# _Ab Initio_ Study of the Mechanism of Proton Migration in Perovskite LaScO\({}_{3}\)
## 2 Materials and Methods  
All calculations were performed using the SIESTA software package. These ab initio molecular dynamics calculations are based on the density functional theory and are implemented within the plane wave model. For all considered atoms, only valence electrons were taken into account in the calculations. The calculation of the exchange-correlation functional was based on the Perdew-Burke-Ernzerhof (PBE) formalism [17]. In our calculations, the periodic supercell consisted of 16 La atoms, 16 Sc atoms, and 48 oxygen atoms. The initial configuration of the perfect LaScO\({}_{3}\) system is shown in Figure 1. The resulting parameters of the initial lattice were: a = 11.6003997 A, b = 8.107899684 A, c = 11.328470354 A, \(\alpha=\beta=\gamma=90^{\circ}\), and the volume of the created system was 1068.87 A\({}^{3}\). Projecting the plane-wave onto specific orbitals allowed us to compute local states and populations. The Monkhorst-Pack algorithm [18] was used to generate all 10 \(\times\) 10 \(\times\) 1 k-points. The cutoff energy of plane waves was 400 Ry. This was found to be sufficient to achieve convergence in the context of calculating relative energies. The integration of the equals of motion in realization of ab initio molecular dynamics was carried out by the Verlet method with a time step of 1 fs. The calculations were performed according to the block diagram shown in Figure 2. The input data are the charges of the nuclei, the number of electrons, and the coordinates of the atoms. Plane waves were used as basic functions. The calculation procedure was completed in accordance with the specified criterion for the accuracy of determining the density. Before each ab initio molecular dynamics calculation, geometric optimization was performed. The reference value for the change in the total energy of the system during the dynamic relaxation of atoms was 0.001 eV. As a result of this preliminary preparation, a structure relaxed to a stable state was obtained, the final stress (pressure) in which was equal to 0.0007 Ry/Bohr3. Next, a proton was placed in the resulting system, which occupied different positions:  
1. in the Sc octahedron plane between the oxygen atoms at point 1, which has average coordinates (0.44, 0.75, 0.23),
2. in the Sc octahedron plane near the oxygen atom at point 2, which has average coordinates (0.56, 0.75, 0.23),
3. in the plane of finding La atoms at localization near the oxygen atom at point 3, which has average coordinates (0.33, 0.75, 0.94). In each of these three cases of a specific initial position of the proton, five coordinates were used, differing in \(x\) and \(z\) coordinates. Accordingly, for each case (1-3), five different molecular dynamics calculations were performed. The average values of the initial coordinates of the proton (mentioned above) are also the initial coordinates of the proton for one of the runs in each series. Figure 1: Initial crystal structure of LaCoO\({}_{3}\) in the absence of a proton: large circles denote the A-site cations, the regions bounded by octahedra shows the localization of the B-site cations, and small circles denote oxygen atoms. First-principle molecular dynamics calculations were performed for systems containing a proton. The initial velocity of the proton varied from 0 to 0.37418 \(\times\) 10\({}^{5}\) m/s (from 0 to 0.5 Bohr/fs). An electric field acted on the proton, and the intensity vector was in the \(xz\) plane. The directions of this vector varied, and the magnitude of the tension varied from 0 to 4.95 \(\times\) 10\({}^{10}\) V/m (from 0 to 4.95 V/A). Note that the maximum value of the electric field strength specified here approaches the value of the atomic electric field, which appears as a result of the interaction of atoms in perovskite oxide SrTiO\({}_{3}\) (4 \(\times\) 10\({}^{11}\) near the oxygen atom and 10\({}^{12}\) V/m near the Ti and Sr atoms) [19].