## 3 Results and Discussion

Table 2: Calculated mechanical properties of RFeO\({}_{3}\) (R = La, Eu, Gd). In this section, the topological analysis of electron density is performed by employing quantum theory of atoms in molecules (QTAIM) to investigate the role of chemical bonds on the mechanical properties of the studied materials [21; 22; 23]. A deep understanding between structure (chemical bonds) and mechanical and electronic properties can help to design of low-cost, and scalable materials with enhanced properties. To carry out QTAIM (Bader's analysis), a single-point calculation of the optimized RFeO\({}_{3}\) structure is done to generate the full electron density (Table 5). The simulations are conducted by CRITIC2 code which used the core and valence electron densities as input parameters [24; 25]. Firstly, the critical points (CPs) of the electron density are determined by QTAIM analysis. A critical point together with all downward-gradient lines, all of which have their origins in the critical points, constitute an atomic basin, and are the most important of the regions fulfilling the zero-flux condition [26]. There are four types of non-degenerate critical points, which are defined by the number of non-zero eigenvalues of the curvature (Hessian) matrix (\(\lambda_{\text{i}}\)) and the sum of the algebraic signs of \(\lambda_{\text{i}}\). These four critical points of electron density are: maxima (3, \(-\)3), minima (3, \(+\)3), and two types of saddle points (3, \(+\)1) and (3, \(-\)1), which are corresponding to the nuclear positions, cages, rings, and bonds, respectively [26; 27]. In solids, atomic basins are equivalent to polyhedral with curved edges and faces. The crystal partitioning can lead to predict the mechanical properties of RFeO\({}_{3}\) on a local scale interdependent with the atoms in the crystal. The analysis is based on variation of hydrostatic pressure  
Figure 3: 3D contours of Young’s modulus, shear modulus, and Poisson’s ratio for (**a**) LaFeO\({}_{3}\); (**b**) EuFeO\({}_{3}\), and (**c**) GdFeO\({}_{3}\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Number and**} & \multicolumn{2}{c}{**\(\boldsymbol{\rho}\)(rc)**} & \multicolumn{2}{c}{\(\boldsymbol{\
abla}\boldsymbol{2}\)**(rc)**} & \multicolumn{2}{c}{\(\boldsymbol{\Lambda}\)**1**} & \multicolumn{2}{c}{\(\boldsymbol{\lambda}\)**2**} & \multicolumn{2}{c}{\(\boldsymbol{\lambda}\)**3**} \\ \cline{3-8}  & \multicolumn{2}{c}{**Type of CPs**} & \multicolumn{2}{c}{**(e.A\(-\)3)**} & \multicolumn{2}{c}{**(e.A\(-\)5)**} & \multicolumn{2}{c}{**(e.A\(-\)5)**} & \multicolumn{2}{c}{**(e.A\(-\)5)**} & \multicolumn{2}{c}{**(e.A\(-\)5)**} \\ \hline \multirow{4}{*}{LaFeO\({}_{3}\)} & Bond & 6 & 0.66 & 13.33 & \(-\)2.19 & \(-\)2.19 & 17.73 \\  & Bond & 12 & 0.22 & 2.59 & \(-\)0.73 & \(-\)0.68 & 4.01 \\  & Ring & 24 & 0.12 & 1.71 & \(-\)0.20 & 0.93 & 0.98 \\  & Cage & 8 & 0.11 & 1.72 & 0.29 & 0.29 & 1.34 \\  & Cage & 3 & 0.06 & 0.85 & 0.24 & 0.24 & 0.37 \\ \hline \multirow{4}{*}{EuFeO\({}_{3}\)} & Bond & 6 & 0.73 & 13.34 & \(-\)3.07 & \(-\)3.07 & 19.49 \\  & Bond & 12 & 0.21 & 2.75 & \(-\)0.65 & \(-\)0.61 & 4.02 \\  & Ring & 24 & 0.12 & 1.77 & \(-\)0.16 & 0.86 & 1.08 \\  & Cage & 8 & 0.12 & 1.82 & 0.30 & 0.30 & 1.22 \\  & Cage & 3 & 0.06 & 0.80 & 0.26 & 0.26 & 0.26 \\ \hline \multirow{4}{*}{GdFeO\({}_{3}\)} & Bond & 6 & 0.74 & 13.57 & \(-\)2.88 & \(-\)2.88 & 18.46 \\  & Bond & 12 & 0.2 & 2.56 & \(-\)0.69 & \(-\)0.63 & 4.02 \\ \cline{1-1}  & Ring & 24 & 0.12 & 1.75 & \(-\)0.17 & 0.85 & 1.03 \\ \cline{1-1}  & Cage & 8 & 0.12 & 1.83 & 0.31 & 0.31 & 1.32 \\ \cline{1-1}  & Cage & 3 & 0.06 & 0.83 & 0.25 & 0.25 & 0.27 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Properties of electron density critical points of RFeO\({}_{3}\) (R = La, Eu, Gd). (P) compared to the changes in partition of basin volumes (V\(\Omega\)).