2 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: 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 E = 9 BG/ (3 B + G ) , υ = (3 B − 2 G ) / [2(3 B + G )] . (5) 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: ϵ 1 = ( δ, δ, δ, 0 , 0 , 0) , ϵ 2 = ( δ, 0 , δ, 0 , 0 , 0) , ϵ 3 = ( δ, δ, 0 , 0 , 0 , 0) , ϵ 4 = (0 , 0 , δ, 0 , 0 , 0) , ϵ 5 = (0 , 0 , 0 , δ, δ, 0) , ϵ 6 = (0 , 0 , 0 , 0 , 0 , δ ) (4) ϵ 1 = ( δ, δ, δ, 0 , 0 , 0) , ϵ 2 = ( δ, 0 , δ, 0 , 0 , 0) , ϵ 3 = (0 , 0 , 0 , δ, δ, δ ) . (3) ε$_{ij}$ =   e$_{1}$ 1 $_{2}$e$_{6}$ 1 $_{2}$e$_{5}$ 1 $_{2}$e$_{6}$ e$_{2}$ 1 $_{2}$e$_{4}$ 1 $_{2}$e$_{5}$ 1 $_{2}$e$_{4}$ e$_{3}$   (2) E ( V, ϵ ) = E ( V$_{0}$, 0) + V$_{0}$ 6 ∑ i =1 σ$_{i}$e$_{i}$ + V$_{0}$ 2 6 ∑ i,j =1 C$_{ij}$e$_{i}$e$_{j}$ + O ( { e 3 $_{i}$} ) , (1) FIG. 1: (Color online) Crystal structures of (a) bct ThH 2 , (b) fcc ThH 2 , and (c) bcc Th 4 H 15 . Here, larger cyan spheres stand for Th atoms and the smaller white H. Note that the magenta and red spheres in (c) stand for H1 and the white and grey spheres H2. GLYPH<s32> FIG. 2: Comparison of total energy vs the primitive cell vol- ume for ThH 2 in bct and fcc phases. GLYPH<s51>GLYPH<s48> GLYPH<s51>GLYPH<s53> GLYPH<s52>GLYPH<s48> GLYPH<s52>GLYPH<s53> GLYPH<s53>GLYPH<s48> GLYPH<s45>GLYPH<s49>GLYPH<s53>GLYPH<s46> GLYPH<s54> GLYPH<s45>GLYPH<s49>GLYPH<s53>GLYPH<s46> GLYPH<s50> GLYPH<s45>GLYPH<s49>GLYPH<s52>GLYPH<s46> GLYPH<s56> GLYPH<s45>GLYPH<s49>GLYPH<s52>GLYPH<s46> GLYPH<s52> GLYPH<s45>GLYPH<s49>GLYPH<s52>GLYPH<s46> GLYPH<s48> GLYPH<s69>GLYPH<s110>GLYPH<s101>GLYPH<s114>GLYPH<s103>GLYPH<s121> GLYPH<s32> GLYPH<s40>GLYPH<s101>GLYPH<s86>GLYPH<s41> GLYPH<s80>GLYPH<s114>GLYPH<s105>GLYPH<s109> GLYPH<s105>GLYPH<s116> GLYPH<s105>GLYPH<s118>GLYPH<s101>GLYPH<s32> GLYPH<s99> GLYPH<s101>GLYPH<s108>GLYPH<s108>GLYPH<s32> GLYPH<s118>GLYPH<s111>GLYPH<s108>GLYPH<s117>GLYPH<s109> GLYPH<s101>GLYPH<s32> GLYPH<s40>GLYPH<s65> GLYPH<s51> GLYPH<s41> GLYPH<s32> GLYPH<s98>GLYPH<s99> GLYPH<s116> GLYPH<s32> GLYPH<s84> GLYPH<s104>GLYPH<s72> GLYPH<s50> GLYPH<s32> GLYPH<s102> GLYPH<s99> GLYPH<s99> GLYPH<s32> GLYPH<s84> GLYPH<s104>GLYPH<s72> GLYPH<s50> GLYPH<s111> III. RESULTS A. Atomic and electronic structures of thorium hydrides