A multi-populations multi-strategies differential evolution algorithm for structural optimization of metal nanoclusters

Abstract

Theoretically, the determination of the structure of a cluster is to search the global minimum on its potential energy surface. The global minimization problem is often nondeterministic-polynomial-time (NP) hard and the number of local minima grows exponentially with the cluster size. In this article, a multi-populations multi-strategies differential evolution algorithm has been proposed to search the globally stable structure of Fe and Cr nanoclusters. The algorithm combines a multi-populations differential evolution with an elite pool scheme to keep the diversity of the solutions and avoid prematurely trapping into local optima. Moreover, multi-strategies such as growing method in initialization and three differential strategies in mutation are introduced to improve the convergence speed and lower the computational cost. The accuracy and effectiveness of our algorithm have been verified by comparing the results of Fe clusters with Cambridge Cluster Database. Meanwhile, the performance of our algorithm has been analyzed by comparing the convergence rate and energy evaluations with the classical DE algorithm. The multi-populations, multi-strategies mutation and growing method in initialization in our algorithm have been considered respectively. Furthermore, the structural growth pattern of Cr clusters has been predicted by this algorithm. The results show that the lowest-energy structure of Cr clusters contains many icosahedra, and the number of the icosahedral rings rises with increasing size.

Introduction

Metallic nanoclusters (less than a few hundred atoms) have attracted increasing attention because of their unique properties and extensive applications in a diversity of fields such as physics, chemistry, biology, and engineering  [1], [2], [3], [4]. Nowadays, the transition metal nanoclusters are of significant interest due to their potential uses in ultrahigh density magnetic recording materials  [5], catalytic particles in the synthesis of carbon nanotubes  [6], [7], [8], [9], and other applications in electronics and optics. Due to the ultrafine size, the nanoclusters may remain in a ‘liquid-like’ state at temperatures far below the bulk melting point, and their magnetic moments may exceed bulk values at cluster sizes of several hundred atoms  [10]. Among the transition metal clusters, iron (Fe) and chromium (Cr) clusters are the primary target of many research groups owing to their important magnetic properties  [11], [12], [13], low price, and good catalytic performance  [14], [15], [16], [17], [18]. Since the magnetic and catalytic properties of Fe and Cr clusters are strongly dependent on their structures, investigation of the stable structures is crucial for understanding their physical and chemical properties.

Essentially, searching the stable structures of clusters is a typical global optimization problem  [10]. The number of distinct structures corresponding to local minima on the potential energy surface (PES) of a cluster is expected to grow exponentially with the cluster size  [19]. Therefore, the searching procedures are computationally expensive. To date, many methods have been developed to solve the global optimization problem. They can be mainly classified into two classes: Those who use the iteration of single solution and ones who adopt the evolution of one population of candidate structures. The representatives of the first class are Basin-hopping approach  [20], [21], [22], [23], Monte Carlo algorithm  [24], [25], [26], [27], [28] and dynamical lattice searching (DLS) method  [29], [30]. These algorithms are very simple and few control parameters are used to various optimization problems. However, they are less efficient than the population searching because their convergence is too slow. The other class is the evolutionary algorithm  [31], [32], [33], [34], [35], such as genetic algorithm (GA)  [32], [33], ant-colony method  [34], and particle swarm optimization  [35]. These algorithms are widely applied in study of various clusters, and have achieved remarkable successes in structural optimization of clusters. Nevertheless, among these algorithms, it is difficult to select the optimal solutions from many control parameters, and it needs about tens of thousands of local optimization steps for large clusters. Therefore, it is not effective and very time-consuming for the large clusters. Furthermore, a parallel method has been recently introduced into the evolutionary algorithm during the structural optimization of clusters  [36], [37]. Although the parallel scheme keeps the diversity of the solutions and enhances the global searching capability to some extent, it makes the main features of previous implementation lose since it only reserves an optimal solution.

To search the globally steady structure of the clusters more efficiently, in this article we propose a multi-populations multi-strategies differential evolution (DE) algorithm to perform the structural optimization of Fe (Cr) clusters. First, to reduce the calculation time and improve the searching efficiency, a local optimization strategy is introduced by transferring the potential energy surface into a set of basins of attraction. Secondly, the multi-populations with an elite pool method is applied to keep the diversity of the solutions and avoid prematurely trapping into local optima. Meanwhile, to improve the convergence speed and lower the computational demand, the multi-strategies such as growing method in initialization and three differential strategies in mutation are introduced into DE algorithm. Importantly, an adjusting strategy is applied to modify the broken structures caused by crossover operator. Additionally, the Finnis–Sinclair potentials are adopted to describe the interatomic interactions between iron (chromium) atoms in the global optimization. Furthermore, to test and verify the accuracy of the proposed algorithm, some benchmark data published by Cambridge Cluster Database (CCD) have been used for comparison with the results of global optimization for Fe clusters. The lowest-energy structures of Cr clusters have also been predicted by the multi-populations multi-strategies DE algorithm. This article is structured as follows. Section  2 describes the Finnis–Sinclair potentials for Fe and Cr clusters and introduces the DE algorithm with multi-population evolution. Section  3 presents the calculated results and discussion. The main conclusions are summarized in Section  4.