## 3 Results
### Optical Properties  
The optical properties for the absorption coefficient, refractive index, and dielectric function were analyzed for the undoped and the X(=Cr, Mo, W):TiO\({}_{2}\) substitutional cases. In the optical calculations, we used the Hubbard + U parameter and concluded that the present results are in good agreement with the experiment, although they are highly dependent on this parameter. In Figure 4a-f, the complex dielectric function is presented for photon energies up to 15 eV. In Figure 4g, the dielectric function of the undoped case is shown. The dielectric constant \(\varepsilon_{\rm oo}\) is predicted in the limit of zero photon energy of the real part of the dielectric function, at 5. This agrees well with other computational works, which predict a value of 5.10 [57] and slightly higher than the experimental value of 4.5 [58]. The imaginary part of the dielectric function for the undoped case shows a peak at about 5 eV. This peak is in good agreement with the literature [57] and was attributed to the electronic transition from Ti-3d to O-2p states at the conduction and the valence band. For the Cr doping case, the dielectric function is presented in Figure 4a. The dielectric constant is calculated at the value of 6 from the real part of the dielectric function. From the imaginary part of the dielectric function, we show that it has a similar form to the pure TiO\({}_{2}\). In Figure 4b, the dielectric function for the Mo:TiO\({}_{2}\) is presented. Focusing on the real part, the dielectric constant is estimated at the value of 9. From the imaginary part, a peak in the low region is created (0.31 eV), which comes from the electronic intra-band transition of Mo 4d and Ti-3d orbitals inside the conduction band. The present results are similar with other theoretical works [59; 60]. For the W-case, we predict the dielectric constant at 6. As for the imaginary part, it is seen that 0.23 eV is created from the W-4d transition with the Ti-3d in the conduction band. The computed refractive index value for the undoped TiO\({}_{2}\) is 2.35. The present results agree with other theoretical studies that calculate a value of 2.21 [57]. [PERSON] [61], performed experiments and estimated that the refractive index of undoped TiO\({}_{2}\) is computed at the value of 2.62 at 550 nm. Here, we calculate a value of 2.62 at 550 nm (which is equal to an energy of 2.25 eV), identical to the experimental value. The imaginary part of the refractive index corresponds to the extinction coefficient. The significant peaks from the extinction coefficiency are located at 5.30, which is close to other provided values [57]. It is seen that the real part of the refractive index gives a value of 2.44 for zero photon energy. The available experimental data predict a value of 2.40, which is similar with our data [62]. From the real part of the refractive index of the Mo:TiO\({}_{2}\), we observe that at the zero photon energy, the index is calculated near 4. This is a significant increase compared to the undoped case. Similar to the present study, [PERSON] et al. [60] also showed an increase of 1.5 units to the refractive index, after the 0.92% doping of Mo to the pure TiO\({}_{2}\). An increase in the refractive index of TiO\({}_{2}\) upon Mo doping boosts its anti-reflective (AR) ability and might be beneficial for application in solar harvesting devices. For the extinction coefficient, k(\(\omega\)), in this case, it is seen that a strong peak is created at 5.2 eV. Finally, with regards to the W doped TiO\({}_{2}\), from the real part of the refractive index, it is seen that at zero photon energy, the index is calculated at 2.40, almost equal to the experimental value of 2.3 provided by [PERSON] [63].