Monte-Carlo tree search for stable structures of planar clusters

Abstract

Illustrated by the case of the planar clusters, we propose a new method to search the possible stable structures by combining the structural identification and Monte-Carlo tree algorithm. We adopt two kinds of model-potential to describe the interaction between atoms:the pair interaction of Lennard-Jones potential and three-body interaction based on the Lennard-Jones potential. Taking the possible triangular lattice fragment as candidates, we introduce a new nomenclature to distinguish the structures, which can be used for the rapid congruence check. 1) We label the atoms on the triangular lattice according to the distances and the polar angles. where a given triangular structure has a corresponding serial number in the numbered plane. Note that the congruent structures can have a group of possible serial numbers. 2) We consider all the possible symmetrical operations including translation, inversion and rotation, and obtain the smallest one for the unique nomenclature of the structure. In conventional search of magic clusters, the global optimizations are performed for the structures with given number of atoms. Herein, we perform the Monte-Carlo tree search to study the evolution of stable structures with various numbers of atoms. From the structures of given number of atoms, we sample the structures according to their energy with the importance sampling, and then expand the structures to the structures with one more atom, where the congruence check with the nomenclature is adopted to avoid numerous repeated evaluations of candidates. Since the structures various numbers of atoms are correlated with each other, a searching tree will be obtained. In order to prevent the over-expansion of branches, we prove the “tree” according to energy to make the tree asymmetric growth to retain the low energy structure. The width and depth of search is balanced by the control of temperature in the Monte-Carlo tree search. For the candidates with lower energies, we further perform the local optimization to obtain the more stable structures. Our calculations show that the triangular lattice fragments will be more stable under the pair interaction of Lennard-Jones potential, which are in agreement with the previous studies. Under the three body interaction with the specific parameter, the hexagonal lattice fragments will be more stable, which are similar to the configurations of graphene nano-flakes. Combining the congruence check and Monte-Carlo tree search, we provide an effective avenue to screen the possible candidates and obtain the stable structures in a shorter period of time compared with the common global optimizations without the structural identification, which can be extended to search the stable structure for materials by the first-principles calculations.