## 3 Results and Discussion  
### Structural, and Electronic Band Structure  
The calculated equilibrium lattice constants, \(a\), \(b\), _c/a_ ratio, unit cell volume V\({}_{0}\), atomic coordination O\({}_{12}\), O\({}_{2\omega}\), O\({}_{2\gamma}\), and bandgap are listed in Table 1. It is obvious that the _c/a_ ratio for the Pb\({}_{2}\)AsI\({}_{0}\), Pb\({}_{2}\)SbIO\({}_{6}\), Pb\({}_{2}\)BiBrO\({}_{6}\), Pb\({}_{2}\)BiO\({}_{6}\), and Pb\({}_{2}\)SeTeO\({}_{6}\) is 1.413 which is very close to the ideal value of \(\sqrt{2}\). The _c/a_ ratio of Pb\({}_{2}\)AsBrO\({}_{6}\) and Pb\({}_{2}\)SbBrO\({}_{6}\) were 1.282 and 1.286, respectively. We also found that Pb\({}_{2}\)AsI\({}_{0}\) has the smallest equilibrium volume whilst Pb\({}_{2}\)BiIO\({}_{6}\) has the largest equilibrium volume. It is related to the difference in wigner-seitz radius of the As and Bi atom. The calculated bandgap is all zero without SOC, while it becomes nonzero in Pb\({}_{2}\)BiBrO\({}_{6}\) and Pb\({}_{2}\)SeTeO\({}_{6}\) double perovskites with SOC. The bandgap for the Pb\({}_{2}\)BiBrO\({}_{6}\) and Pb\({}_{2}\)SeTeO\({}_{6}\) double perovskites is indirect and the magnitude is 0.390 eV and 0.181 eV. Displayed in Figure 2 is the electronic band structures of the Pb\({}_{2}\)AA'O\({}_{6}\) double perovskites without, upper panel, and with, bottom panel, spin-orbit coupling. Let us take Pb\({}_{2}\)BiBrO\({}_{6}\), see Figure 2e, as an example. Without SOC, the band structures of Pb\({}_{2}\)BiBrO\({}_{6}\) at \(\Gamma\) point is contributed by the Bi-p and Br-p orbitals. The Bi-p and Br-p orbitals are separated into \(p_{z}\) and \(p_{y}\) (\(p_{x}\)) groups. About 1 eV at \(\Gamma\) point, the bands are mainly contributed from the Bi-s and Br-s orbital. It satisfies the first condition given by [PERSON] et al. [30] that there is a _s_- and _p_-band inversion that occurs at the \(\Gamma\) point without SOC. When the SOC is taken into account, see Figure 2l, the p-orbital of the Bi and Br atom split to \(-\)0.6 eV and 0.55 eV with respect to the Fermi level. The orbital contribution at \(\Gamma\) point is still contributed from the p-orbital of the Bi and Br atom. Away from \(\Gamma\) along \(\Gamma\)-X direction, the band's contribution transferred from _p_- to _s_-orbital At X point, the bands become mostly from the _s_-orbital of the Bi and Br atom. To make it clearer, we demonstrate the Bi (Se) and Br (Te), _s_- and _p_-orbital projected band structures in Figure 3. Therefore, we conclude that Pb\({}_{2}\)BiBrO\({}_{6}\) and Pb\({}_{2}\)SeTeO\({}_{6}\) are possible topological insulators. The SOC-induced bandgap for the Pb\({}_{2}\)BiBrO\({}_{6}\) is 0.390 eV, being higher than most 3D TIs. We also found that the Z\({}_{2}\) index [32] (\(v_{0}\); \(v_{1}\)\(v_{2}\)\(v_{3}\)) of Pb\({}_{2}\)BiBrO\({}_{6}\)and Pb\({}_{2}\)SeTeO\({}_{6}\) are (1; 0 0 0) showing that they are strong topological insulators. We also demonstrate the atom and orbital decomposed density of states in Figure 4. We focus on the Bi, Br, Se, and Te atoms' -, \(p_{x}\)-, \(p_{y}\)- and \(p_{z}\)-orbitals. In Pb\({}_{2}\)BiBrO\({}_{6}\) double perovskite, the density of states of Bi's _s_-orbital is larger than Br's _s_-orbital. In contrast, the density of states for Br's p orbitals is larger than Bi's p orbital. The peak for both Bi's and Br's _s_-orbital lies between \(-\)1 eV and \(-\)2 eV. It is quite different compared with the density of states of Se's and Te's _s_-orbital. The Te's _s_-orbital is closer to the Fermi level than Se's _s_-orbital. We also note that Br and Te's _p_x_- and _p_y_-orbital completely degenerate between \(-\)6 eV to 2 eV. To further confirm that Pb\({}_{2}\)BiBrO\({}_{6}\) and Pb\({}_{2}\)SeTeO\({}_{6}\) are possible topological insulators, we present in detail our calculations for \(Z_{2}\) invariants. We first calculate the bulk band structure using the maximally localized Wannier functions as implemented in the WANNIER90 package [37].