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(No. 225) is considered as metastable phase for ThH$_{2}$ [see Fig. 1(b)]. The fluorite structure is the stable struc- ture of all actinide dioxides. Moreover, this fcc structure can be obtained from the bct stable structure by modu- lating the base vectors. In fcc structure, each Th atom is surrounded by eight H atoms forming a cube and each H connects with four Th atoms to build a tetrahedron. Our optimized lattice constant ( a ) for fcc ThH$_{2}$ is 5.4851 ˚ A (see Table I), in excellent agreement with the measured 5 values of 5.489 ˚ A. The structure of Th$_{4}$H$_{15}$ is more com- plicated. It crystallizes in a body-centered cubic (bcc) structure with space group I ¯ 43 d (No. 220) [see Fig. 1(c)]. The unit cell has sixteen Th (16 c ), twelve H1 (12 a ), and forty eight H2 (48 e ) atoms. Each H1 is surrounded by four Th atoms to build the tetrahedral structure and each H2 is located at the center of triangle formed by three Th 

atoms. As a result, each Th is surrounded by three H1 atoms and nine H2 atoms. In Fig. 1(c), we label clearly the H1 atoms (red spheres) and H2 atoms (grey spheres) connected to the Th atom located at center. The current optimized structural parameters of Th$_{4}$H$_{15}$ are listed in Table I, where one can find that our calculation results are in good agreement with experiment $^{20}$. 

To investigate the stability of ThH$_{2}$, we have optimized the structural parameters of its bct and fcc structures at different pressures. To avoid the Pulay stress problem, we perform the structure relaxation calculations at fixed volumes rather than constant pressures. For fcc struc- ture, due to its high symmetry, the structure relaxation calculations are performed at fixed volumes with no re- laxation of coordinates and cell shape. However, for bct structure, the cell shape is necessary to be optimized due 

