The structure of atomic and molecular clusters, optimised using classical potentials

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Abstract

The problem of the determination of the minimum energy configuration of an arrangement of N point particles under the interaction of their interatomic forces is discussed. The interatomic forces are described by classical many body potentials. Different optimisation methods are considered, multi level single link, topographical differential evolution and a genetic algorithm but it is shown that genetic algorithms combined with an efficient local optimisation method is especially quick and reliable for this task. In addition to comparing some different optimisation methods, the structures of clusters of atoms described by interatomic potential functions containing up to a few hundred atoms are calculated including some with some special symmetries. A number of applications are given including covalent carbon and silicon clusters, close-packed structures such as argon and silver and the two-component carbon–hydrogen system.

Introduction

This paper addresses the problem of determining the lowest energy configuration of a system of particles. This is an old problem related to that of determining the optimal arrangement of close-packed spheres, which has attracted the attention of pure mathematicians for many years. The “orange box” arrangement is generally thought to be the best way of packing the most number of spheres together into the least possible volume but a formal proof of this has proved to be very difficult. In this arrangement, the first layer consists of spheres in a close-packed triangular lattice. The second layer is formed by placing a sphere in the centre of the depressions left in the first layer, forming a lattice shifted with respect to the first layer. In the third layer the spheres can lie either directly above those in the first layer or in the unused depressions from either layer. In the first case this is known as hexagonal close packing (hcp) and in the second face-centred-cubic (fcc) because the spheres are arranged in a periodic cubic unit cell at the faces of a cube. In many materials, especially metals and frozen rare gases, the atoms also arrange themselves into this fcc structure. For other types of materials where directional bonding is important, such simple close-packed structures are not so common and complex geometrical arrangements occur depending precisely on the nature of the chemical interactions.

In this paper we address the problem from another angle. The spheres are now considered to be atoms that are regarded as point particles which interact through their interatomic forcefields. The forcefields are given as the starting point for the problem and are determined from interatomic potentials. The problem is therefore to find the optimal arrangement, i.e., the minimum potential energy, when N such particles interact. At the most sophisticated level, the forcefields are determined by the solution of Schrödinger's equation under various levels of approximation. This is a very complicated and time-consuming procedure which although becoming more standard can still take many days of computing time even to calculate the forces between 10s of atoms.

To overcome this problem, semi-empirical interatomic potential functions have been developed. They are of various forms and levels of detail depending on the type of material. For example, covalent materials such as silicon and carbon have strong directional bonds and can be modelled by so-called bond order potentials [1], [2], [3]; metals can be described by a model which describes the atomic nuclei as being embedded in a sea of electrons, the so-called embedded atom potentials [4], [5]. Some rare gases [6] can even be described by simple potentials that are pairwise additive such as the well-known Lennard-Jones potential. Multi-component materials can also be described. For more details on the forms of these types of potential function see the review article by Carlsson [7].

The energy minimisation task is an extremely challenging global optimisation problem for even moderately sized clusters, because the number of variables as well as the number of local minima are very high. The problem can therefore be used as a sample test problem for global optimisation. In order to calculate the minimum potential energy for small number of particles using some of the potential functions described below, we first attempted to use several recent stochastic global optimisation techniques. They are aspiration based simulated annealing [8], multi-level single linkage [9], topographical multi-level single linkage [10] and controlled random search [11]. Our numerical experiments with these algorithms suggested that they are successful and robust for problems with a small number of dimensions as well as problems with a small number of local minima. For problems containing up to five particles most of these algorithms were robust in locating the best minimum. However, as the number of particles increases their robustness diminishes. We will present the results of these algorithms in the comparison section. The results obtained by the algorithms suggested that for the energy minimisation problems with high dimensions, special types of algorithms are required. Therefore in this paper we will concentrate on two recent global optimisation techniques designed for energy minimisation problems. These are the topographical differential evolution [12] and the genetic algorithm [13]. There are several genetic algorithms have been designed for chemical cluster optimisation previously. For more details of these see the review articles by Judson [14], [15].

The most extensively tested energy minimisation problem is that using the Lennard-Jones potential. Extensive numerical studies have been carried out on this pair potential function, for some recent studies see [16], [17], [18]. The number of local minima for this problem is believed to be exponential in the number of atoms or molecules. The Lennard-Jones (L-J) function is simple to state and easily programmable, yet challenging and complex in the behaviour of its solutions.

Section snippets

Potential functions

In this section, we describe some well-known potential functions that can be used as challenging test problems for global optimisation.

Global optimisation algorithms

In this section we describe four algorithms that are suitable for use in solving the problem described above. They are multilevel single linkage (MSL) [9], topographical multilevel single linkage (TMSL) [10], topographical differential evolution (TDE) [12] and the genetic algorithm (GA) [13]. All are iteration (or generation) based algorithms. Of these algorithms, MSL and TMSL have a similar algorithmic structure. They generate random points from the search space per iteration. On the other

The search region Ω and initial atom positions

The search region for all the problem is constructed in the following way in a Cartesian system (x,y,z). Distances are measured in Å. The first atom is fixed at the origin. The second atom has y=z=0. The x co-ordinate position due to the second atom is taken as x1[0,4]. The third atom has z=0 and the second and third variables are such that its x and y co-ordinates are given by x2[0,4] and x3[0,4]. The coordinates of any other atom i are taken randomly to lie in[414i43,4+14i43],

Conclusions

Different algorithms have been compared for the important problem of determining the most stable configuration of a system of particles in space which interact through classical force-fields which are both attractive at large separation and repulsive when the particles lie close together. Modern global optimisation algorithms have been shown to obtain some of the correct configurations when the number of particles is small but the GA outperforms these for larger systems.

The algorithm would also

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