## 3 Results and Discussion  
Table 1 summarizes the results of the optimized structural parameters for the perovskite-type material CaTi\({}_{1-x}\)Cr\({}_{x}\)O\({}_{3}\). The theoretical data show good agreement with the experimental results, with the overestimation of the lattice parameters not exceeding 2.9% and the deviation in the ratio between them less than 2.2%. The energetic stability of the structures corresponding to the perovskite-type material CaTi\({}_{1-x}\)Cr\({}_{x}\)O\({}_{3}\), with x = 0.0, 0.25, 0.5, 0.75, and 1.0, representing the Cr substitution in the crystallographic sites of Ti, was analyzed by considering different structural configurations. These configurations are presented in Figure 1, which shows the highest energetic stabilities. Figure 1a shows the orthorhombic structure belonging to the Pbnm space group (#62) [42]. This space group was discovered by [PERSON] in 1839 [43], from which the perovskites were named. It is characterized by a distribution in which the Ti cation is octahedrally coordinated with six oxygen anions, while the Ca cation adopts a cuboctahedral coordination with 12 oxygen anions. When 25% of the Ti is substituted by Cr, in the same oxidation state, the space group is maintained, so that there is one out of four octahedra containing a Cr instead of a Ti, as exemplified in Figure 1b. It is necessary to note that Cr is substituted for Ti with the same 4+ oxidation state. With octahedral coordination, it does not present crystal field splitting, so Jahn-Teller type distortions are not expected [44], although a small decrease in the unit cell is expected due to its slightly smaller ionic radius--that of Ti\({}^{4+}\) is 0.60 A and Cr\({}^{4+}\) is 0.55 A. For 50% substitution, the energetically most favorable state is that of a superstructure in which Ti and Cr cations alternate successively along the three crystallographic axes as shown in Figure 1c. In the case represented in Figure 1d, the partial substitution of Cr in the Ti sites reaches 75%, corresponding to the inverse case to that of Figure 1b, since now Cr is the majority in the cell, such that three out of four cations in this crystallographic site are occupied by Cr and only one by Ti. Finally, Figure 1e corresponds to the orthorhombic cell for CaCrO\({}_{3}\), that is, for 100% Cr substitution in the octahedral Ti sites. Figure 2 illustrates the variation in the total energy of the perovskite-type material as a function of the different Cr concentrations studied in this work. It is observed that, as the Cr percentage increases, the total energy of the composite decreases linearly. This trend suggests that the material becomes more stable with higher Cr concentrations, which is consistent with the observed stability of the crystal structures obtained in this study. Although Cr\({}^{3+}\) is the most stable and commonly observed oxidation state for chromium, the stability of the ABO\({}_{3}\) perovskite structure strongly depends on the oxidation states of both the A and B cations. In particular, perovskites with A\({}^{2+}\)B\({}^{4+}\)O\({}^{2-}\)\({}_{3}\) valence distribution are known to exhibit a higher stability compared to those with A\({}^{3+}\)B\({}^{3+}\)O\({}^{2-}\)\({}_{3}\) oxidation states [45]. Given that the starting point of this study is the Ca\({}^{2+}\)Ti\({}^{4+}\)O\({}^{2-}\)\({}_{3}\) perovskite, the substitution of Ti by Cr necessitates the adoption of a +4 oxidation state by Cr to preserve the stability and charge balance of the crystalline structure. This approach aligns with established methods for predicting perovskite-type crystal structures, for which the ox