# Development of a CALPHAD Thermodynamic Database for Pu-U-Fe-Ga Alloys
## 2 Thermodynamic Modeling
### Electronic Structure Calculations  
The results from all electronic-structure calculations come from applying density-functional theory (DFT) that took shape in the seminal papers by [PERSON], [PERSON], and [PERSON] [12; 13]. This theory is in principle exact, but it relies on a fundamental approximation, namely, the so-called electron exchange and correlation functional. For actinides as well as iron [14] the generalized gradient approximation (GGA) is currently the best approximation and is applied here. Specifically, the assumption for all DFT calculations is GGA either in its original form (PW91) [15] or the similar but simplified; Perdew, Burke and Ernzerhof (PBE) functional form [16]. Before detailing the electronic-structure methods used here, it is necessary to mention that theories which go beyond DFT have been applied to _5f_-electron systems including uranium and plutonium metals. [PERSON] and [PERSON] [17] applied intra-atomic Coulomb correlations with a large Hubbard U of the order of 4-4.5 eV in their DFT + U approach for plutonium and many other works using similar U (see references in [18]). Also, for uranium metal DFT + U has been applied [19] but it has been shown that with carefully executed DFT calculations this empirical parameter is not required for metallic uranium or plutonium (or alloys with iron) [18; 20]. For the plutonium system it is required for good accuracy to extend the DFT to include orbital-orbital coupling. This can be done rather straightforwardly by adopting the orbital-polarization scheme [18]. The orbital-orbital interaction enhances the magnitude of the orbital magnetic moment (caused by the spin-orbit interaction) and results in more realistic magnetic and chemical bonding properties. Here all-electron full-potential linear muffin-tin orbital (FPLMTO) calculations include spin-orbit coupling and orbital polarization. The orbital polarization tends to increase the stability of the respective compound relative to the spin-orbit only case. In terms of the technical details of the calculations, for the most part, the FPLMTO method is applied, which has been described in detail [21]. In this method, no approximations are made for the electron core states that lie deeper in energy than the valence states, unlike the so-called pseudopotential method. The present implementation does not make any assumptions beyond that of the electron exchange and correlation functional. Basis functions, electron densities, and potentials are calculated without any geometrical approximation and these are expanded in spherical harmonics inside non-overlapping (muffin-tin) spheres surrounding each atom and in Fourier series in the region between these muffin-tin spheres. The theory includes all relativistic corrections including spin-orbit coupling for \(d\) and \(f\) states but not for the \(p\) states [18]. The actinide metals have been shown to be very well described with 6s and 6\(p\) semi-core states and 7s, 7p, 5f, and 6\(d\) valence states [22] and this setup is replicated here.For the calculation of the formation enthalpy one calculates the energies for the An\({}_{6}\)Fe (An = Pu, U) compound and subtract the energies of the constituents in their ground-state phase (\(\alpha\)-An and \(\alpha\)-Fe). Ideally, if computationally feasible, these structures need to be optimized (adjusted or relaxed) so that the calculations produce the lowest total energy of the phase. For the FPLMTO calculations this has been done by calculating numerical forces from small atomic displacements [23]. For An\({}_{6}\)Fe, \(\alpha\)-Pu, \(\alpha\)-U, \(\alpha\)-Fe, a total of 128, 54, 256, and 2000 k points are included, respectively in the electronic-structure calculations. Each energy eigenvalue is broadened with a Gaussian having a width of 20 mRy. When calculating the formation enthalpy, it is important to know precisely what the crystal structure is. For the studied compounds they are known (Table 2), except for Pu\({}_{6}\)Fe where detailed experimental information is lacking.