## 3 Results and Discussion  
### Adsorption of Hydrogen Molecules on the Surface of Iron  
The Fe(110) surface, owing to its unique atomic arrangement and surface properties, exhibits significant representativeness in studies of the interaction between hydrogen and iron. The top, bridge, and hollow sites--these high-symmetry positions--are critical in exploring hydrogen's adsorption behavior on iron surfaces, often displaying strong surface activity [39]. Previous studies indicate that the adsorption energies of hydrogen atoms on the Fe(110) surface amount to \(-\)1.98 eV for the top site, \(-\)2.63 eV for the bridge site, and \(-\)2.98 eV for the hollow site, highlighting the hollow site as hydrogen atoms' preferred stable adsorption position, with the top site being relatively unstable [40]. Based on these findings, this study concentrates on analyzing the adsorption of hydrogen molecules at these two extremes--namely, the most stable and least stable positions. By positioning hydrogen molecules parallel to the Fe(110) surface and incrementally adjusting the distance between the hydrogen molecule and the iron surface, the simulation captured the approach of hydrogen molecules towards the iron surface, enabling the system to achieve an energy-minimized state at each step. The total-energy profile of the H\({}_{2}\)/Fe(110) system was obtained by varying the vertical distance d between the geometric center of the hydrogen molecule and the nearest surface Fe atoms. At each separation, the adsorption energy was evaluated as  
\[E_{ads}=E_{sys}-E_{Fe}-\frac{1}{2}E_{H2} \tag{1}\]  
where \(E_{sys}\) is the total energy of the combined system, \(E_{Fe}\) is the energy of the clean surface, and \(E_{H2}\) is it the energy of an isolated H\({}_{2}\) molecule. Determining the minimum \(E_{ads}\) allowsthe most favorable adsorption site to be identified, thereby providing the starting point for full geometry optimization [41]. The calculated values are summarized in Table 1.  
The analysis results (refer to Figure 1) show that as the hydrogen molecule approaches the iron surface to a proximity of approximately 2 A, the system's energy change begins to stabilize, indicating the initiation of interactions between the hydrogen molecule and the iron surface atoms. Further comparison of the energy changes at the top and hollow sites yields a definitive conclusion: the adsorption behavior of the hydrogen molecule at these two distinct symmetrical sites shows significant differences. The fitted quadratic polynomial equation in Figure 1 is  
\[E=33.4429D^{2}-126.2266D-10272.3340 \tag{2}\]  
The fitted quadratic polynomial equation in Figure 2 is  
\[E=1.0716D^{2}-5.9927D-10375.1198 \tag{3}\]  
where \(D\) is in A and \(E\) is in eV. It's critical to acknowledge that the analysis presented herein relies on system energy calculations that do not incorporate geometric optimization, thereby omitting the potential effects of chemical bonding between hydrogen molecules and iron atoms. For the purposes of this preliminary analysis, it is assumed that the hydrogen molecule retains its original molecular structure without alteration, and no geometry optimization is performed during this step. Observations indicate a marked reduction in system energy as the distance between the hydrogen molecule and the Fe(110) surface extends from 0.5 A to 1.75 A. This implies that, within this specified range and in the absence of chemical bonding, the interaction is predominantly influenced by van der Waals forces or alternative short-range repulsive forces. Notably, at the distance of 1.75 A from the iron surface, where the system energy reaches a global minimum, this specific distance could denote the most favorable position for the physical adsorption of the hydrogen molecule. Beyond the distance of