## 3 Results and Discussion
A two-dimensional coordinate system S\({}^{\prime}\) can be established by making a transformation as Equation (16) for the moduli \(\Delta M^{dis}\):  
\[ \begin{array}{c}\Delta M^{dis}_{EG}=\Delta E^{dis}\cdot\sin(\theta)+\Delta G ^{dis}\cdot\cos(\theta)\\ \theta=180^{\circ}(1-\text{acos}(R)/\pi)\end{array}  \tag{16}\]  
We set \(\Delta M^{dis}_{EG}\) and \(\Delta B^{dis}\) as the values on the \(x\) and \(y\) axes, clustering the transformed points, and \(\overline{g}(r)\) could then be calculated. Ignoring groups 4, 6, and 7 with too few structures, the results are shown in Figure 6. Figure 6a shows that \(\Delta M^{dis}_{EG}\) was proportional to \(\Delta B^{dis}\) with a ratio of R\({}^{\prime}\) = 0.076. From Figure 6b,  
Figure 4: Mean-shift clustering for the differences of the elastic moduli in generated structures. (**a**) The projection of points in the S system and the clustering results on the \(\Delta E^{dis}\)-\(\Delta G^{dis}\) plane, which shows the linear correlation of \(\Delta E^{dis}\) and \(\Delta G^{dis}\). (**b**) The projection of points in the S system and the clustering results on the \(\Delta E^{dis}\)-\(\Delta B^{dis}\) plane, which shows that group 0 contained the most structures, and that there was no linear relationship of \(\Delta E^{dis}\) and \(\Delta B^{dis}\). Figure 5: The averaged partial pair-correlation function (PCF) \(\overline{g}(r)\) for structures in clustered groups. \(\Delta E^{dis}\) and \(\Delta G^{dis}\) increased with the hydrogen atoms distributed in the 5 th nearest neighbor (5 NN) and 10 th nearest neighbor (10 NN) sites. the \(\overline{g}(r)\) of the structures in groups 1, 0, 2, and 3 at the 7 th nearest neighbor (7 NN) clearly decreased with the increase of \(\Delta B^{dis}\). Since the larger the B is, the more difficult it is for the material to be compressed, the results indicate that in \(\alpha\)-Fe, the fewer hydrogen atoms are distributed at the positions of 7 NN (s\({}^{7\text{NN}}\)), the more difficult the compression. Most mechanical properties of metallic materials are generally influenced by shears and slips; however, what we observed was that E, B, and G were affected by the distribution of the amount of hydrogen atoms occupied at 5 NN, 7 NN, and 10 NN in \(\alpha\)-Fe, which is also likely related to the effects of hydrogen and slip systems. For this reason, we analyzed the relationship between the Tet-sites that can be occupied by hydrogen atoms and the specific crystal faces. As shown in Figure 7a, if any Tet-site is selected as the reference position Z, the Tet-sites on its neighbors are distributed on concentric spherical shells, and the radius of the nth spherical shell is the distance between one of its n nearest neighbor (nNN) sites and Z. We cut any spherical shell with a crystal plane with an index of (hkl). If the number of Tet-sites on the cross-section was greater than or equal to three, then we say that they fell on the crystal plane with index (hkl). Dividing the number of these sites by the area of the crystal face determined by them, we obtained the area density of the hydrogen-occupied positions on the crystal face, which is decided by the crystal index (hkl). This means that at the same \(c_{H}\), if hydrogen is more distributed in a certain spherical shell, this may affect the difficulties of shears and slips of different crystal planes and cause different effects. Since the crystal faces in the [hkl] family are equivalent by the symmetry of the lattice of \(\alpha\)-Fe, the effect of hydrogen on elastic moduli of [hkl] should be the same on \(\rho\). We considered all (hkl) crystal planes in the [hkl] family when calculating \(\rho_{[\text{hkl}]}\) of the nNN spherical shell. As the major slip planes of \(\alpha\)-Fe belong to the [110] and [112] families [56], we calculated the areal densities \(\rho_{[110]}\) and \(\rho_{[112]}\) of Tet-sites on planes at different NN locations. As shown in Figure 7b, there were more NNs where \(\rho_{[110]}\
eq 0\) than \(\rho_{[112]}\
eq 0\), and \(\rho_{[110]}>\rho_{[112]}\) in most NNs.