## 1 Introduction
[58; 59; 60; 61] calculated MAE of the YCo\({}_{5}\) magnet (\(K_{1}\) = 2.83 MJ/m\({}^{3}\)) as well as Y(Co\({}_{1\text{-}\text{x}}\)Fe\({}_{\text{x}}\))\({}_{5}\) and Y(Co\({}_{1\text{-}\text{x}}\)Cu\({}_{\text{x}}\))\({}_{5}\) alloys. Calculated MAE of the Y(Co\({}_{1\text{-}\text{x}}\)Fe\({}_{\text{x}}\))\({}_{5}\) compounds is in qualitative accord with the results of the previously mentioned LCAO calculations [56]. MAE of the Y(Co\({}_{1\text{-}\text{x}}\)Fe\({}_{\text{x}}\))\({}_{5}\) magnets with 6-7 % doping increases by about 1.9 MJ/m\({}^{3}\) before falling rapidly with a larger Fe doping. The total magnetic moment gradually increases from \(m^{(tot)}\) = 7.2 \(\upmu_{\text{B}}\)/f.u. (YCo\({}_{5}\)) to -10 \(\upmu_{\text{B}}\)/f.u. (Y(Co\({}_{1}\)Fe\({}_{4}\))) then falls to 9 \(\upmu_{\text{B}}\)/f.u. for YFe\({}_{5}\). When doping the YCo\({}_{5}\) magnet with copper, the total moment gradually decreases with increasing copper concentration, and finally the moment disappears for the Y(Co\({}_{1\text{-}\text{x}}\)Cu\({}_{\text{x}}\)) compound. The calculations performed by [PERSON] and [PERSON] [61] show that, for the Y(Co\({}_{1\text{-}\text{x}}\)Cu\({}_{\text{x}}\))\({}_{5}\) alloys, the total magnetic moment decreases linearly with Cu doping, identical to the results of Ni doping of the Y(Co\({}_{1\text{-}\text{x}}\)Ni\({}_{\text{x}}\))\({}_{5}\) alloys [51]. However, in contrast to the Y(Co\({}_{1\text{-}\text{x}}\)Ni\({}_{\text{x}}\))\({}_{5}\) compounds, the magnetic moments of the cobalt atoms are reasonably local and barely impacted by Cu doping of the Y(Co\({}_{1\text{-}\text{x}}\)Cu\({}_{\text{x}}\))\({}_{5}\) alloys until \(x\) \(\sim\)0.5, where the moment promptly falls to a small value before steadily decreasing [61]. The authors [61] concluded that cobalt in the doped YCo\({}_{5}\) compounds can occur either in a high- or a low-spin state, depending on the local surroundings. A similar conclusion (the high spin to the low spin transition) is presented in the paper of [PERSON] et al. [63], where VASP-PAW formalism is applied to study the Y(Co\({}_{1\text{-}\text{x}}\)Ag\({}_{\text{x}}\))\({}_{5}\) alloys. The ASW-ASA formalism has been applied for study magnetic and electronic properties of the Y(Co\({}_{1\text{-}\text{x}}\)Fe\({}_{\text{x}}\))\({}_{5}\) compounds: YCo\({}_{5}\), Y(Fe\({}_{2}\)Co\({}_{3}\)), Y(Fe\({}_{3}\)Co\({}_{2}\)), and YFe\({}_{5}\)[65]. FPLMTO calculations for the Y(Co\({}_{1\text{-}\text{x}}\)Fe\({}_{\text{x}}\))\({}_{5}\) compounds with \(x\) = 0, 0,1, 0.2, and 0.3 [67] confirm that iron atoms choose to occupy 3g sites, and total magnetization increases from 7.50 \(\mu_{\text{B}}\)/f.u. to 8.97 \(\mu_{\text{B}}\)/f.u. with increasing \(x\) from 0 to 0.3 because iron atoms have a much bigger magnetic moment (\(\sim\)2.4 \(\mu_{\text{B}}\)) than the magnetic moment of cobalt atoms (\(\sim\)1.5 \(\mu_{\text{B}}\)). A real-space pseudopotential formalism with SOC incorporated by norm-conserving pseudopotentials has been used to compute MAE of the YCo\({}_{5}\) magnet in different structures [78]. It appears that the authentic CaCu\({}_{5}\)-type structure (Space Group P6/mmm) possesses the largest total magnetic moment \(m^{(tot)}\) = 7.13 \(\mu_{\text{B}}\)/f.u. and MAE \(K_{1}\) = 2.68 MJ/m\({}^{3}\). The latter is, however, significantly smaller than the experimental value of \(K_{1}\) = 6.5 MJ/m\({}^{3}\)[6]. [PERSON] et al. [79] studied the dependence of MAE of the YCo\({}_{5}\) magnet with intra-atomic Coulomb interaction (the Hubbard \(U\) parameter, DFT+\(U\) scheme) and strength of SOC within the PAW-VASP method. These calculations showed that, for GGA+\(U\) calculations, with 1.4 eV \(\leq U\leq\) 1.47 eV and \(J\) = 0.8 eV, MAE of the YCo\({}_{5}\) compound changes within the experimentally observed range. Similar results are presented in the work of [PERSON] et al. [70], where LDA+DMFT computational scheme is used to replace the OP correction [37].