## 3 Results And Discussion
### Mechanical Properties  
The mechanical stability and the related physical properties of the APoO\({}_{3}\) perovskites are listed in Table 2. Due to the cubic symmetry of the perovskite structure, only three independent elastic constants C\({}_{11}\), C\({}_{12}\), and C\({}_{44}\) need to be taken into account. These elastic constants are determined from the stress-strain relations, where some finite distortions of the lattice are calculated. In these calculations, we used a k-mesh of 14 \(\times\) 14 \(\times\) 14, and the self-consistent total energy criterion was set to be 1.0 \(\times\) 10\({}^{-8}\) eV. Listed in Table 2 are the elastic constants C\({}_{11}\), C\({}_{12}\), and C\({}_{44}\) of the APoO\({}_{3}\) perovskites together with the bulk modulus, \(B\), Voigt shear modulus, \(G_{V}\), [PERSON] shear modulus, \(G_{R}\), shear modulus, \(G\), [PERSON]'s modulus, \(E\), anisotropy factor, \(A_{f}\), [PERSON] ratio, \(\
u\), and [PERSON]'s ratio, \(B/G\). These physical properties are calculated from the following equations:  
\[B=\frac{C_{11}+2C_{12}}{3}, \tag{1}\]  
\[G_{V}=\frac{C_{11}-C_{12}+3C_{44}}{5}, \tag{2}\]  
\[G_{R}=\frac{5(C_{11}-C_{12})C_{44}}{4C_{44}+3(C_{11}-C_{12})}, \tag{3}\]  
Figure 5: Calculated electric polarization P\({}_{e}\), in C/m\({}^{2}\), and fully relativistic bandgap, in eV, for the (**a**) CaPoO\({}_{3}\), (**b**) SrPoO\({}_{3}\), (**c**) BaPoO\({}_{3}\) and (**d**) RaPoO\({}_{3}\) as a function of the lattice constant with respect to its equilibrium lattice constant \(a_{\alpha\Omega}\) (Å). \[G = \frac{G_{R}+G_{V}}{2}, \tag{4}\] \[E = \frac{9 GB}{3B+G},\] (5) \[A_{f} = \frac{2C_{44}}{C_{11}-C_{12}},\] (6) \[\
u = \frac{E}{2G}-1, \tag{7}\]  
The mechanical stability criteria [42] for the cubic structure are \(C_{44}>0\), \(C_{11}>B>C_{12}\), and \(B>0\). We found that the calculated \(C_{11}\) for the APoO\({}_{3}\) perovskites was positive and ranged from 220 GPa to 260 GPa. Similarly, \(C_{12}\) was positive and ranged between 47 GPa and 61 GPa. In all APoO\({}_{3}\) perovskites, the criterion \(C_{11}>C_{12}\) was satisfied, and thus the anisotropy factor was positive. If the anisotropy factor is negative in the cubic structure, which indicates the criterion \(C_{11}>C_{12}\) is not satisfied, the structure is unstable. The \(C_{11}>C_{12}\) criterion can be regarded as an elastic property under the orthogonal distortion with a fixed volume. We notice that the largest \(C_{44}\) of 49.6 GPa in RaPoO\({}_{3}\) was two times larger than the corresponding one in BePoO\({}_{3}\). However, all C\({}_{44}\) were positive. The bulk modulus describes a measure of resistance to the volume change under external pressure. Our calculated bulk modulus was positive and ranged from 110 GPa to 125 GPa. This bulk modulus was larger, which implies a strong resistance to the volume change under pressure. We conclude that all stability criteria are satisfied, indicating that APoO\({}_{3}\) perovskites are mechanically stable. Besides, the stiffness is also an important material qualtiy. We present [PERSON]'s modulus \(E\) in Table 2. If \(E\) is higher, it implies the material is stiffer. The \(E\) increased monotonically from 103.5 GPa in BePoO\({}_{3}\) to 155.5 GPa in RaPoO\({}_{3}\). The calculated [PERSON]'s modulus was large, being over 100 GPa. To understand the bonding characteristics in a material, we calculated the Poisson's ratio and listed it in \[B=\frac{C_{11}+2C_{12}}{3}, \tag{1}\]0 The Poisson's ratio \(\
u\) describes the expansion or contraction in the direction perpendicular to the external force's direction. It has the value \(-1<\
u<0.5\). It is hard to find a material with a negative Poisson ratio in nature, though it is possible in mathematics. However, there are still experimental suggestions to synthesize these materials [43; 44]. Therefore, \(\
u\) was between 0 and 0.5. It is shown that the lower limit of \(\
u\) was 0.25 for the central force solid, and \(\
u=0.5\) was the upper limit of the infinite elastic anisotropy.