# _Ab Initio_ Study of the Mechanism of Proton Migration in Perovskite LaScO\({}_{3}\)
## 3 Results  
Let us trace the trajectories of a proton in cases, where its initial placement corresponds to its average initial coordinates obtained in five calculations of the same type. Horizontal projections of the configurations of the LaScO\({}_{3}\) crystal corresponding to the time instant of 1 ps are shown in Figure 4.  
The upper figures reflect the above system obtained in series 1, and the lower figures show a similar system resulting from the series 3. In series 1, a constant electric field with an intensity \(E=0.7071\times 10^{10}\) V/m acted on the proton, the series 3 was performed in the absence of an electric field. The trajectory of a proton is enclosed in a more limited space under the effect of the electric field. It is noteworthy that a large part of the path of the proton does not pass along the direction determined by the strength of the electric field (line in the figure), but in the direction forming a certain angle with the direction of the vector E. In either case, the trajectory of the ion turns out to be rather entangled. It can get into the vicinity of another (different from the original) oxygen atom and stay there for some time. It can also approach adjacent La and Sc atoms, which are not initially adjacent to it. In addition, the proton can not only move away from its initial location, nor approach it again, moving in the direction opposite to the direction of the passed path. Thus, the electric field does not affect the choice of the preferred location for the proton in the LaScO\({}_{3}\) crystal. Most likely, the place of its virtual location is determined by the attraction of a proton to any closest oxygen atom. Figure 4: Horizontal projections of the configurations of the LaScO\({}_{3}\) crystal, corresponding to the time instant of 1 ps, (**a**) under the electric field with an intensity of \(E=0.7071\times 10^{10}\) V/m and (**b**) without the electric field; the figure also shows the trajectories of the proton for the specified time interval, and the symbols \(i\) and \(f\) denote the beginning and the end of the trajectory, respectively. The temperature dependence of the average potential energy of perovskite LaSo\({}_{3}\) is not monotonic (Figure 5). Sharp changes in the energy \(U\) are observed at temperatures \(T<153\) K and in the temperature range \(669\leq T\leq 685\) K. In this temperature range, the highest energy \(U\) corresponds to the state with the highest value of the electric field and the least energy appears when the field is missing. The largest value of the energy \(U\) is also observed at \(T=1099\) K at the maximum value of the electric field strength. The behavior of a proton in a system essentially depends on the temperature of the perovskite. We can say with confidence that no proton jumps are observed at temperatures below 400 K; the proton vibrates around the oxygen atom near which it was originally located. In the temperature range of 600-800 K, random proton jumps are observed between oxygen atoms. Most of the proton jumps are observed at the temperatures above 800 K, when the proton has a sufficient speed to overcome the potential barrier. A visual representation of the proton motion in the LaScO\({}_{3}\) perovskite is provided in Supplementary data. This film representation shows the behavior of the proton in the calculation 1, when the temperature in the model was 685 K.  
The behavior of a proton in a perovskite system in the temperature range of 680-850 K makes it possible to distinguish the time interval, during which the proton jumps to a neighboring oxygen atom, or to another place of temporal localization. In the time sweep of the potential energy, such an interval is always included between the local extrema of the function \(U(t)\). Figure 4 shows the \(U(t)\) functions obtained in the series 1-3, defined by Table 1. The time intervals of the proton jump to a new sedentary position are highlighted in Figure 6 with red stripes. In all cases, the duration of the interval for the jump did not exceed 100 fs, i.e., 100 time steps.