## 3 Results and Discussions  
The quaternary Heusler compounds have a LiMgPdSb-type structure, which belongs to the F\(\bar{4}\)3m (No.216) space group. There are four crystallographic sites in Heusler compounds. For quaternary Heusler compounds, there are three possible atomic arrangements depending on each crystallographic site occupation. In this present study, the atomic arrangement properties of ZrRhTiZ (Z = Al, Ga) compounds were investigated by calculating the total energy in ferromagnetic (FM) and non-ferromagnetic (NM) states. The calculated results show that for the ZrRhTiZ (Z = Al, Ga) compound, the most stable case is that Zr and Ti atoms with less valence electrons enter into the Wyckoff sites 4a(0, 0, 0) and 4c(0.25, 0.25, 0.25), Rh atoms with more valence electrons occupy the 4b(0.5, 0.5, 0.5) site, and the main group of atoms, Z atoms, tend to locate at the 4d(0.75, 0.75, 0.75) site. The corresponding simulated crystal structure is shown in Figure 1. The stable arrangement of ZrRhTiZ (Z = Al, Ga) Heusler compounds is like the combination of inverse Zr\({}_{2}\)RhZ [14] and Ti\({}_{2}\)RhZ (Z = Al, Ga) [15] Heusler compounds and found to be similar to the other Zr-based SQH compounds [16; 17]. Figure 2 shows the dependence of the total energy on the lattice parameters in FM and NM states for ZrRhTiZ (Z = Al, Ga) compounds. One can see that both compounds are more stable in FM than NM states according to the viewpoint of the total energy minimization. The equilibrium lattice constants corresponding to the ground state energy are 6.47 A and 6.45 A for ZrRhTiAl and ZrRhTiGa, respectively. The spin-projected band structures in the reduced Brillouin zone along the high symmetry directions were calculated under the equilibrium lattice constant and plotted in Figure 3 for ZrRhTiAl  
Figure 1: Crystal structure of ZrRhTiZ (Z = Al, Ga) quaternary Heusler compounds. Figure 2: Total energy as a function of the lattice constant in the non-ferromagnetic (NM) and ferromagnetic (FM) states for ZrRhTiAl (**a**) and ZrRhTiGa (**b**) compounds. The lowest energy in the FM state is set as zero point. and ZrRhTiGa compounds. From Figure 3, an indirect band gap can be clearly observed in the spin-down channel, and the Fermi level is located within the band gap. In the spin-up channel, the valence and conduction bands are overlapping, and the Fermi level is intersecting them for ZrRhTiZ (Z = Al, Ga) compounds, which indicates that both compounds are half-metallic materials. The band gap (\(E_{bg}\)) and HM band gap (\(E_{HM}\)) in the spin-down channel are listed in Table 1. The HM band gap is the minimum energy required to flip a minority of spin electrons from the valence band maximum (VBM) to the majority spin Fermi level. The \(E_{\rm bg}\) is 0.432 eV for ZrRhTiAl and 0.541 eV for ZrRhTiGa. Usually, the large \(E_{\rm bg}\) can be considered as evidence of the robustness of half-metallicity to lattice distortion. The Mulliken atomic populations quantify the charge transfer from one atom to another one and are listed in Table 2 for ZrRhTiZ (Z = Al, Ga) compounds. It is clear that the charge transfer from Ti to Rh is 0.69e and Zr to Al is 0.08e in ZrRhTiAl, and the charge transfer from Ti to Zr is 0.27e and Ga to Rh is 0.93e in ZrRhTiGa. Figure 4 shows the total density of state (TDOS) and partial density of state (PDOS) patterns. It is clear that the main contributions to the TDOS near the Fermi level are the strong d-d hybridization among the Zr-4d state, Ti-3d state, and Rh-4d state.