## 3 Results and Discussion
### Optical Properties  
The real part \(\epsilon_{1}(\omega)\) and the imaginary part \(\epsilon_{2}(\omega)\) of the dielectric function for Mo\({}_{2}\)CF\({}_{2}\), Mo\({}_{2}\)C(OH)\({}_{2}\), and Mo\({}_{2}\)CO\({}_{2}\) with different layers are presented in Figure 4. The calculated values of \(\epsilon_{1}(0)\) were 5.76, 7.65, and 10.60 for the Mo\({}_{2}\)CF\({}_{2}\), Mo\({}_{2}\)C(OH)\({}_{2}\), and Mo\({}_{2}\)CO\({}_{2}\) monolayers, respectively. Our calculation results for the real part \(\epsilon_{1}(\omega)\) of all Mo\({}_{2}\)C-structures show that, from 0 to 6 eV, the intensity of the peaks increased with the increasing number of layers. The maximum value of \(\epsilon_{1}(\omega)\) was achieved at about 0.67 eV, 0.21 eV, and  
Figure 2: Electronic band structures of monolayer and multilayer for Mo\({}_{2}\)CF\({}_{2}\), Mo\({}_{2}\)C(OH)\({}_{2}\), and Mo\({}_{2}\)CO\({}_{2}\) MXenes. 1.48 eV for the 2D sublayers of Mo\({}_{2}\)CF\({}_{2}\), Mo\({}_{2}\)C(OH)\({}_{2}\), and Mo\({}_{2}\)CO\({}_{2}\), respectively. Obtaining higher values of \(\varepsilon_{1}(\omega)\) indicated a large polarizability of the incident photon energy as the number of layers increased. The higher value of \(\varepsilon_{1}(\omega)\) for Mo\({}_{2}\)CO\({}_{2}\) showed the greater polarization ability of the -O functional groups compared to -F, -OH. In particular, when moving toward higher incident photon energies, we observed inflection point positions, where \(\varepsilon_{1}(\omega)\) began to take negative values. One inflection point was observed for the Mo\({}_{2}\)CF\({}_{2}\) material at about 6 eV, while two inflection points were observed for Mo\({}_{2}\)C(OH)\({}_{2}\) at 1.28 eV and 4.76 eV and for Mo\({}_{2}\)CO\({}_{2}\) at 1.78 eV and 4.15 eV. We found that increasing the number of layers led to the magnitude of the \(\varepsilon_{1}(\omega)\) peaks decreasing and becoming more negative starting at these inflection point values. The dependences of the imaginary part of the dielectric function \(\varepsilon_{2}(\omega)\) on the number of Mo\({}_{2}\)C-MXene layers is also shown in Figure 4, whereby increasing the Mo\({}_{2}\)C-MXenes layers led to an increase in the intensity of \(\varepsilon_{2}(\omega)\). The electronic band structure of Mo\({}_{2}\)C-, especially the transition between the valence band and the conduction band, is related to the imaginary component. Therefore, the optical band gap energy of the 2D Mo\({}_{2}\)C-sublayer can be estimated from \(\varepsilon_{2}(\omega)\). Above the optical bandgap energy, we obtained a sharp increase in the intensity of \(\varepsilon_{2}(\omega)\), with its maximum occurring at 1.12 eV and 0.78 eV for Mo\({}_{2}\)CF\({}_{2}\) and Mo\({}_{2}\)C(OH)\({}_{2}\), respectively. These results show that Mo\({}_{2}\)CO\({}_{2}\) had a relatively large visible-light absorption area under the \(\varepsilon_{2}(\omega)\) curve. In the photon energy interval 1.5-3 eV, the intensity of \(\varepsilon_{2}(\omega)\) increased, with its peak occurring at 1.88 eV for Mo\({}_{2}\)CF\({}_{2}\) and Mo\({}_{2}\)CO\({}_{2}\). Figure 3: Total and partial density of states for 2D layered Mo\({}_{2}\)CF\({}_{2}\), Mo\({}_{2}\)C(OH)\({}_{2}\) and Mo\({}_{2}\)CO\({}_{2}\). both layers of Mo\({}_{2}\)CO\({}_{2}\). The electron transition between the Mo-4d state to the Mo-5d state was attributed to the peak, as represented in the density of states of Mo\({}_{2}\)CF\({}_{2}\), Mo\({}_{2}\)C(OH)\({}_{2}\), and Mo\({}_{2}\)CO\({}_{2}\) (see Figure 3). It was found that the =O functional group of Mo\({}_{2}\)C- had a larger absorption coefficient than the -F and -OH terminated ones, making them suitable for applications in the field of transparent electrodes [36]. These findings suggest that the optical characteristics of the 2D layered Mo\({}_{2}\)C- can be engineered by the functional groups and the number of layers [37].