# High-Temperature Thermodynamics of Uranium from Ab Initio Modeling
## 2 Ab Initio Computational Methodology  
We calculate the Gibbs free energy, \(F\), and heat capacity, \(C_{v}\) or \(C_{p}\), for \(\gamma\)-uranium using parameter-free first-principles density-functional theory for the electronic structure. The thermodynamic quantities are here divided into contributions from the lattice dynamics and electronic excitations. No explicit electron-phonon coupling term is included in the formulation of the free energy, simply because there is no straightforward way to compute this contribution. We introduce the most general expression for the Gibbs free energy at zero pressure that we consider for \(\gamma\)-U as a function of volume and temperature (\(V\),\(T\)):  
\[F(V,T)=F_{lat}(V,T)+F_{el}(V,T)+F_{mag}(V,T) \tag{1}\]  
Here, \(F_{lat}\) is the free energy from lattice vibrations, \(F_{el}\) is the electronic free energy, including entropy, and \(F_{mag}\) is the magnetic contribution due to classic entropy of magnetic disorder, \(S_{mag}\);  
\[F_{mag}(V,T)=-k_{B}TS_{mag}=-k_{B}Tln(2\mu+1) \tag{2}\]  
where \(k_{B}\) is the Boltzmann constant and \(\mu\) is the total magnetic moment (spin and orbital contributions) [77; 78]. In an analogous fashion to the free energy, we express and calculate the heat capacity:  
\[C_{p}(T)=C_{p}^{lat}(T)+C_{p}^{el}(T)+C_{p}^{mag}(T) \tag{3}\]  
As mentioned in the introduction, under the Sommerfeld assumption [70], the electronic contribution to the specific heat, \(C_{p}^{el}\), is derived from the electronic density of states, \(D\) (\(E\)), at the highest occupied energy state (Fermi level, \(E_{F}\)), \(D\) (\(E_{F}\)):  
\[C_{p}^{el}(T)=\frac{\pi^{2}}{3}D(E_{F})k_{B}^{2}T=\gamma_{e}T \tag{4}\]  
Note that \(D\) (\(E\)) depends on temperature in two ways. First, it is broadened by a Fermi-Dirac temperature distribution. Second, it is sensitive to the atomic volume that is increasing with temperature due to thermal volume expansion. We account for both these effects when determining \(C_{p}^{el}\). Finally, the magnetic term in Equation (3), \(C_{p}\)\({}^{mag}\), is acquired from the following thermodynamic equation at constant pressure [79]:  
\[C_{p}^{mag}(T)=k_{B}T\bigg{[}\frac{dS_{mag}(T)}{dT}\bigg{]}_{p} \tag{5}\]where the magnetic entropy, \(S_{mag}\), is expressed in Equation (2). Analogous to the electronic contribution, the magnetic term depends on temperature via the electronic structure and thermal expansion. The electronic and magnetic terms (\(F_{el}\), \(F_{mag}\), \(C_{p}^{el}\), and \(C_{p}^{mag}\)) are calculated using a very flexible and reliable all-electron full-potential linear muffin-tin orbital (FPLMTO) method [80]. The framework for this method is built on basis functions where the radial part is a linear superposition of atomic-like functions and their derivatives. What makes this implementation different and more flexible than most others is that the basis set can describe energy levels associated with atomic states with separate quantum numbers while having identical magnetic quantum numbers. In other words, for uranium, 6s and 6p states can be represented as well as 7s and 7p states. In some approaches, these states with the same quantum number are defined by a single set of energy parameters in separate energy panels. However, sometimes these panels can intersect, causing numerical problems. Contrarily, in the present FPLMTO approach bases belonging to the same quantum numbers are composed within one fully hybridizing basis set. Some quantities in FPLMTO are represented in series (bases, charge densities, and potentials) of spherical harmonics inside muffin-tin spheres located on the crystal atomic positions. The radial portion of the muffin-tin-sphere basis function is determined by solving a wave equation that acknowledges all relativistic corrections such as the spin-orbit coupling for d and f states only, i.e., spin-orbit coupling is ignored for p states. The reason for this choice has been discussed and evaluated rather thoroughly for actinide materials [81].