## 3 Results and Discussion
### Optical Properties
The contribution of the direct interband transitions to the imaginary dielectric function \(\varepsilon_{2}(\omega)\) can be obtained from the Kohn-Sham particle equation [1]:  
\[\varepsilon_{2}(\omega)=\frac{Ve^{2}}{2\pi\hbar m^{2}\omega^{2}}\int d^{3}k \sum_{nn^{\prime}}\left|\langle kn|p|kn\rangle\right\rangle^{\prime}\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
It is worth noting that \(\varepsilon_{1}\) is negative around 9 eV. A negative value indicates that light cannot be transmitted to the material and that the material exhibits a certain metallic luster. It can be seen from Figure 7b that the thresholds of \(\varepsilon_{2}\) starts from about 1 eV and 1.4 eV respectively. These thresholds are called basic absorption edges [35] and are related to the direct interband transition between VBM and CBM in CsGeI\({}_{3}\), which is consistent with the band values we calculated previously. \(\varepsilon_{2}\) reaches the first peak at about 4.0 eV, the second peak (maximum) at about 6.5 eV, and the third peak at about 8.5 eV. The first peak value is 3.7 and 4.1, the second peak value is 4.3 and 4.9, and the third peak value is 3.4 and 3.6 when the applied pressure is 0 GPa and \(-\)0.5 GPa. Combined with the DOS in Figure 5, it can be seen that the first peak is mainly caused by the transition of electrons in Ge atom and I atom, the second peak is mainly caused by the transition of electrons in Cs atom and I atom, and the third peak is determined by the complex transition of electrons in the deep energy levels of Cs, Ge, and I. By comparison, we find that the corresponding peak value when the pressure is \(-\)0.5 GPa is higher than the peak value when the pressure is 0 GPa. This indicates that CsGeI\({}_{3}\) has better dielectric properties after applied pressure.  
Figure 8 shows the real and imaginary parts of the conductivity of the perovskite CsGeI\({}_{3}\). We know that the real part of the conductivity represents the actual dissipated energy, and that when the moving electrons hit the lattice and get dragged down, they transfer their kinetic energy to the lattice and eventually they dissipate as joule heat. The imaginary part of the conductivity represents the conversion of the energy of the electric field to the kinetic energy of the electrons, which makes the electrons do simple harmonic vibrations under the action of an external alternating electric field. We can see from Figure 8a that the real part of the conductivity is similar to the imaginary part of the dielectric function. The value of the threshold and the position of the peak value coincide with the imaginary part of the dielectric function. At 6.5 eV, the real part of the conductivity has a second peak (maximum) of 3.3 (0 GPa) and 3.9 (\(-\)0.5 GPa), respectively. Figure 8b shows that the imaginary part of the conductivity of them is not significantly changed. The peak value of the real part of the conductivity is slightly larger after applied pressure, indicating that the energy loss is larger.  
Figure 9 shows the absorption coefficient, from which it is obvious that there are three absorption peaks, which appear in positions consistent with the imaginary part of the dielectric function and the real part of the conductivity. It can be seen that the dielectric function, conductivity, and absorption coefficient are closely related. The absorption coefficient can be described by the real and imaginary parts of the dielectric function as follows [36]:  
\[I(\omega)=2\omega(\frac{\left[\varepsilon_{1}^{2}(\omega)+\varepsilon_{2}^{2} (\omega)\right]^{1/2}-\varepsilon_{1}(\omega)}{2})^{1/2} \tag{4}\]  
Figure 7: The dielectric function of CsGeI\({}_{3}\) of (**a**) real and. (**b**) imaginary under the pressure of 0 GPa and \(-\)0.5 GPa. It can be seen from Figure 9 that the absorption peak under pressure is slightly larger than that without pressure.