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notice from Table II that our calculated Th − H distances are all well consistent with the previous theoretical 5 and experimental 19 results. As for Bader analysis, it is a well established analysis tool for studying the topology of the electron density and thus is particularly suitable for discussing the mixed ionic/covalent character of a compound. The charge (Q$_{B}$) enclosed within the Bader volume (V$_{B}$) is a good approximation to the total elec- tronic charge of an atom. In the present study, the de- fault charge density grids for one unit cell are 56 × 56 × 72, 56 × 56 × 56, and 112 × 112 × 112 for bct ThH$_{2}$, fcc ThH$_{2}$, and bcc Th$_{4}$H$_{15}$, respectively. To check the precision, the charge density distribution for fcc ThH$_{2}$ is calculated with a series of n times finer grids ( n = 2, 3, 4, 5, 6). The deviation of the effective charge between the five and the six times finer grids is less than 0.02%. Thus we perform the charge density calculations using the six times finer grid (336 × 336 × 432, 336 × 336 × 336, and 672 × 672 × 672 for bct ThH$_{2}$, fcc ThH$_{2}$, and bcc Th$_{4}$H$_{15}$, respectively). The calculated results are presented in Table II. Note that although we have included the core charge in charge density calculations, since we do not expect variations as far as the trends are concerned, only the valence charge are listed. From Table II the following prominent fea- tures can be seen: (i) The Bader charges and volumes for bct and fcc ThH$_{2}$ are almost equal to each other. This shows the same ionic character, through a flux of charge (about 1.5 electrons for each Th atom) from cations to- wards anions, for thorium dihydride both in their sta- ble phase and metastable phase. (ii) The Bader charges and volumes for bcc Th$_{4}$H$_{15}$ are different comparing to thorium dihydride. This originates from their different hydrogen concentration. For bcc Th$_{4}$H$_{15}$, about 2.1 elec- trons transfer from each Th atom to H. This indicates that the Th atoms in Th$_{4}$H$_{15}$ are more ionized than that in thorium dihydrides. 

## B. Mechanical properties of thorium hydrides 

Our calculated elastic constants, various moduli, pres- sure derivative of the bulk modulus B $^{′}$, and Poisson’s ratio υ for bct ThH$_{2}$, fcc ThH$_{2}$, and bcc Th$_{4}$H$_{15}$ are col- lected in Table III. Obviously, bct ThH$_{2}$ is mechanically 

Reference $^{19}$, Reference $^{5}$. 

stable due to the fact that its elastic constants satisfy the following mechanical stability criteria 23 of tetragonal structure: 

Th$_{4}$H$_{15}$ is also mechanically stable because its elas- tic constants satisfy the following mechanical stability criteria 23 of cubic structure: 

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However, the ThH$_{2}$ in its metastable cubic phase is me- chanically unstable. In fact, one can see from Table III that C$_{11}$ is much smaller than C$_{12}$ for fcc ThH$_{2}$. Therefore, the mechanical stability criteria 23 C$_{11}$> | C$_{12}$ | can not be satisfied and the shear modulus, Young’s modulus, and Poisson’s ratio also can not be obtained. As for bulk modulus B , each derived value from VRH approximations 16 for all the thorium hydrides turns out to be very close to that obtained by fitting Murnaghan EOS $^{12}$. Moreover, we notice that our calculated B for fcc ThH$_{2}$ is approximately equal to previous LMTO the- oretical result $^{5}$. For shear modulus G and Young’s mod- ulus E , their values for bcc Th$_{4}$H$_{15}$ are approximately 2.5 times of those for bct ThH$_{2}$. For Poisson’s ratio, the values for both bct ThH$_{2}$ and bcc Th$_{4}$H$_{15}$ are within the range from 0.25 to 0.45 for typical metals. 

## C. Phonon dispersion curves of thorium dihydrides 

Employing the Hellmann-Feynman theorem and the direct method, we have calculated the phonon curves along some high-symmetry directions in the Brillouin zone (BZ), together with the phonon PDOS. For the phonon dispersion calculation, we use the 2 × 2 × 2 bct (fcc) supercell containing 48 (96) atoms for bct (fcc) ThH$_{2}$. To calculate the Hellmann-Feynman forces, we displace four and eight atoms, respectively, for bct and fcc ThH$_{2}$ from their equilibrium positions and the amplitude of all the displacements is 0.03 ˚ A. The calculated phonon dispersion curves along the 

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