## 3 Results and Discussions
### Computation and Analysis  
Table 2 shows the lattice parameters of pure TiO\({}_{2}\) and RE/TiO\({}_{2}\). Parameters a and b almost have no changes compared to pure TiO\({}_{2}\) (2 \(\times\) 2 \(\times\) 1 supercell), while the \(c\) is longer than pure TiO\({}_{2}\), resulting in the increase of cell volume (the radius of La\({}^{3+}\), Ce\({}^{3+}\), Gd\({}^{3+}\), and Yb\({}^{3+}\) is larger than Ti\({}^{4+}\)). The result is inconsistent with the reference's calculation [23; 24].  
The calculated result indicates the indirect band gap of 2.105 eV for anatase TiO\({}_{2}\) (unit cell), which is significantly lower than the laboratory data (approximately 3.2 eV). However, the electronic structure is essentially the same [24]. The band gaps of calculated doped TiO\({}_{2}\) are 1.951 eV, 2.059 eV, 0.475 eV, and 2.029 eV for La/TiO\({}_{2}\), Ce/TiO\({}_{2}\), Gd/TiO\({}_{2}\), and Yb/TiO\({}_{2}\) (Figure 5), respectively, all less than pure anatase TiO\({}_{2}\), which is in agreement with the laboratory result in the study. Gd/TiO\({}_{2}\) has the biggest changes on band gap according to the poor stability of Gd's electrons. According to [PERSON]'s statement, the energy state forms of RE dopants inside the CB and VB are divided into three parts including deep impurity states (electron transfer from RE energy state to CB), shallow impurity states (electron transfer from VB to RE energy state) and both states [25]. It could be concluded that element Yb belongs to deep impurity states, La and Ce belong to shallow impurity states, and Gd belongs to both.  
Similar to the pure anatase TiO\({}_{2}\), there also exists the hybridization of O 2p and Ti 3d states in the valence band region for RE/TiO\({}_{2}\) observed from the calculated PDOS (Figure S2). As seen from Figure 6, there exists impurity energy localized at about 0.743 eV above the Fermi level for Gd/TiO\({}_{2}\) (2 \(\times\) 2 \(\times\) 1 supercell), which indicates that some of the Gd 4f states are partially occupied. Simultaneously, the formation of Gd-O bonds leads to the partial hybridization of the Gd 4f orbitals with O 2p states below the Fermi level (Figure 6b). Therefore, the introduced Gd 4f orbital mainly contributes to extend the impurities' energy level into the forbidden band and produce a wider valence band in comparison with the pure TiO\({}_{2}\). Besides, the top of the valence band shifts obviously while the bottom of the conduction band shifts slightly with respect to the Fermi level, which results in the narrower band gap and the consequent redshift response of the optical absorption edge [38].  
\begin{table}
\begin{tabular}{c c c c c} \hline  & _a/_nm** & _b/_nm** & _c/_nm** & _V/_nm\({}^{3}\)** \\ \hline TiO\({}_{2}\) (unit cell) & 3.8064 & 3.8064 & 9.6716 & 140.13 \\ TiO\({}_{2}\) (2 \(\times\) 2 \(\times\) 1 supercell) & 7.6126 & 7.6126 & 9.6710 & 560.46 \\ La/TiO\({}_{2}\) (2 \(\times\) 2 \(\times\) 1 supercell) & 7.5917 & 7.5919 & 10.3202 & 594.81 \\ Ce/TiO\({}_{2}\) (2 \(\times\) 2 \(\times\) 1 supercell) & 7.6552 & 7.6555 & 9.9428 & 582.66 \\ Gd/TiO\({}_{2}\) (2 \(\times\) 2 \(\times\) 1 supercell) & 7.6386 & 7.6502 & 9.9168 & 579.48 \\ Yb/TiO\({}_{2}\) (2 \(\times\) 2 \(\times\) 1 supercell) & 7.6290 & 7.6291 & 9.9296 & 577.93 \\ \hline \end{tabular}
\end{table}
Table 2: Lattice parameters of rare-earth-doped anatase TiO\({}_{2}\).