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Γ − N − P − X − Γ − Z directions for bct ThH$_{2}$ and along the Γ − X − K − Γ − L − X − W − L directions for fcc ThH$_{2}$ are displayed in Fig. 6(a) and Fig. 6(b), respectively. 

For both bct and fcc ThH$_{2}$, there are only three atoms in their primitive cells. Therefore, nine phonon modes exist in the dispersion relations. Due to the fact that thorium is much heavier than hydrogen atom, the vibra- tion frequency of thorium atom is apparently lower than that of hydrogen atom. Therefore, evident gap between the optic modes and the acoustic branches exits and the phonon DOS of bct (fcc) ThH$_{2}$ can be viewed as two parts. One is the part lower than 4.4 (4.1) THz where the main contribution comes from the thorium sublattice, while the other part range from 26.0 to 32.1 (26.4 to 31.8) THz is dominated by the dynamics of the light hydrogen atoms. In experimental measurements, Dietrich et al. reported that the acoustic branches of ThH$_{2}$ are lower than 4.8 THz and optic modes range from 23.4 to 36.2 THz $^{4}$. Their measured data of optic modes are presented in Fig. 6(a) for comparison and the acoustic-phonon data are unable obtained from their time-of-flight spectrum. Obviously, our calculated results are on the whole con- sistent with the experimental data. In addition, while all frequencies are positive for bct ThH$_{2}$, which assures a dynamical stability of the bct phase against mechanical perturbations. For fcc ThH$_{2}$, one can see from Fig. 6 (b) that the transverse acoustic (TA) mode close to Γ point becomes imaginary along the Γ − K and Γ − L directions. The minimum of the TA branch occurs along the Γ − K direction. This indicates instability of fcc phase of ThH$_{2}$ 

compared to stable bct phase, which is well consistent with our previous mechanical stability analysis of ThH$_{2}$. Moreover, there exists LO-TO splitting at Γ point in bct phase. But in fcc phase there is no LO-TO splitting. 

In summary, we have used the first-principles DFT- GGA method to calculate the structural, electronic, me- chanical, and thermodynamic properties of ThH$_{2}$ in its stable bct phase and metastable fcc phase and Th$_{4}$H$_{15}$ in bcc phase. Our optimized structural parameters are well consistent with experiments. The occupation characters of electronic orbitals also accord well with experiments and previous calculations. Through Bader analysis, we have found that the Th − H bonds in all thorium hydrides exhibit weak covalent character, but the ionic property for ThH$_{2}$ and Th$_{4}$H$_{15}$ are different. While ∼ 1.5 electrons transfer from each Th atom to H in ThH$_{2}$, in Th$_{4}$H$_{15}$ about 2.1 electrons deviate from each Th atom. In ad- dition, our calculated phonon curves of fcc ThH$_{2}$ have shown that the TA mode becomes imaginary close to Γ point and the subsequent instability of this phase. 

This work was supported by the Foundations for De- velopment of Science and Technology of China Academy 

## IV. CONCLUSION 

## Acknowledgments 

