## 3 Results and Discussions
The spin polarization (P) at Fermi energy is  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Compound** & **Total** & **Zr** & **Rh** & **Ti** & **Z** & **a (Å)** & **CBM** & **VBM** & \(E_{bg}\) & \(E_{HM}\) & \(Z_{t}\) & **S-P rule** & **P (\%)** \\ \hline ZrRhTiAl & 2.00 & 1.26 & \(-\)0.32 & 1.18 & \(-\)0.14 & 6.47 & 0.350 & \(-\)0.082 & 0.432 & 0.082 & \multirow{2}{*}{20} & \multirow{2}{*}{\(M_{t}=Z_{t}-18\)} & \multirow{2}{*}{100} \\ \cline{1-1} \cline{6-11  
100% for ZrRhTiZ (Z = Al, Ga), for the Fermi level lies in a band gap in the spin-down channel. For Heusler compounds with half-metallicity, the origin of the half-metallic band gap is usually owed to the hybridization of the d electrons between the transition metal atoms. Through the classical molecular orbital approach, Zr-4d and Rh-4d hybridization within tetrahedral symmetry has been considered, as presented in Figure 5. In ZrRhTiZ (Z = Al, Ga) Heusler compounds, the 4d orbitals hybridization of Rh and Zr atoms generated five bonding bands (\(3t_{\rm 2g}\) and \(2e_{\rm g}\)) and five non-bonding bands (\(2e_{\rm u}\) and \(3t_{\rm 1u}\)). Then, the 3d orbitals of Ti atom hybridize with the five bonding 4d hybridized orbitals (\(3t_{\rm 2g}\) and \(2e_{\rm g}\)) of Rh and Zr atom, while the five non-bonding hybridized 4d orbitals (\(2e_{\rm u}\) and \(3t_{\rm 1u}\)) still hold with no hybridization. Finally, the distribution of the 15 d orbitals in the spin-down channel can be determined, i.e., \(3\times t_{\rm 2g}\), \(2\times e_{\rm g}\), \(2\times e_{\rm u}\), \(3\times t_{\rm 1u}\), \(3\times t_{\rm 2g}\), and \(2\times e_{\rm g}\) are arranged from high energy to low energy. The main-group of \(Z\) atoms generate \(1\times s\) orbital and \(3\times p\) orbitals which are distributed below the 15 hybridized d orbitals and totally occupied in ZrRhTiZ (Z = Al, Ga) compounds. The triple-degeneracy \(t_{\rm 1u}\) states are not occupied in the spin-down direction, and a band gap is formed between \(t_{\rm 1u}\) (non-bonding) and \(t_{\rm 2g}\) (bonding) orbitals in the spin-down channel of ZrRhTiZ (Z = Al, Ga) compounds. Similarly, from Table 1, it can be seen that the number of valence electrons is 20 per ZrRhTiZ (Z = Al, Ga) unit cell, and the calculated total magnetic moment is 2 \(\upmu_{\rm B}\), which follows the Slater-Pauling rule \(M_{t}=Z_{t}-18\) and is also consistent with the reported Zr-based SQH materials [4; 5; 6; 7; 16; 17; 18]. Figure 4: Calculated total density of states and partial density of states of ZrRhTiAl (**a**) and ZrRhTiGa (**b**). Figure 6 displays total magnetic moment and atomic magnetic moments as a function of lattice constant for ZrRhTiAl and ZrRhTiGa compounds. It can be noted that the total magnetic moment is constant at 2\(\mu_{\text{B}}\) over the range of the lattice constant, from 6.09 (6.10) to 6.61 (6.75) A for ZrRhTiAl (ZrRhTiGa) compounds. The atomic magnetic moment is appreciably increasing with the lattice constant values for the Rh and Ti atoms, whereas the atomic magnetic moment of Zr as well as Al/Ga was not found to change appreciably over the wide range of lattice constants. The constant total magnetic moment can be attributed to the antiparallel arrangement of the magnetic moments of Rh and Ti atoms. The main contributors to the magnetic moment are Zr and Ti atoms. The Rh and Z atoms only carry small magnetic moments antiparallelly aligned to those of the Zr and Ti atoms. The half-metallicity is very much sensitive to the change of lattice constant. Therefore, it is very useful to acquire knowledge about the sensitivity of the half-metallicity to the uniform strain (US) for the ZrRhTiZ (Z = Al, Ga) compounds. Here, the CBM (conduction band minimum) and VBM in the spin-down channel at different lattice constants have been recorded to show the effects of the US on half-metallic behavior, as plotted in Figure 7. Obviously, the half-metallicity can be maintained in the lattice constant range of 6.09-6.61 A for ZrRhTiAl and 6.10-6.75 A for ZrRhTiGa.