## 3 Results and Discussion
To calculate the bulk modulus B and its derivative with respect to pressure \(B_{0}^{\prime}=\frac{\partial\text{B}_{0}}{\partial\text{P}}\), the third-order Birch-Murnaphan (BM) equations of state as [28; 29]:  
\[\text{P}=\frac{3}{2}\text{B}_{0}\bigg{(}\text{n}^{-\frac{7}{3}}-\text{n}^{- \frac{5}{3}}\bigg{)}\bigg{[}1+\frac{3}{4}\left(\text{B}_{0}^{\prime}-4\right) \bigg{(}\text{n}^{-\frac{2}{3}}-1\bigg{)}\bigg{]} \tag{16}\]  
here \(\eta=\) V/V\({}_{0}\) is the ratio of volume V at pressure P to equilibrium V\({}_{0}\) at zero pressure, and B\({}_{0}\) is bulk modulus and B\({}_{0}^{\prime}=\frac{\partial\text{B}_{0}}{\partial\text{P}}\). The analysis is based on the partition of the cell volume (V) into basin volumes (V\({}_{\Omega}\)), which compress differently upon the application of hydrostatic pressure (P). The local bulk modulus B\({}_{\Omega}\) and the bulk modulus (B) of the bulk crystal are defined as:  
\[\text{B}_{\Omega}=-\text{V}_{\Omega}\bigg{(}\frac{\partial\text{P}}{\partial \text{V}_{\Omega}}\bigg{)} \tag{17}\]  
\[\text{B}=\sum_{\Omega}\text{f}(\Omega)\text{B}_{\Omega} \tag{18}\]  
where f(\(\Omega\)) is the ratio of total volume (V) to basin volume (V\({}_{\Omega}\)). As shown in Table 6, the local bulk modulus is specified for each basin, which is calculated using Equation (17). The incremental amount of the local bulk modulus is maintained by reducing the atomic radius, which is in agreement with our previous results presented in Table 3. The local bulk modulus of Fe and O in changes from one structure to the other one due to the fact that the atomic environment is not the same in the LaFeO\({}_{3}\), EuFeO\({}_{3}\), and GdFeO\({}_{3}\) structures. Using Equation (18), the bulk modulus of RFeO\({}_{3}\) (R = La, Eu, Gd) structures are obtained by averaging the local bulk modulus (see Total row of B column), which confirms our previous results in Table 3.  
In addition, as can be seen in Table 6, Q is topological charges of atoms in the crystal, which are calculated using from the integration of each basin corresponding to each atom using [PERSON] and Trinkle method [30]. Obtained charges for each basin help to specify an important parameter that characterizing the whole charge transfer by averaging the ratios between topological charges Q (\(\Omega\)) and nominal oxidation states OS (\(\Omega\)):  
\[\text{C}=\frac{1}{\text{N}}\sum_{\Omega}\frac{\text{Q}(\Omega)}{\text{OS}( \Omega)} \tag{19}\]  
where \(\Omega\) denotes each basin and N is the total number of basins. In this equation C = 1 means the total iconicity, while C = 0 shows total covalency. Table 6 shows the topological volumes, topological charges, and the C indexes are calculated for LaFeO\({}_{3}\), EuFeO\({}_{3}\), and GdFeO\({}_{3}\). The C index is an \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & **V** & **f (\(\Omega\))** & **B(\(\Omega\))** & **Q(\(\Omega\))** & **OS(\(\Omega\))** \\ \hline \multirow{4}{*}{LaFeO\({}_{3}\)} & **La** & 17.18 & 0.308 & 268.83 & 2.071 & +3 \\  & **Fe** & 7.16 & 0.128 & 221.59 & +1.475 & +3 \\  & **O** & 10.45 \(\times\) 3 & 0.188 \(\times\) 3 & 176.37 & \(-1.182\times\) 3 & \(-2\times\) 3 \\  & **Total** & 55.70 & 1.00 & 210.66 & 0 & 0 \\ \hline \multirow{4}{*}{EuFeO\({}_{3}\)} & **Eu** & 15.08 & 0.287 & 256.59 & +1.896 & +3 \\  & **Fe** & 6.83 & 0.130 & 191.23 & +1.527 & +3 \\  & **O** & 10.19 \(\times\) 3 & 0.194 \(\times\) 3 & 169.23 & \(-1.141\times\) 3 & \(-2\times\) 3 \\  & **Total** & 52.54 & 1.00 & 196.99 & 0 & 0 \\ \hline \multirow{4}{*}{GdFeO\({}_{3}\)} & **Gd** & 14.80 & 0.283 & 249.64 & +1.882 & +3 \\  & **Fe** & 6.78 & 0.130 & 185.91 & +1.537 & +3 \\ \cline{1-1}  & **O** & 10.23 \(\times\) 3 & 0.196 \(\times\) 3 & 167.25 & \(-1.139\times\) 3 & \(-2\times\) 3 \\ \cline{1-1}  & **Total** & 52.29 & 1.00 & 193.04 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The topological charges (Q), volume of each basin, and C index of RFeO\({}_{3}\) (R = La, Eu, Gd).