## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Structural Parameters** & **0\%** & **25\%** & **50\%** & **75\%** & **100\%** \\ \hline a (Å) & 5.4067 & 5.3878 & 5.3929 & 5.3476 & 5.3221 \\  & 5.3928 [39] & & 5.3511 [40] & & 5.2886 [41] \\ b (Å) & 5.5056 & 5.4935 & 5.4596 & 5.4748 & 5.4668 \\  & 5.4494 [39] & & 5.3936 [40] & & 5.3172 [41] \\ c (Å) & 7.6949 & 7.6552 & 7.5989 & 7.5720 & 7.5341 \\  & 7.6582 [39] & & 7.5874 [40] & & 7.4844 [41] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimized structural parameters of the system perovskite CaTi\({}_{1-x}\)Cr\({}_{x}\)O\({}_{3}\). idation state is fundamental for the calculation of the bond valence sum [46]. The bond valence sum, in combination with the tolerance factor, allows for the calculation of the global instability index, which is crucial in determining the likelihood of crystallization for the material [47]. Figure 3 shows the density of electronic states (DOS) for all cases considered for the CaTi\({}_{1-\times}\)Cr\({}_{x}\)O\({}_{3}\) system. The upper part of each of the pictures is the representation of spin-up polarization contributions and the lower ones correspond to spin-down orientations. Conventionally, the value E = 0 eV is established as the Fermi level. In Figure 3a-c, the total density of states, the contributions of the 3d-Ti and 2p-O orbitals, respectively, is exemplified for the CaTiO\({}_{3}\) material, i.e., CaTi\({}_{1-\times}\)Cr\({}_{x}\)O\({}_{3}\) (x = 0). It is observed from the total and partial DOS that the states obtained from the calculation are symmetric for the two spin orientations, both in the valence and conduction bands, so that the bandgap is the same for the two cases, reaching a value of E\({}_{\text{g}}\) = 2.34 eV. This value, which is typical of semiconductor materials, has been observed in CdS films produced using the techniques of chemical bath deposition and pulsed direct current magnetron sputtering [48], and is among the bandgap values expected for CaTiO\({}_{3}\)[49]. It is observed in Figure 3a-c that, in the vicinity of the Fermi level, the predominantly contributing states in the valence band  
Figure 1: Representation of the crystalline cell for the CaTi\({}_{1-\times}\)Cr\({}_{x}\)O\({}_{3}\) perovskite, for the following different substitutions: (**a**) CaTiO\({}_{3}\), (**b**) CaTi\({}_{0.75}\)Cr\({}_{0.25}\)O\({}_{3}\), (**c**) CaTi\({}_{0.5}\)Cr\({}_{0.5}\)O\({}_{3}\), (**d**) CaTi\({}_{0.25}\)Cr\({}_{0.75}\)O\({}_{3}\), and (**e**) CaCrO\({}_{3}\). correspond to the 2p-O orbitals, while in the conduction band, the available states are due to the 3d-Ti orbitals. When Cr\({}^{4+}\) is partially and fully incorporated into the structure (Figure 3d-o), the valence band is not substantially affected, so that the 2p-O orbitals continue to provide the charge carriers in the material for both spin-up and spin-down polarizations. Thus, in this band, neither the total DOS nor the partial DOS show changes, as shown in Figure 3c,f,i,l,o. Meanwhile, a systematic decrease in the small contributions of 3d-Ti orbitals, as opposed to a low but recurring increase in the contributions of 4d-Cr orbitals, is observed in this band as the proportion of Cr\({}^{4+}\) in the structure increases, as shown in the Figure 3b,e,h,k,n. Due to the magnetic character of Cr\({}^{4+}\), the symmetry between the up and down states is slightly modified, not only in the valence band but also in the conduction band, when this cation is included in the structure of the material. Above the Fermi level, the changes introduced by the inclusion of Cr\({}^{4+}\) in the structure are more drastic. This effect is due to the appearance of 3d-Cr states that contribute to the decrease in the bandgap for the spin-up orientation. This decrease is initially large but tends to saturate for high Cr\({}^{4+}\) concentrations. Likewise, the magnitude of the DOS increases for these 3d-Cr orbitals in the conduction band as the proportion of Cr\({}^{4+}\) in the material is increased.