## 3 Results and Discussion  
The calculated optimum values for the lattice parameters of the cubic RFeO\({}_{3}\) (R = La, Eu, Gd) structures as well as the previous experimental and theoretical values are given in Table 1. There is befitting compliance between our calculated results with the reported experimental data the literature with a nearly 0.5% underestimation. The weakness of DFT in describing the dispersion forces causes an underestimation in the prediction of the electronic, structural, and physical properties of materials. It can be seen in Table 1, the cell parameter of the unit cells for all three structures decreases by reducing the atomic radius of the central cation. This is in agreement with the other computational studies [5; 6; 7; 12; 13], as listed in Table 1.  
In the first step of obtaining the mechanical properties, the elastic constants of RFeO\({}_{3}\) are calculated. These constants are calculated by applying strain in the harmonic region. The relation between the applied strain under three different deformation moods and the total energy changes is presented in Figure 2. The other elastic properties such as bulk modulus, Young's modulus, shear modulus, and Poisson's ratio are then identified from the elastic constants [13]. In the [PERSON] theory, the bulk modulus (B\({}_{\mathrm{V}}\)), and shear modulus (G\({}_{\mathrm{V}}\)) can be written as [14]:  
\[\mathrm{B_{V}}=\frac{1}{3}[\mathrm{C_{11}}+2\mathrm{C_{12}}] \tag{3}\]  
\[\mathrm{G_{V}}=\frac{1}{5}[\mathrm{C_{11}}-\mathrm{C_{12}}+3\mathrm{C_{44}}] \tag{4}\]  
While, in Reuss theory, the bulk modulus (B\({}_{\mathrm{R}}\)), and shear modulus (G\({}_{\mathrm{R}}\)) are written as [15]:  
\[\frac{1}{\mathrm{B_{R}}}=(\mathrm{S_{11}}+\mathrm{S_{22}}+\mathrm{S_{33}})+2( \mathrm{S_{12}}+\mathrm{S_{13}}+\mathrm{S_{23}})] \tag{5}\]  
\[\frac{15}{\mathrm{G_{R}}}=4[(\mathrm{S_{11}}+\mathrm{S_{22}}+\mathrm{S_{33}})- 4(\mathrm{S_{12}}+\mathrm{S_{13}}+\mathrm{S_{23}})+3(\mathrm{S_{44}}+\mathrm{ S_{55}}+\mathrm{S_{66}})] \tag{6}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Experiment Data** & **Theoretical Data** \\ \hline \multirow{2}{*}{**LaFeO\({}_{3}\)**} & 3.89 [10] & 3.82 \\  & 3.92 [11] & 3.92 [5] \\  & 3.93 * [9] & 3.95 [12] \\ \hline \multirow{2}{*}{**EuFeO\({}_{3}\)**} & \multicolumn{2}{c}{3.75} \\  & 3.68 * [9] & 3.83 [5] \\ \hline \multirow{2}{*}{**GdFeO\({}_{3}\)**} & \multicolumn{2}{c}{3.74} \\  & - & 3.82 [5] \\ \hline \end{tabular}  
* Pseudocubic structure.  
\end{table} 
Table 1: Comparison between our calculated, and reported the experimental and theoretical results in the literature for the lattice parameters of LaFeO\({}_{3}\), EuFeO\({}_{3}\), and GdFeO\({}_{3}\). where \(\rm S_{ij}\), which is entitled the elastic compliance coefficient, is the inverse of \(\rm C_{ij}\). Furthermore, in Hill theory, bulk modulus (\(\rm B_{H}\)), and shear modulus (\(\rm G_{H}\)) are described as follows [16]:  
\[\rm B_{H}=\frac{\rm B_{V}+B_{R}}{2} \tag{7}\]  
\[\rm G_{H}=\frac{\rm G_{V}+G_{R}}{2} \tag{8}\]  
Finally, the [PERSON]'s modulus (E), and [PERSON]'s ratio (\(\rm v\)) are calculated using Equations (9) and (10), respectively:  
\[\rm E=\frac{9 BG}{3B+G} \tag{9}\]  
\[\rm v=\frac{3B-2G}{[2(3B+G)]} \tag{10}\]  
The calculated elastic constants of LaFeO\({}_{3}\), EuFeO\({}_{3}\), and GdFeO\({}_{3}\) are reported in Table 2. To the best of our knowledge, there are limited theoretical and experimental efforts on the prediction of the elastic constants of LaFeO\({}_{3}\), EuFeO\({}_{3}\), and GdFeO\({}_{3}\) with the cubic structure. Therefore, to validate our obtained results, we compare the mechanical properties of rare earth Ferritic perovskites with other oxide perovskites. This helps to achieve a better understanding of the mechanical properties of rare earth Ferritic perovskites.