## 3 Results and Discussion
### Electronic Structure and Slater-Pauling Rule  
The partial and total (\(M_{\rm t}\)) element magnetic moments, and total number of valence electrons (\(Z_{\rm t}\)) of this compound at the most stable configuration are shown in Table 3. The \(M_{\rm t}\) of the ScFeRhP EQH compound is 1 \(\mu_{\rm B}\), while its total number of valence electrons is 25. This EQH compound obeys the Slater-Pauling rule: \(M_{\rm t}=Z_{\rm t}-24\)[16].  
Figure 4 shows the calculated band structures for the ScFeRhP compound. The band structure reveals the half-metallicity of this compound near the Fermi level. The spin-up channel exhibits a metallic character, while the spin-down channel exhibits a semiconducting character. Therefore, based on the obtained band structures and magnetism, we can conclude that the ScFeRhP EQH compound is a HM.  
We analyze the origin of the band gap at the spin-down channel in Figure 5. According to the study of [PERSON] et al., P has completely occupied \(1s\) and \(3p\) states. We need to consider only the hybridization character of the \(3d\) and \(4d\) states of the ScFeRhP EQH compound, as shown in Figure 5. For this alloy, the \(d_{4}\) and \(d_{5}\) orbits of the Fe and Sc atoms couple forming antibonding \(e_{u}\) and bonding \(e_{g}\) states. The \(d_{1}\), \(d_{2}\), and \(d_{3}\) orbits of the Fe and Sc atoms couple forming antibonding \(t_{1u}\) and bonding  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**States** & **Calculated Value** & **Type 1** & **Type 2** & **Type 3** \\ \hline \multirow{2}{*}{FM} & Equilibrium lattice constant & 5.97 Å & 6.14 Å & 6.18 Å \\  & Total energy & \(-\)2931.40 eV & \(-\)2930.75 eV & \(-\)2930.72 eV \\ \hline \multirow{2}{*}{NM} & Equilibrium lattice constant & 5.96 Å & 6.08 Å & 6.10 Å \\  & Total energy & \(-\)2931.23 eV & \(-\)2929.80 eV & \(-\)2930.06 eV \\ \hline \end{tabular}
\end{table}
Table 2: The equilibrium lattice constant and minimum total energy of each type for both FM and NM magnetic states.  
Figure 4: Band structures of the (**a**) spin-up and (**b**) spin-down channels of the ScFeRhP EQH compound at its equilibrium lattice constant.  
\(t_{2g}\) states. The same states of the Rh atom hybridize with the above orbits, yielding 15 orbits (3 \(t_{2g}\), 2 \(e_{g}\), 2 \(e_{u}\), 3 \(t_{1u}\), 3 \(t_{2g}\), and 2 \(e_{g}\)). There are 8 occupied orbits under the Fermi level (Figure 5). Combined with the 3\(p\) and 1s orbitals generated by P, there are a total of 12 orbits below the Fermi level; therefore, this compound follows the Slater-Pauling rule: \(M_{\rm t}=Z_{\rm t}-24\).  
In order to explain the origin of the band gap [17], we analyzed the partial densities of states (PDOS) and total density of states (TDOS) of ScFeRhP; the results are shown in Figure 6. The PDOS indicate that the TDOS at the Fermi level are mostly attributed to the Fe 3\(d\) and Rh 4\(d\) states. The PDOS of the P atoms and rare-earth element Sc are significantly lower near the Fermi level compared with those of the Fe and Rh atoms. Figure 6 reveals that the Fe element exhibits a stronger spin splitting at \(-1.4\) eV in the majority channel and \(-0.8\) eV in the other channel for the bonding state. For the Rh element, the bonding state was observed mostly in the range of \(-3.5\) eV to \(-4\) eV in the majority channel and in the range of \(-3\) eV to \(-3.5\) eV in the other channel.  
Figure 5: \(d\)-Orbit hybridization of the Rh (4\(d\)) and Fe/Sc (3\(d\)) transition elements in the ScFeRhP EQH compound.  
Figure 6: Total density of states (TDOS) and partial densities of states (PDOSs) for ScFeRhP.