## 3 Results and Discussion
important indicator for measure the degree of ionicity of the crystal, where 1\(-\)C shows the degree of covalency. The ratio between Q (\(\Omega\)) and OS (\(\Omega\)) provides a measurement of the deviation from the ideal ionic model for a given basin \(\Omega\). The topological charges (Q) are not close to nominal charges (R\({}^{(+3)}\)Fe\({}^{(+3)}\)O\({}_{3}^{(-2\times 3)}\)) due to the existence of different topological charges for the same atoms in different atomic environments. The calculated C index is 0.59, 0.55, and 0.57 for LaFeO\({}_{3}\), EuFeO\({}_{3}\), and GdFeO\({}_{3}\) compounds, respectively. Another important mechanical property, which is of great importance in the materials design category, is the elastic limit. The elastic limit is the maximum stress that can be applied on the solid material before the permanent or plastic deformation starts. Prior to the elastic limit, by unloading, the material returns to its original undeformed shape. Passing this limit reduces the material's stiffness and leads to deflections and buckling. This is very important, particularly in thin film applications, where there is a substrate, which strains the film due to the lattice mismatch between the substrate and the thin film. The change of the second Piola-Kirchhoff (PK2) stress with Lagrangian strain is represented in Figure 4 for all LaFeO\({}_{3}\), EuFeO\({}_{3}\), and GdFeO\({}_{3}\) structures. To obtain the Piola-Kirchhoff (PK2) stress, we applied the cubic deformation tensor on the structures. The PK2 is related to the Cauchy (true) stress (\(\sigma\)), and can be obtained from Equation (20) [31]:  
\[\Sigma=\mathrm{jF}^{-1}\sigma\Big{(}\mathrm{F}^{-1}\Big{)}^{\mathrm{T}} \tag{20}\]  
where J is the determinant of the deformation gradient tensor F. The Cauchy stresses are identified based on DFT calculations in the framework of the PBEsol approximation. The maximum stress in this curve is called the ultimate stress and the corresponding strain at this point is called the ultimate strain. The strain range prior to this ultimate strain point is the uniform elastic region of the material. It should be emphasized that the ultimate stress and strain obtained by this method is under the ideal condition, where there is no crystal defect or thermal effects, which are usually present in the real conditions and lower the strength of the material. The ultimate stress and strain values obtained from the PK2 stress curve are given in Table 7. From the PK2 stress, it can be seen that both materials behave in a symmetric manner, which is due to the cubic crystal lattice. In addition, from the steep slope in the compression region, it can be concluded that the structures possess higher resistance in compression compared to tension. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\Sigma^{u}\) **(eV/Å).** & **LaFeO\({}_{3}\)** & **EuFeO\({}_{3}\)** & **GdFeO\({}_{3}\)** \\ \hline \(\Sigma^{u}_{\mathrm{a}}\) & 0.126 & 0.115 & 0.111 \\ \(n^{u}_{\mathrm{a}}\) & 8.5\% & 8.5\% & 8.5\% \\ \(\Sigma^{u}_{\mathrm{b}}\) & 0.126 & 0.115 & 0.111 \\ \(n^{u}_{\mathrm{b}}\) & 8.5\% & 8.5\% & 8.5\% \\ \(\Sigma^{u}_{\mathrm{c}}\) & 0.126 & 0.115 & 0.111 \\ \(n^{u}_{\mathrm{c}}\) & 8.5\% & 8.5\% & 8.5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Ultimate stresses and ultimate strain of RFeO\({}_{3}\) (R = La, Eu, Gd) in a, b, and c directions under cubic deformation.