## 3 Results and Discussion
Although all the structures had a uniform distribution of hydrogen atoms randomly arranged, when \(c_{H}>0.2\%\), the differences in the microscopic distribution still led to significant changes in the results. Combined with Table 1, these differences can reach up to 15% and increase with the \(c_{H}\). Such significant changes caused by the deviations of hydrogen atoms from uniform distribution conditions are likely to occur during HE and other processes and significantly impact the material service security. According to the interstitial hydrogen atom spacing condition in iron [27; 45], the relaxed structures were screened. When \(c_{H}=0.8\%\), there are enough hydrogen atoms in the system for statistics and comparison, the structures that meet the conditions are the most present (101 groups in total), and the elastic moduli of each structure are significantly different at this concentration. Thus, we selected them for the clustering calculation. Clustering algorithms are increasingly used in materials science and condensed matter physics research for complex data [32; 54; 55]. In this work, we used the mean-shift algorithm [47], which does not require manually setting the clustering centers and the number of classifications and can handle arbitrary shapes of data distributions. We established a coordinate system S, with \(\Delta E^{dis}\), \(\Delta G^{dis}\), and \(\Delta B^{dis}\) as the values on the \(x\), \(y\), and \(z\) axes, and the values of the structure q as the coordinates of the point Mq. All points were clustered, and the results were projected to the \(\Delta E^{dis}\)-\(\Delta G^{dis}\), and \(\Delta E^{dis}\)-\(\Delta B^{dis}\) planes, as shown in Figure 4a,b. It can be seen from Figure 4a that \(\Delta E^{dis}\) was proportional to \(\Delta G^{dis}\), the proportional coefficient R = 2.2, and there were obvious gaps between different groups. The \(\Delta E^{dis}\) and \(\Delta G^{dis}\) in the second, third, and fourth groups were less than 0. The elastic moduli of the structures in these groups were lower than the average results when \(c_{H}=\)0.8%, and more obvious softening occurred; correspondingly, the 1, 5, 6, and 7 groups had obvious hardening. It can be seen from Figure 4b that the number of structures in group 0 was the most, and the \(\Delta E^{dis}\) and \(\Delta G^{dis}\) of these structures were also the closest to 0, indicating that the microscopic distribution of hydrogen atoms in most randomly generated structures deviated little from the ideal uniform distribution. In addition, there was no linear relationship between \(\Delta E^{dis}\) and \(\Delta B^{dis}\). Figure 3: Solute hydrogen atom effects on the elastic moduli of the \(\alpha\)-Fe single crystal. **(a)** Averaged elastic moduli changes. This shows that [PERSON]’s modulus E, the bulk modulus B, and shear modulus G were all slightly decreased by the hydrogen solution, which is a combination of the decrease by volume and the increasing by cohesion. **(b)** The hydrogen distribution influences on the moduli in each structure, which indicates that the slight differences in the microscopic distribution of hydrogen atoms might cause different effects from the averaged results. We calculated g(r) for each structure according to Equation (14), and then averaged it for groups 2, 3, and 4, group 0, and groups 1, 5, 6, and 7, respectively, to obtain \(\overline{g}(r)\). As shown in Figure 5, the peaks at the 5 th nearest neighbor (5 NN) and the 10 th nearest neighbor (10 NN) between hydrogen atoms were clearly related to \(\Delta E^{dis}\) and \(\Delta G^{dis}\). The values of the three groups of \(\overline{g}(r)\) obtained at 5 NN, were 3.434, 4.226, and 4.568, and, at 10 NN were 2.023, 2.105, and 2.659, all of which increased gradually. This indicates that the increase of \(c_{H}\) at positions 5 NN (s\({}^{5\text{NN}}\)) and 10 NN (s\({}^{10\text{NN}}\)) between hydrogen and hydrogen increased E and G, thus inducing hardening.