Exploring a potential energy surface by machine learning for characterizing atomic transport

Abstract

We propose a machine-learning method for evaluating the potential barrier governing atomic transport based on the preferential selection of dominant points for the atomic transport. The proposed method generates numerous random samples of the entire potential energy surface (PES) from a probabilistic Gaussian process model of the PES, which enables defining the likelihood of the dominant points. The robustness and efficiency of the method are demonstrated on a dozen model cases for proton diffusion in oxides, in comparison with a conventional nudge elastic band method. PACS numbers: 31.50.Bc,66.30.Lw,89.20.Ff Atomic transport plays a key role in a variety of phenomena related to physics, chemistry, and materials science. Concerning the transport of a mobile atom governed by thermally activated processes in a crystal, the kinetics is fully characterized by the entire potential energy surface (PES) of the mobile atom in the host crystal. The most important region in the entire PES is the optimal path, which is defined as the lowest-energy path between two global minimum points separated by a lattice translation vector. Therefore, the optimal path is identical to a valley line, which generally passes through several saddle points. Based on transition state theory (TST) [1][2][3][4], the kinetics is determined primarily by the potential barrier of the optimal path, i.e., the difference in potential energy (PE) between the global minimum point and the bottleneck point, which is defined as the point having the highest PE on the optimal path.The nudged elastic band (NEB) method [5, 6] is a wellestablished and powerful technique for identifying the optimal path and its energy profile. The NEB method, however, has a serious drawback in that it requires a given initial trajectory to identify the optimal path. When the optimal path consists of several elementary paths, all local minimum points (including the global minimum point) in the entire PES are found in advance, because these points may be the initial and final points of the elementary paths. For each of the possible initial trajectories derived from the local minimum points, an elementary path is analyzed using the NEB method, and the optimal path formed by some of the elementary paths is finally identified. Moreover, physical and chemical prior knowledge, e.g., ionic radii, chemical bonding states, and electrostatic interaction, has usually been used to obtain local minimum points and initial trajectories. However, the excessive dependence on the prior knowledge may cause us to miss a key elementary path. Thus, a robust and efficient alternative method to the NEB-based analysis is desired.In the present study, we propose a machine-learning (ML) method for robustly and efficiently estimating the potential barrier of the optimal path. The basic strategy is to focus only on finding the two dominant points, i.e., the global minimum and bottleneck points. To this end, a probabilistic Gaussian process (GP) model [7,8] of the PES is introduced, and is iteratively updat...

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