha\)-U [81], it shall not be ignored for \(\gamma\)-U. The orbital-polarization coupling can only happen if there is a charge current, and it is thus not addressed in traditional current-free density-functional theory. Generally, results from the present DFT + OP scheme (see more technical details and references in [84] or [85]) compare excellently with those computed from an approach where orbital polarization is rigorously formulated within an advanced current density-functional theory [86]. The lattice-dynamics contributions to the Gibbs free energy and heat capacity (\(F_{lat}\) and \(C_{p}^{lat}\)) are obtained from the SCAILD method that was described in [75]. The method requires calculations of forces on thermally perturbed atoms in a supercell representation of the crystal phase. In this case, we set up a \(3^{3}\) (27 atoms) supercell of bcc (\(\gamma\)) uranium and perform 400 SCAILD iterations to ensure very good convergence of the free energy and heat capacity. The supercell size is consistent with our previous SCAILD works [13; 53; 60; 65] and is assumed to be sufficient. Three strategic volumes and six temperatures are simulated with this approach. The SCAILD free-energy volume dependence is obtained by interpolating a cubic polynomial between the three volumes. Atomic forces can be determined from applying the FPLMTO method [13], but it is computationally far easier to utilize a pseudopotential plane-wave-based method, namely, the Vienna ab initio simulation package (VASP) in conjunction with the projector-augmented-wave (PAW) method [87; 88]. One important aspect of the calculation is the energy cut-off that defines the expansion of the plane waves in the method. In our previous studies involving SCAILD and force calculations using VASP-PAW [65], we found that a plane-wave energy cut-off of about 400 eV is reasonably accurate. However, in the present case of \(\gamma\)-uranium, we checked the influence of a greater energy cut-off (500 eV) on one temperature and volume (1000 K and 20.86 A\({}^{3}\)). The lattice contribution to the free energy changed with these more accurate calculations, but only by a small amount of about 0.24 meV. In analogy with the all-electron FPLMTO treatment, we assumed ferromagnetic order in the VASP calculations of the forces for SCAILD. Furthermore, again consistent with the all-electron approach, the VASP calculations relied on the generalized gradient approximation for the electron interactions. For the spin-orbit interaction, we implemented [81] the same correction as with the FPLMTO and included the orbital-polarization scheme. One difference with VASP relative to FPLMTO is that we apply a fixed 5f-electron Racah parameter (0.056 eV) in VASP.