# Thermodynamics and Magnetism of SmFe\({}_{12}\) Compound Doped with Co and Ni: An Ab Initio Study
## 2 Computational Methods  
The fundamental framework for the present modeling is based on density-functional theory (DFT). In this approach, there is an unavoidable approximation for the electron exchange and correlation interactions. Here, we apply the generalized gradient approximation for this purpose. This exchange and correlation functional works extremely well for many elements and compounds in the periodic table but cannot determine the localized nature of the electrons. This is no issue for the 3\(d\) transition elements Fe, Co, and Ni that we study, while for the rare-earth element Sm, the 4\(f\) electrons are localized and require special treatment. The 3\(d\) transition metals Fe, Co, and Ni are generally very well described within DFT, but for very good orbital moments one can include a simple parameter-free scheme that couples the orbital moments, i.e., an orbital polarization (OP) mechanism, which is otherwise not included in conventional DFT. This recipe results in excellent spin and orbital moments for Fe, Co, and Ni [154]. Samarium metal poses a more challenging problem for DFT. Here, the 4\(f\) electrons are known to be localized (atomic-like and essentially non-bonding) with magnetic properties closer to the free Sm\({}^{3+}\) atom but with a magnetic moment largely quenched by crystal-field effects. DFT-GGA-type calculations are not able to describe the nature of these 4\(f\) states but exaggerate their itineracy or band-like character. For energetics such as the structural stability, the DFT-GGA approximation works better than one would expect, and the formation energies are reproduced via experiments for the SmCo\({}_{5}\) (doped with Fe and Ni) very nicely [149; 150]. For spectroscopic or magnetic properties, however, one must go beyond DFT-GGA to achieve an accurate theory. In the literature, DFT in connection with large intra-atomic Coulomb repulsion and a Hubbard U parameter (DFT + U) has been applied for Sm systems [155]. The more advanced dynamical mean-field theory (DMFT) has also been utilized for SmCo\({}_{5}\)[156]. A more practical approach, which allows for accurate MAE for SmCo\({}_{5}\), is the application of the \"standard rare-earth model\" (SRM), where the 4\(f\) electrons are treated as localized by being constrained to occupy core states that do not hybridize with the band states. This technique is very similar to DMFT in the Hubbard I approximation [157]. In the SRM, the magnetic-moment contribution from the Sm 4\(f\) electrons is constrained to zero, while a small induced magnetic moment still forms on the Sm atom (\(\sim\)0.3 \(\mu_{B}\)) from other valence electrons. This is reasonable because for Sm metals the net magnetic moment is estimated to be very small due to spin and orbital moment cancelation (0.1 \(\mu_{B}\)) [158]. The SRM approach, coupled with an orbital polarization extension of DFT (see the next section), proved to be an excellent model for the magnetic anisotropy energies in SmCo\({}_{5}\) and related systems [149; 150], and we adopt it here for all MAE calculations. Here, we employ two methods (described below)--one is very well suited for alloy calculations, and one is appropriate for MAE calculations within the standard rare-earth model. The calculations we refer to as EMTO are carried out using [PERSON]'s function based on the improved screened Korringa-Kohn-Rostoker formalism, where the one-electron potential is described by the optimized overlapping muffin-tin (OOMT) potential spheres [159; 160]. Inside the potential spheres, the potential is spherically symmetric, while it is constant between the spheres. The radius of the potential spheres, the spherical potential inside these spheres, and the constant value in the interstitial region are determined by minimizing (i) the difference between the exact and overlapping potentials and (ii) the inaccuracy that is caused by the overlap between the spheres. Within the EMTO formalism, the one-electron states are resolved exactly for the OOMT potentials.