## 3 Results And Discussion
### Structural and Electronic Band Structures
The lowest energy of conduction band near R point decreased as the lattice constant increased, resulting in a decrease in bandgap. In the cubic BaPoO\({}_{3}\) perovskites, the Ba \(d\) bands were split into two-fold \(e_{g}\) and three-fold \(t_{2g}\) orbitals under cubic symmetry. At \(\Gamma\) point the topmost valence bands were composed of Ba\({}_{p}\), Po\({}_{p}\), and O\({}_{p}\) orbitals. The lowest conduction bands at \(\Gamma\) point were composed of Ba\({}_{e_{g}}\) and O\({}_{s}\) orbitals. In other words, the energy for the Ba\({}_{t_{2g}}\) orbitals is higher than the Ba\({}_{e_{g}}\) orbitals. Due to Ba and Po hybridization, the Ba\({}_{t_{2g}}\) bands were located below the Ba\({}_{e_{g}}\) bands at the \(R\) points. At \(R\) point at the Fermi level, the band compositions were in triplets with Ba\({}_{t_{2g}}\) states, Po\({}_{p}\) states, and O\({}_{s}\) states. Near \(R\) point, the valence bands and conduction bands were both composed of Ba\({}_{t_{2g}}\) states, Po\({}_{p}\) states, and O\({}_{s}\) and O\({}_{p}\) states. These band compositions were very  
Figure 2: Scalar relativistic band structures, (**a–f**) **upper panel**, and fully relativistic band structures, (**g–l**) **bottom panel** for APoO\({}_{3}\) perovskite. The Fermi level is shifted to zero as the reference. Without SOC effect, (**a**) BePoO\({}_{3}\), (**b**) MgO\({}_{3}\), (**c**) CaPoO\({}_{3}\), (**d**) SrPoO\({}_{3}\), (**e**) BaPoO\({}_{3}\), and (**f**) RaPoO\({}_{3}\) are metals while there are Dirac corn-like band appeared at R point. The highest occupied band at R point is singlet s = 1/2 and the lowest two bands are \(k_{eff}\) = 3/2 and 1/2. When the SOC effect is included, (**g**) BePoO\({}_{3}\) and (**h**) MgO\({}_{3}\) are metals, however (**i**) CaPoO\({}_{3}\), (**j**) SrPoO\({}_{3}\), (**k**) BaPoO\({}_{3}\), and (**l**) RaPoO\({}_{3}\) are insulators. The metal to insulator transition appears due to the SOC effect and the s = 1/2 and \(k_{eff}\) = 1/2 bands flipped, see (**e**) and (**k**). different from the corresponding one at the \(\Gamma\) point, where the topmost valence bands were composed of Ba\({}_{p}\), Po\({}_{p}\), and O\({}_{p}\) orbitals, and the lowest conduction bands were composed of Ba\({}_{\sigma_{S}}\) and O\({}_{s}\) orbitals. When the SOC effect was taken into account, we found that at the equilibrium lattice constant, BePoO\({}_{3}\) and MgPoO\({}_{3}\) were still metals. At the same time, CaPoO\({}_{3}\), SrPoO\({}_{3}\), BaPoO\({}_{3}\), and RaPoO\({}_{3}\) became insulators, and the corresponding indirect bandgaps were 0.861, 0.871, 0.820, and 0.810 (eV), respectively. Calculated bandgap is a very large TI bandgap in oxide perovskites. Especially for the PBE result in the DFT calculation usually underestimates the bandgap [29]. The band inversion induced by the SOC effect evidences the topology. To confirm, we also calculated the \(Z_{2}\) topological numbers. The \(Z_{2}\) index is defined as (\(\upsilon_{0}\),\(\upsilon_{1}\upsilon_{2}\upsilon_{3}\)) as reported by [PERSON] et al. [7]. Our calculated \(Z_{2}\) index for the APoO\({}_{3}\) perovskites was (1000), which suggests a strongly topological insulator. Figure 3: Calculated (**a**) fully relativistic band gap, in eV, and (**b**) external pressure (GPa) as a function of the lattice constant with respect to its equilibrium lattice constant \(a_{eq}\) (Å). It is fascinating if the cubic perovskites are electrically polarized. Nonetheless, the ideal cubic perovskite has a space inversion structure. Hence the electric polarization, \(P_{e}\), for all cubic perovskites is zero. However, there were kinds of defects or structural distortions in experiments. These defects or distortions will break the space inversion of the cubic perovskite, which produces a nonzero electric polarization. One way to study the electric polarization is to calculate the induced \(P_{e}\) as a function of the atom's displacement. Plotted in Figure 5 is the electric polarization as a function of the Po atom's displacement.