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ment with experimental spectra [2, 3]. Furthermore, if the excitonic effect is taken into account, the oscillator strength in ε$_{2}$ will shift to lower energies, we expect that the agreement will become better. 

The calculated projected DOS and dielectric function spectra obtained by the GW + U method are also showed in Fig. 1(e) and Fig. 1(f). It is clear that the Ce 4 f orbitals are further localized compared to the HSE+ U results. As for the gaps, the p − d gap is close to exper- imental result, whereas the p − f gap of about 3.9 eV is enlarged. Notice that the band gaps for semiconductors or insulators are usually overestimated in GW scheme. This is because the screening is reduced upon updating the eigenvalues in W [17]. As expected, the first peak in ε$_{2}$ spectrum will shift to a higher energy and our cal- culated ε$_{2}$ spectrum shows that it is located at 6.0 eV, while other two peaks at 8.1 and 10.3 eV, respectively. 

Comparing the three results showed in Fig. 1, one can conclude that the results obtained by the HSE+ U method are better than the other two. Furthermore, for ε$_{2}$ spectrum, we expect the agreement will be become better if the excitonic effect is taken into account. Based on this, the important adsorption spectrum and energy- loss function deduced from the frequency-dependent dy- namical dielectric function obtained by HSE+ U , together with the corresponding experimental data [3], are showed in Fig. 2. It is obvious that the main structures are well reproduced. Similar to the ε$_{2}$ spectrum, the first peak 

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in adsorption spectrum has a shifting of about 1.2 eV to higher energy. As for the energy-loss function, the two peaks at about 5.5 and 15.5 eV also shift to high energies by about 1.1 eV compared to the experimental results [3]. 

