## 2 Computational Methods  
The calculations we denote as EMTO are performed while using the Green's function formalism based on the improved screened Korringa-Kohn-Rostoker technique, where the one-electron potential  
Figure 1: Crystal structure of SmCo\({}_{5}\). The schematic is taken from ref. [62]. is defined by the optimized overlapping muffin-tin (OOMT) potential spheres [98; 99]. Inside the potential spheres, the potential is spherically symmetric, while it is permanent between the spheres. The radius of the potential spheres, the spherical potential inside these spheres, and the permanent value in the interstitial region are resolved by minimizing (i) the deviation between the exact and overlapping potentials and (ii) the errors that are caused by the overlap between the spheres. Within the EMTO formalism, the one-electron states are calculated exactly for the OOMT potentials. As a product of the EMTO calculation, one can determine the self-consistent Green's function of the system and the complete, non-spherically symmetric charge density. Lastly, the total energy is calculated while using the full charge-density technique [100]. We consider, as the valence states, _6s_, _5p_, _5d_, and _4f_ states for Sm and _4s_ and _3d_ states for Co and Fe. The corresponding Kohn-Sham orbitals are expanded in terms of _spdf_ exact muffin-tin orbitals, i.e., we select an orbital momentum cutoff \(l_{max}=3\). The EMTO orbitals, consequently, consist of the _spdf_ partial waves (solutions of the radial Schrodinger equation for the spherical OOMT potential wells) and the _spdf_ screened spherical waves (solutions of the Helmholtz equation for the OOMT muffin-tin zero potential). The completeness of the muffin-tin basis was discussed in detail in ref. [99]. The generalized gradient approximation (GGA-PBE) [101] is selected for the electron exchange and the correlation energy functional. Integration over the Brillouin zone is performed using the special _k_-point technique [102] with 784 _k_-points in the irreducible wedge of the zone (IBZ). The moments of the density of states, needed for the kinetic energy and valence charge density, are computed by integrating the Green's function over a complex energy contour with 2.2 Ry diameter while using a Gaussian integration technique with 30 points on a semi-circle enclosing the occupied states. In the case of the implementation of the FR-EMTO formalism, SOC is included through the four-component Dirac equation [103]. In order to treat compositional disorder, the EMTO method is coupled with the coherent potential approximation (CPA) [104; 105]. The ground-state properties of the Y(Co\({}_{1\text{-x-y}}\)Fe\({}_{\text{x}}\)Ni\({}_{y}\))\({}_{5}\) alloys are obtained from EMTO-CPA calculations that include the Coulomb screening potential and energy [106; 107; 108]. The paramagnetic state of the Y(Co\({}_{1\text{-x-y}}\)Fe\({}_{\text{x}}\)Ni\({}_{y}\))\({}_{5}\) is modeled within the disordered local moment (DLM) approximation [109; 110]. The equilibrium atomic density of the alloy is obtained from a Murnaghan fit to the total energy versus atomic volume curve [111]. The highest level of theory (least approximations) is implemented in a full-potential scheme, i.e., no structural approximations, which includes spin-orbit interaction as explained [112] for early and late lanthanides [96; 113]. The full-potential linear muffin-tin orbital (FPLMTO) accomplishes this in ways that are detailed [114]. For Y, we use two energy parameters coupled with each basis function, and these parameters have different values for pseudocore states (_4s_ and _4p_) and valence states (_5s_, _5p_, _4d_, and _4f_). Test calculations including the _3d_ states as semi-core states for Y do not significantly change any result. For the TM atom, we use a similar set-up. The spin-orbit interaction and the orbital polarization (OP) operate on the \(d\) and \(f\) orbital. OP is introduced in FPLMTO in a self-consistent fashion so there are no fixed parameters.