## 1 Introduction
Substituting cobalt atoms with nickel or copper atoms gradually decreases magnetization and magnetic anisotropy. By connecting diffraction techniques and electron microscopy, [PERSON] et al. [66] studied the solubility restriction of Ge in the Y(Co\({}_{1\text{-}\text{x}}\)Ge\({}_{\text{x}}\))\({}_{5}\) alloys. A preferential substitution of Ge atoms that were larger than Co atoms on the 3g positions was observed. Numerous theoretical approaches have been used to study the YCo\({}_{5}\) magnets. These include augmented-plane-waves (APW), point charge, linearized muffin-tin orbitals within atomic sphere approximation (LMTO-ASA), Haydok recursion (HR), full-potential linearized augmented-plane-waves (FLAPW), linear combination of atomic orbitals (LCAO), pseudopotential projected augmented waves (VASP-PAW), augmented spherical waves (ASW-ASA), full-potential linear muffin-tin orbitals (FPLMTO), dynamical mean field theory in conjunction with density functional theory (LDA-DMFT), and Korringa-Kohn-Rostoker multiple-scattering formulation in conjunction with the coherent potential approximation and disordered local moment approximation (KKRASA-CPA-DLM) [21; 26; 27; 30; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 74; 75; 76; 77; 78; 79; 81; 82; 83; 84]. According to the self-consistent APW calculations [21], the total moment of the YCo\({}_{5}\) magnet is equal to \(m^{(tot)}\) = 7.31 \(\upmu_{\text{B}}\)/f.u., which is in accordance with the initially reported experimental data, \(m^{(tot)}\) = 7.52 \(\upmu_{\text{B}}\)/f.u. [20]. Using the point charge model, Inomata [30] confirmed the conclusion presented in the experimental papers [8; 16; 23; 25; 28], namely, the orbital moments of Co atoms located on the 2c sites contribute to the axial (positive) MAE of YCo\({}_{5}\), and the orbital moments of Co atoms located on the 3g sites contribute to its planar (negative) contribution. The importance of the orbital polarization (OP) has been emphasized in LMTO-ASA papers [37; 38; 39; 40; 45; 46; 47]. The magnetization of the YCo\({}_{5}\) compound doped by Ni, Y(Co\({}_{5\text{-}\text{x}}\)Ni\({}_{\text{x}}\)) has been studied within the Haydok recursion method [44]. According to these calculations, when the amount of Ni dissolved in the YCo\({}_{5}\) magnet reaches 60%, the YCo\({}_{2}\)Ni\({}_{3}\) compound loses its magnetization. The LMTO-ASA formalism has also been applied to study the Y(Co\({}_{1\text{-}\text{x}}\)Ni\({}_{\text{x}}\))\({}_{5}\) alloys [48; 51]. These calculations confirm that the nickel atoms choose to occupy 2c sites. LCAO calculations by [PERSON] et al. [49] reveal the total magnetic moment of the YCo\({}_{5}\) magnet is equal to \(m^{(tot)}\) = 8.70 \(\upmu_{\text{B}}\)/f.u. [PERSON] et al. [55; 56] performed fully relativistic LCAO calculations of MAE of YCo\({}_{5}\) and Y(Co\({}_{1\text{-}\text{x}}\)Fe\({}_{\text{x}}\))\({}_{5}\) compounds. Their calculated MAE of the YCo\({}_{5}\) magnet is \(K_{1}\) = 1.11 MJ/m\({}^{3}\) and \(K_{1}\) = 8.40 MJ/m\({}^{3}\) for SOC and SOC+OP theory (where SOC means spin-orbit coupling), correspondingly. They furthermore obtained within the SOC+OP scheme the orbital moments that are 0.33 \(\upmu_{\text{B}}\) and 0.26 \(\upmu_{\text{B}}\) for 2c and 3g sites, correspondingly. These moments are in accordance with experimental measurements of 0.26 \(\upmu_{\text{B}}\) (2c) and 0.24 \(\upmu_{\text{B}}\) (3g) [28]. Calculated MAE for the Y(Co\({}_{1\text{-}\text{x}}\)Fe\({}_{\text{x}}\))\({}_{5}\) compounds show qualitative agreement with the experimental data [34]. The calculated Curie temperature of the YCo\({}_{5}\) magnet \(T_{c}\) = 998 K [57] is in good accord with results of the previous calculations by [PERSON] [26; 27] and the experimental data, \(T_{c}\)\(\sim\)987 K [6]. Using FLAPW (without OP), [PERSON] et al.