# _Ab Initio_ Study of the Mechanism of Proton Migration in Perovskite LaScO\({}_{3}\)
## 3 Results
Obviously, during such a time interval, the proton completely \"forgets\" the initial direction of its velocity. The further speed of the proton is determined by the interatomic interaction, temperature, and, in part, the direction of the electric field strength. The electric field does not completely determine the direction of motion of the proton, but it affects the instantaneous values of the potential energy: the greater electric field results in the more significant fluctuations of the \(U(t)\) function. Comparison of Figure 6a,b testifies in favor of the fact that the frequency of the proton jumps can be influenced by the direction and magnitude of its initial velocity. In particular, at a higher value of the velocity, the time intervals between the proton jumps increase. At  
Figure 5: Dependence of the average potential energy on the average temperature for the calculated states of the LaScO\({}_{3}\) perovskite. The numbers indicate the series numbers from Table 1.  
the same time, a comparison of Figure 4a,c shows that the number of proton jumps in the presence of an electric field is likely to decrease. The types of possible jumps made by a proton in a perovskite are shown in Figure 7. The directions of the proton movement are shown by light segments connecting the light circles (proton designations). In case I, the proton changes its position, passing from the oxygen atom belonging to one octahedron to the oxygen atom assigned to another octahedron, and the transition occurs between La atoms. In case II, the proton moves along only one La atom and passes from one oxygen atom to another along the edge of the octahedron, with both oxygen atoms belonging to the same octahedron. In cases III and IV, the proton also passes one La atom, but the oxygen atoms are separated by an octahedron, i.e., together they belong to three adjacent octahedra. The forward and backward paths of the proton in these cases are not equivalent due to the difference in the atomic environment. Therefore, we consider them separately, assuming that case III corresponds to a direct transition, i.e., the proton hits the plane of the octahedron, and case IV corresponds to a reverse transition. Figure 6: Change in the potential energy of the perovskite system over time in calculations (**a**) 1, (**b**) 2, and (**c**) 3; red stripes show the proton jumps among oxygen atoms; the average potential energy is shown by the dashed line; the initial placement of a proton corresponds to its average initial coordinates for each selected series of five runs. Figure 7 shows the characteristics of proton hopping classified according to the four types defined above. The jumps of the first type can be characterized as the migration of a proton in the lanthanum plane (the plane between the octahedra). The second type of jumps takes place on the plane of scandium (in the plane of the octahedrons). The jumps of the third type include the movement of a proton from the plane of lanthanum to the plane of scandium (transition to the plane of the octahedron). The fourth type of transition expresses the exit of the proton from the scandium plane and its return to the lanthanum plane. The characteristics of jumps of different types are shown in Figure 8. Jumps of the first type are characterized by the maximum energy barrier height, the shortest jump time (since in this case the proton acquires the highest kinetic energy), and the minimum number of jumps of this type. To make a jump of the second type, the proton also needs to acquire a sufficiently large kinetic energy to overcome the second highest energy barrier. In this case, the jump time is the longest. The number of such jumps is a quarter of all jumps performed. The barrier for the jumps of the third type is slightly less than that for the jumps of the second type, and the time needed for such jumps is noticeably shorter than that for the jumps of the second type, but it is slightly longer than for the jumps of the first type. The proportion of the jumps of the third type turns out to be the largest.