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bulk modulus B , and pressure derivative of the bulk mod- ulus B ′ are obtained by fitting the energy-volume data in the third-order Birch-Murnaghan equation of state (EOS) $^{12}$. In order to calculate elastic constants, a small strain is applied onto the structure. For small strain ϵ , Hooke’s law is valid and the crystal energy E ( V, ϵ ) can be expanded as a Taylor series $^{13}$, 

where E ( V$_{0}$, 0) is the energy of the unstrained system with the equilibrium volume V$_{0}$ , ϵ is strain tensor which has matrix elements ε$_{ij}$ ( i, j =1, 2, and 3) defined by 

and C$_{ij}$ are the elastic constants. For cubic structures, there are three independent elastic constants ( C$_{11}$ , C$_{12}$ , and C$_{44}$ ). So, the elastic constants for fcc ThH$_{2}$ and bcc Th$_{4}$H$_{15}$ can be calculated from three different strains listed in the following: 

The strain amplitude δ is varied in steps of 0.006 from δ = − 0.036 to 0.036 and the total energies E ( V, δ ) at these strain steps are calculated. After obtaining elastic con- stants, we can calculate bulk and shear moduli from the Voigt-Reuss-Hill (VRH) approximations $^{14–16}$. The Voigt (Reuss) bounds on the bulk modulus B$_{V}$ ( B$_{R}$ ) and shear modulus G$_{V}$ ( G$_{R}$ ) for these two cubic crystal systems are deduced from the formulas of elastic moduli in Ref. $^{17}$. As for bct ThH$_{2}$, the six independent elastic constants ( C$_{11}$ , C$_{12}$ , C$_{44}$ , C$_{13}$ , C$_{33}$ , and C$_{66}$ ) can be obtained from six different strains listed in the following: 

and the formulas of elastic moduli in VRH approximations 16 are from Ref. $^{18}$. Based on Hill approximation $^{16}$, B = 1 $_{2}$( B$_{R}$ + B$_{V}$ ) and G = 1 $_{2}$( G$_{R}$ + G$_{V}$ ). The Young’s modulus E and Poisson’s ratio υ are given by the following formulas: 

E = 9 BG/ (3 B + G ) , υ = (3 B − 2 G ) / [2(3 B + G )] . (5) 

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## A. Atomic and electronic structures of thorium hydrides 

At ambient condition, the stable thorium dihydride crystallizes in a body-centered tetragonal (bct) ionic structure with space group I 4/ mmm (No. 139). Its unit cell is composed of two ThH$_{2}$ formula units with the tho- rium atoms and the hydrogen atoms in 2 a (in Wyck- off notation) and 4 d sites, respectively [see Fig. 1(a)]. Each Th atom is surrounded by eight H atoms forming a tetragonal and each H connects with four Th atoms to build a tetrahedron. The present optimized lattice parameters ( a and c ) are 4.0670 ˚ A and 4.9125 ˚ A (see Table I), in good agreement with the experimental 19 val- ues of 4.10 ˚ A and 5.03 ˚ A. Besides, the face-centered cu- bic (fcc) fluorite-type structure with space group Fm ¯ 3 m 

## III. RESULTS 

