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to their internal degrees of freedom. The total energies (per primitive cell) of the two structures at different vol- umes are calculated and shown in Fig. 2. Obviously, the bct ThH$_{2}$ is more stable than fcc ThH$_{2}$ under am- bient pressure. The equilibrium volumes of bct and fcc structures are 40.63 ˚ A 3 and 41.26 ˚ A $^{3}$, respectively. Thus the equilibrium volumes of both bct and fcc phases are approximately equal and the total energy difference at their equilibrium states is ∼ 0.032 eV. This conclusion is consistent well with the previous FLAPW results $^{2}$. 

Basically, all the macroscopical properties of materi- als, such as hardness, elasticity, and conductivity, origi- nate from their electronic structure properties as well as chemical bonding nature. Therefore, it is necessary to perform the electronic structure analysis of thorium hy- drides. The calculated total densities of states (DOS) and the orbital-resolved partial densities of states (PDOS) of bct ThH$_{2}$, fcc ThH$_{2}$ and bcc Th$_{4}$H$_{15}$ are shown in Fig. 3. Overall, the occupation properties of ThH$_{2}$ in bct and fcc phases are similar. The occupied DOS near the Fermi level is featured by the three well-resolved peaks. The one near − 1.0 eV is principally Th 6 d in character, while the other two peaks respectively near − 4.5 and − 6.0 eV are mostly H 1 s hybridized with Th 6 d and Th 5 f orbitals. These well-separated orbital peaks have been observed in the photoelectron spectroscopy (PES) measurement $^{21}$. In addition, the band width of the Th 6 d valence band near the Fermi level is 3 eV, consistent with the FLAPW calculation 2 and experimental data $^{21}$. The H 1 s valence band width is of 5.0 eV, also in ac- cord with the experimental data (6.0 eV) and previous FLAPW result (5.2 eV). Moreover, the low bands of bct (fcc) ThH$_{2}$ covering from − 18.8 ( − 19.0) to − 17.2 ( − 17.5) eV is mainly featured by Th 6 p state mixed with a little H 1 s state and the conduction bands is principally occu- pied by Th 5 f states with admixtures of the Th 6 d and H 1 s states and has a width of 5.5 (5.1) eV. For Th$_{4}$H$_{15}$, our calculations reproduce all the features calculated by Shein et al . 2 and agree well with PES measurements $^{21}$. Note that our calculated conduction band width is 1.7 eV, consistent with the PES experimental value 21 (0.9 eV) and evidently smaller than that of FLAPW calcula- tion result $^{2}$. The main occupation of conduction bands is Th 5 f orbital, mixed with a little Th 6 d and H 1 s states. In addition, we have also presented in Table I the partial and total DOSs at the Feimi level N ( E$_{F}$ ) for tho- rium hydrides to study the occupation of the conduction band. Clearly, our results are in good agreement with the FLAPW calculation $^{2}$. 

In order to gain more insight into the bonding nature of ground state thorium hydrides, we also investigate the valence charge density distribution. The calculated va- lence charge density maps of the bct ThH$_{2}$ in (100) plane, fcc ThH$_{2}$ in (1 ¯ 10) plane, and bcc Th$_{4}$H$_{15}$ in plane estab- lished by three Th and one H2 atoms are plotted in Fig. 4. Clearly, the charge densities around Th and H ions are all near spherical distribution with slightly deformed to- wards the direction to their nearest neighboring atoms. 

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To describe the ionic/covalent character quantitatively and more clearly, we plot in Fig. 5 the line charge density distribution along the nearest Th − H bonds and perform the Bader analysis $^{6,7}$. For bct and fcc ThH$_{2}$, the two charge density curves change with the same way along the Th − H bonds. For Th$_{4}$H$_{15}$, the two curves along the Th − H1 and Th − H2 bonds are almost the same before the bridge locus (indicated by the arrow), after where they split evidently. Besides, one can find a minimum value of charge density in Fig. 5 for each bond at around the bridge. These minimum values are listed in Table II. Although these values are much smaller than 0.7 e / ˚ A 3 for Si covalent bond, they are prominently higher than 0.05 e / ˚ A 3 for Na − Cl bond in typical ionic crystal NaCl. Also we notice that these values are smaller than the minimum values (0.45 e/ ˚ A $^{3}$) of charge density along the Th − O bond in our previous study of ThO$_{2}$ $^{22}$. There- fore, there are weak but clear covalent bonds between Th and H for ThH$_{2}$ in bct and fcc phases and for bcc Th$_{4}$H$_{15}$. The slight difference appears in Th$_{4}$H$_{15}$, where the minimum value of charge density in Th − H1 bond is smaller than that in Th − H2 bond. In addition, one can 

