2 localized Ce 4 f electrons is described by the rotationally invariant approach to the LDA + U method due to Du- darev et al . [12]. In our calculation, we use J = 0.51 eV for the exchange energy and U = 4.5 eV for the spheri- cally averaged screened Coulomb energy U , which is close to the values used in other previous work [13]. Since only the difference between the spherically averaged screened Coulomb energy U and the exchange energy J is im- portant for the total LDA (GGA) energy functional [12], thus, in the following we label them as one single effective parameter U for brevity. Notice that all our calculations are performed at the experimental lattice constants. Before calculating the optical spectra of CeO$_{2}$, we first study the essential electronic structure using the HSE, HSE+ U and GW + U methods, respectively, since the spectra are calculated directly from the interband tran- sitions. Based on this, the frequency-dependent complex dielectric function ε ( ω ) = ε$_{1}$ ( ω )+ iε$_{2}$ ( ω ) for CeO$_{2}$ are cal- culated. In the following, we will show and discuss our results systematically. The calculated results obtained using standard DFT are not present in this letter, be- cause they can be available elsewhere [13–15]. Figure 1(a) shows the site-projected orbital-resolved density of states (DOS) for Ce 4 f , 5 d and O 2 p orbitals using HSE method together with the experimental XPS + BIS spec- tra [16]. From the DOS, it is clear that the valence bands mainly consisted of O 2 p states and a little contribution from Ce 4 f and 5 d states can also be seen. The con- duction bands mainly consisted of Ce 5 d states, and the localized Ce 4 f states are located within the p − d gap. Our calculated width of O 2 p valence bands is about 4 eV, which is in good agreement with the experimen- tal XPS results [16]. Note that the fundamental p − f and p − d gaps govern the optical transitions, therefore, their accurate determination is indispensable to the op- tical spectra. Our calculated p − f gap is about 2.8 eV, which is well consistent with the experimental value of 3.0 eV [16]. Furthermore, the p − d gap of about 6.2 eV is also well determined compared with the experimental value of about 6.0 eV [16]. Another remarkable property is the central position of localized Ce 4 f orbitals. Our DOS showed in Fig. 1(a) shows that the localized Ce 4 f orbitals lie about 3.5 eV above the topmost of the O 2 p valence bands, which is smaller than the XPS-BIS result of about 4.5 eV [16]. Notice the that present electronic structure obtained by HSE are in good agreement with previous study [15] using HSE. Based on the determined electronic structure, the op- tical spectra of the frequency-dependent dynamical di- electric function ε ( ω ) = ε$_{1}$ ( ω ) + i ε$_{2}$ ( ω ) as a function of the photon energy ω up to 12 eV has been calculated and showed in Fig. 1(b). In order to make a compari- son, the experimental result [2] of imaginary part ε$_{2}$ ( ω ) is also showed. Due to the cubic symmetry of CeO$_{2}$, the dielectric tensor only has one independent compo- nent and ε$_{xx}$ = ε$_{yy}$ = ε$_{zz}$ , therefore, only one component is showed in Fig. 1(b). According to our above DOS dis- cussion, the peak at about 3.9 eV in our calculated ε$_{2}$ ( ω ) is assigned to the transition from the O 2 p bands to the localized Ce 4 f states, which is close to the peak in ex- perimental ε$_{2}$ ( ω ) at 3.7 eV [2]. Considering the dipolar selection rule only transitions with the difference Δ l = ± 1 between the angular momentum quantum numbers l are allowed, the p − f transition is forbidden. However, in our above DOS calculation, in valence bands, small hybridization between O 2 p and Ce 4 f exists, which sug- gests that some Ce 4 f electrons transfer to O 2 p states. Notice that Niwano et al. [3] also called the p − f tran- sition as a “charge transfer” transition. As for the O 2 p − Ce 5 d transition, experimentally Marabelli et al. [2] observed a double-peaked very broad structure at about 9 eV in ε$_{2}$ ( ω ); one peak is centered at 8 eV and the other at 10 eV. This corresponds to the transitions from O 2 p to Ce 6 d e$_{g}$ and t$_{2}$ states due to crystal field splitting in fluorite crystal structure. As for the p − d transition indi- cated in ε$_{2}$ ( ω ) in Fig. 1(b), one can see two peaks located at about 8.6 and 10.8 eV, corresponding to p − d ( e$_{g}$ ) and p − d ( t$_{2}$ ) transitions, respectively. The electronic structure and optical spectra obtained by the HSE + U method are showed in Fig. 1(c) and Fig. 1(d). Compared to our pure screened hybrid functional result, it is clear that the Ce 4 f states are more localized because of the Hubbard U correction. Furthermore, the central position of localized Ce 4 f states lies at about 4.5 eV above the topmost of the O 2 p valence bands, which becomes to accord well with the experimental XPS-BIS measurement [16]. As for the p − f and p − d gaps, they are now very close to the experimental data [16]. Over- all, the shape of our calculated DOS curve exhibits the same main features as the experimental XPS-BIS results [16]. Correspondingly, the calculated ε$_{2}$ ( ω ) by the HSE + U is showed in Fig. 1(d). Clearly, there are three peaks in the calculated ε$_{2}$ ( ω ). One peak is at 4.9 eV, and the other two are at 8.1 and 10.3 eV, respectively. Comparing to the three main peaks (located at 3.7, 8 and 10 eV) observed by experiment [2], we notice that the last two peaks are well reproduced except the first peak with a shifting of about 1.2 eV from the experi- mental spectrum. This discrepancy is guessed to be due to the ignorance of the interaction between electron and hole in our calculations. Then, the excitonic effect is not taken into account. As Niwano et al. [3] pointed out that the BIS process is adding one additional elec- tron into the empty conduction bands, whereas optical adsorption process is exciting one electron up to the con- duction bands and a corresponding hole is created; they further deduced the binding energy of an exciton con- tributed by one hole and one 4 f electrons to be about 1 eV according to energy difference between the central position of the localized 4 f states and the first peak in ε$_{2}$ . Notice that compared to the experimental observa- tion [2], our ε$_{2}$ spectrum also exhibits a rising edge at about 6.5 eV, which is close to the p − d gap and indi- cates the beginning of the p − d transitions. Therefore, our calculated ε$_{2}$ spectrum displays the main features as the experimental result and supplies a qualitative agree-