Pauling’s rules guided Monte Carlo search (PAMCARS): A shortcut of predicting inorganic crystal structures

Abstract

The rapid development of the high-throughput calculations and materials genome approaches in recent years has improved researchers’ ability to design advanced materials. Crystal structure prediction techniques play an important role in high-throughput calculations and materials genome approaches. However, the huge computational cost of the widely used crystal structure prediction techniques based on the global optimization algorithms hinders the researchers from predicting the large or medium-sized crystal structures. In recent years, some metastable materials with excellent electrical, magnetic, optical and catalytic properties have attracted a lot of attention. However, there is currently a lack of a structure prediction technique with high energy resolution near the ground-state structures to meet the growing demands for the prediction of low-energy metastable materials. In this article, a set of easy-operation guidelines for constructing crystal structures is extracted from Pauling’s rules. Based on this set of guidelines, we have designed an algorithm called Pauling’s rules guided Monte Carlo search (PAMCARS), which predict the ground-state and meta-stable inorganic crystal structures of a given composition by combining a classical Monte Carlo search in the configuration space constrained by Pauling’s rules with ab initio structural relaxations. In the actual tests, the crystal structures of carbon allotropes, CaSO₄, Ba₂TiSi₂O₈ and BaAlBO₃F₂ are successfully predicted by this crystal structure prediction algorithm. In addition, a number of unreported high-symmetric metastable crystal structures of carbon, CaSO₄ and Ba₂TiSi₂O₈ are also predicted. The successful prediction of graphite layers indicates that PAMCARS is able to predict some layered materials assembled by the Van der Waals forces. Two of the predicted low-energy metastable crystal structures of Ba₂TiSi₂O₈ may explain the property anomaly of Ba₂TiSi₂O₈ around 160 °C. Pauling’s rules guided Monte Carlo search exhibits high efficiency and energy resolution in predicting the ground-state and metastable crystal structures in the tests.

Introduction

The properties of materials are closely related to their composition and crystal structures. The leading experimental method of determining the crystal structures of materials is single crystal X-ray diffraction. In order to obtain accurate and reliable crystal structures by this method, it is necessary to prepare defect-free crystals to obtain high-quality diffraction data. However, a large number of materials are difficult to synthesize, which makes it difficult to determine the accurate crystal structures by experiments. In addition, the traditional strategy to find advanced materials that meet certain properties is to synthesize a large number of unknown substances and perform various tests on them, which wastes a lot of time and resources. Therefore, it has a great advantage to predict the crystal structures of a given composition with little experimental information or prior knowledge. With the continued improvements in computing performance of computers, crystal structure prediction techniques have provided a fast way to materials discovery in recent years [1], [2], [3].

Most materials tend to be in the structures with the lowest free energy. The goal of crystal structure prediction is to find the global minimum and the corresponding structure in the free energy space according to the chemical composition of the material. The dimension of the free energy space is 3N+3 for a crystal with N atoms in a unit cell [4], [5]. It is a great challenge to search the global minimum in the high-dimensional free energy space that contains a large number of local minimums. As the number of the elements in the crystal increases, the search will become much more difficult.

Many algorithms for crystal structure prediction and the corresponding software have been developed, such as random search [6], [7], [8], metadynamics [9], [10], simulated annealing [11], [12], basin hopping [13], [14], minima hopping [15], [16], evolutionary algorithms (EA) [17], [18], [19], [20] and particle swarm optimization (PSO) [21]. These algorithms have been successfully applied to small and medium-sized crystals. Most of the crystals that have been successfully predicted by these algorithms in the literature contain no more than three elements, such as SiH₄ [22], Zr₂Co₁₁ [23], $H_3$S [24], UH₇ [25] and $\delta$-Mg(BH₄)₂ [26], but few successful predictions for crystals with more elements are reported in the literature.

The global optimization techniques, such as EA and PSO, have become the most common methods used in the field of crystal structure prediction. However, there are still two obstacles in this approach to the prediction of large systems. The probability of finding the global minimum through these optimization techniques depends on the population size and the total number of generations. As the number and species of the atoms in crystals increase, the required population size and number of generations will grow significantly, which leads to a huge computational cost for large crystals. In addition, EA and PSO usually start from an initial population that consists of symmetric random crystals, but the symmetries of the descendants of the initial crystals are broken by evolution or PSO operators. In reality, it is hard to relax the complex multi-element structures with space group P1 to structures with higher space group symmetries by material simulation software, such as VASP [27] and GULP [28]. Thus many structures with space group P1 are obtained by the structure relaxation for the descendants. However, these non-symmetrical structures are usually unstable and useless. Due to this fact, many low-energy metastable structures with high symmetry may be missed.

If the mathematical global optimization algorithm is used to predict the structures of large multi-element crystals violently, the amount of computation may become unbearable. If some general rules for crystal structures are applied to crystal structure prediction, we can focus on just the most promising regions of the free energy space to explore the global minimum, which may be a more realistic and efficient strategy for predicting large systems. The guidelines that are easy to program and outline the essence of Pauling’s rules [29], [30] are introduced in the present work. The algorithm of Pauling’s rules guided Monte Carlo searchis designed according to the guidelines.