## 3 Results and Discussion  
### Surface Model of Mo-doped Nb  
In Nb\({}_{16-x}\)Mo\({}_{x}\)H system, Nb\({}_{12}\)Mo\({}_{4}\)H has the highest absolute value of \(\Delta H_{t}\), which is favorable for dehydrogenation. Therefore, Nb\({}_{12}\)Mo\({}_{4}\)H was selected to build the slab model for further study, which is depicted in Figure 1a. There are two high symmetrical interstitial sites in the region between the first and second atomic layer, which are tetrahedral interstitial site (TIS) and octahedral interstitial site (OIS), as shown in Figures 1 and 1. Due to the similar atomic radius of Mo compared to Nb, the crystal keeps the cubic structure after Mo-doping. The equilibrium lattice constant (3.304 A) of bulk Nb agrees well with experimental values (3.305 A) [42]. For the (100) surface, we built a slab model with seven atomic layers, which is depicted in Figure 1d. Surface energy is an important parameter of a metal surface that can be utilized to explain the physical and chemical processes and surface stability. Cutting surfaces from bulk crystal breaking various chemical bonds, which increases the energy of the system, thereby resulting in instability of the surface systems. The surface energy is defined as [14]  
\[\sigma=(E_{slab}-n\times E_{bulk})/(2A) \tag{1}\]  
Here, \(E_{\text{slab}}\), \(E_{bulk}\), \(n\), and \(A\) represent the total energy of the slab, the total energy of the corresponding bulk material, the number of atoms contained in the slab and the area of the surface, respectively. In the structure optimization calculations, the atoms in the top three layers were fully relaxed, and the atoms in the other layers were fixed at their bulk positions. Table 1 lists the calculated surface energies of slab models with different numbers of atomic layers. The surface energy of pure Nb (100) surface is also calculated for comparison. The results agree well with the values from [PERSON] et al. [43]. The surface energy of Nb\({}_{12}\)Mo\({}_{4}\)(100) surface and Nb\({}_{12}\)Mo\({}_{4}\)(111) surface tend to be stable when there are more than five atomic layers. Moreover, the Nb\({}_{12}\)Mo\({}_{4}\)(100) surface is the most  
Figure 1: Theoretical models of Nb\({}_{12}\)Mo\({}_{4}\) used in the calculations: (**a**) bulk model; (**b**) an octahedral interstitial site (OIS), (**c**) a tetrahedral interstitial site (TIS); (**d**) slab model. Nb atoms are represented by green spheres, and Mo atoms are represented by gray spheres. stable one with the smallest surface energy (2.75 J/m\({}^{2}\)). Therefore, we select Nb\({}_{12}\)Mo\({}_{4}\)(100) surface for further study. To determine a practicable number of atomic layers and the vacuum region of Mo-doped Nb (100) surface model, we calculate the changes of the interlayer distances during structure relaxation as the following relationship [44; 45]  
\[\Delta d_{i-j}=d_{i-j}-d_{0} \tag{2}\]  
Here, \(d_{0}\) denotes the layer distance between the i- and j-layers of the unrelaxed slab model established from a geometry-optimized bulk model and \(d_{i-j}\) denotes the corresponding layer distance of the optimized slab model. From the calculated results depicted in Figure 2, we can infer that the interlayer distance between the outermost layer and secondary outer layer tends to be stable when there are more than six atomic layers. According to the previously calculated surface energies, seven atomic layers are sufficient to avoid the effects of external forces in \(z\)-axis. In addition, to ensure that hydrogen permeation process is not influenced by periodicity, a vacuum region between the atomic layers in the (100) surface is required. Thus, the work function of (100) surface model with seven atomic layers and a 12 A vacuum layer along the surface normal direction (z-axis) is calculated, and is depicted in Figure 3. The flat line in the figure indicating that the work function converges in the vacuum region. Therefore, we employed a slab model of seven atomic layers with a 12 A vacuum layer of (100) surface for further hydrogen permeation analysis.