# _Ab Initio_ Study of the Mechanism of Proton Migration in Perovskite LaScO\({}_{3}\)
## 3 Results
The jumps of the fourth type are characterized by the lowest energy barrier and the jumps time is a little longer than that for the jumps of the third type. The number of such jumps, like jumps of the second type, is a quarter of all jumps. Figure 7: Types of proton jumps in perovskite (**a**) I, (**b**) II, (**c**) both, as III as IV. The migration time of a proton between the planes of lanthanum and scandium is practically independent of the direction, while the fraction of transitions into the plane of scandium is much greater than from the plane. Consequently, this jump is more beneficial for the proton than the motion in the lanthanum plane, so it prefers to move in the direction of the scandium plane. On the other hand, the movement of the proton in the scandium plane and exit from it are equally probable (the fractions of jumps of these types are the same). However, to move in the plane, the proton needs to overcome a more significant potential barrier than to exit from it. Therefore, the jump time in the scandium plane is longer. Figure 9 shows the time variation of the hydrogen run in the Cartesian coordinates from the series 3, in which 4 proton jumps were observed by the time of 400 fs. Attention should be paid to the behavior of the y coordinate, according to which it is best to diagnose jumping by a proton. A strong change in the y coordinate ends by the time of 400 fs, and further changes in this coordinate are insignificant. Small fluctuations in the y coordinate near a constant value indicate the absence of proton jumps. Figure 8: Parameters for performing different types of proton jumps: (**a**) the average height of the energy barrier, (**b**) the average time of the jump, and (**c**) the proportion of four types of jumps. The diffusion coefficient \(D\) was determined in terms of the mean square of the proton displacement. According to the Arrhenius law, the dependence of the diffusion coefficient on the reciprocal temperature can be represented as a linear dependence In \(D(1/T)\). To calculate the average diffusion activation energy, the data presented in Figure 8a,c were used, i.e., the average activation energy was determined, taking into account the contribution of each type of proton jump. The calculated dependence ln \(D(1000/T)\), showing the estimated temperature dependence of the proton diffusion coefficient in perovskite LaScO\({}_{3}\), is shown in Figure 4:0 A similar dependence was determined for protons diffusing in yttrium-doped crystalline perovskite BaCeO\({}_{3}\) in a narrow temperature range (from 725 to 770 K) by the neutron scattering method [22]. These values fit satisfactorily into the dependence ln \(D(1000/T)\) obtained by us for LaScO\({}_{3}\). Ceramics BaCeO\({}_{8.5}\)\(Y_{0.2}\)O\({}_{3-8}\) is one of the most highly conductive proton conductors. The perovskite LaScO\({}_{3}\) studied by us can also be attributed to this class of solid proton conductors. Figure 4:1 Dependence of ln \(D\) on the parameter \(1000/T\) at a constant weighted average activation energy \(E_{a}\) = 1.255 eV for the movement of a proton along defect-free perovskite LaScO\({}_{3}\); also presented here are the values of a similar characteristic for BaCeO\({}_{3}\) doped with yttrium obtained in the experiment on neutron scattering [22]. Figure 4:2 Time dependence of the proton Cartesian coordinates related to series 3; the initial placement of a proton corresponds to its average initial coordinates for the series. Figure 11 shows the distributions of the distances between the proton and different types of atoms in the LaScO\({}_{3}\) perovskite system. The first peak of the \"proton-lanthanum\" distance distribution is split into two distinct subpeaks. This testifies the fact that, with a high probability, in addition to the nearest La atom, there are always other La atoms located at a very close distance from the proton. The highest (second) peak in this distribution shows that the majority of other La atoms are located at distances from 4 A to 6 A from the proton.