Elsevier

Chemical Physics Letters

Volume 313, Issues 3–4, 12 November 1999, Pages 655-664
Chemical Physics Letters

Non-additive effects in small gold clusters

https://doi.org/10.1016/S0009-2614(99)00957-4Get rights and content

Abstract

A post-Hartree–Fock second-order perturbational Møller–Plesset method was used to determine the many-body contributions in the interaction energy of Aun (n=3–6), clusters. Non-additive effects in these clusters were studied by decomposing the cluster binding energy in their n-body energy terms. It was found that the non-additive forces in small gold clusters are larger than the additive 2-body interactions and are the main factor to stabilize planar two-dimensional geometries as the lowest energy isomers. The effect of electron correlation in the interaction energy n-body decomposition was also analyzed and found to be important.

Introduction

Understanding the bonding of metal clusters is the starting point in the study of physical and chemical behaviors of these systems. Quantum-mechanical methods have been extensively used to calculate, through several levels of approximations, structural and electronic properties of the lowest energy isomers of small metal clusters [1]. In a typical calculation, the most stable isomers are determined through optimization of the total energy by varying the cluster geometries. One way to gain insight into the bonding and relative stability of isomers of small clusters is to analyze the cluster binding energy using its many-body contributions 2, 3. The decompostion of the cluster energy into 2-, 3-, and n-body terms allows for a systematic and quantitative description of the different effects which contribute to the cluster stability 2, 3.

Previous studies on the non-additive effects in small metal clusters have shown that many-body forces are fundamental to stabilize the cluster structure, as the case of Be4, in which the stronger attractive 3-body forces are decisive, since the 2- and 4-body forces are repulsive 4, 5, 6, 7. For some isomers of small lithium clusters, it was found that not only the 3-body but even the 4-body interaction energies are greater than the 2-body ones 7, 8. Also, for lithium and beryllium clusters, it was recently concluded that a significant contribution to their non-additive forces is the electron correlation [7]. In Agn (n=4–6), clusters, the most stable geometries are determined by competition of attractive additive and repulsive non-additive forces [9]. The larger magnitude of non-additive forces for three-dimensional (3D) conformations in comparison with two-dimensional (2D) ones, is the reason why for Agn (n=4–6), clusters the most stable geometries are planar [9].

Knowledge of the magnitude and character of the n-body terms is also important to construct ab initio many-body (AIMB) model potentials 10, 11, 12. A methodology to generate these potentials, based in the fitting of each n-body contribution of the ab initio calculated potential energy surface to an analytical expression, has been recently proposed and applied to the Ag6 cluster 10, 11, 12. Long-time molecular dynamics (MD) simulations (on a nanosecond timescale) have been performed using an AIMB potential to study the thermal stability, isomerization, and melting of Ag6[12]. This approach, which takes into account the n-body decomposition of the cluster binding energy, represents a real alternative to perform long MD simulations, in comparison with other ab initio MD methods 13, 14 which at present are only able to reach the picosecond time regime [14].

Investigation of the bonding properties of small gold clusters, from the decomposition in its n-body terms, is the first step to construct an AIMB potential for these systems and investigate their dynamical and thermal stability properties. On the other hand, the analysis of the n-body effects in the interaction energy and stability of small gold clusters and their isomers is useful to gain insight into the bonding properties of larger gold nanoclusters in which amorphous-like and ordered structures have been recently found as the lowest energy configurations 15, 16.

In this Letter, we study the non-additive effects on the binding energy of Aun (n=3–6), clusters using quantum-mechanical methods, which are briefly described in Section 3. In the next section the basic equations, on which the many-body decomposition of the cluster binding energy is based, are presented. Results of the latter decomposition for the Aun isomers are presented in Section 4. In Section 5, we present a summary of the present work.

Section snippets

Basic formulas

The total energy of an n-particle system can be represented as a finite sumE(n)=E1(n)+E2(n)+…+En(n)of its n-body terms, and is exact [9]. E1 represents the sum of energies of the separated particles,E1(n)=a=1{n}E(a).E2(n)=a<b{n}εab,whereεab=E(ab)−[E(a)+E(b)]=E(ab)−E1(ab),is the contribution of all pair n(n−1)/2 interactions εab and is normally defined as the additive energy. Higher-order terms, En(n), n>2, are the so called non-additive contributions and are represented in a similar way [9].

Computational methods

A post-Hartree–Fock (HF) second-order perturbational Møller–Plesset (MP2) method was used to determine the m-body energy terms of the lowest-lying isomers of Aun (n=3–6), clusters. A relativistic effective-core potential (RECP) of Hay and Wadt for the [Xe]4f14 inner electrons of Au (11-RECP) was introduced while the 11 5d106s outermost electrons were explicitly described through a double-zeta (DZ) basis set (3s3p3d/2s2p2d) [18]. We also tested the 19 valence-electron pseudopotential (19-RECP) as

Results

The lowest-lying isomers of the Aun (n=3–6), clusters optimized at the MP2 level of approximation are displayed in Fig. 1. Preliminary results on the optimization of these structures were reported elsewhere [20]. Several trial geometries of known planar and 3D structures of the Ia- 25, 26, 27, 28 and Ib-metal clusters 21, 22, 29 were used for the optimizations and, in most cases, geometry restrictions were relaxed during the optimizations. For some structures, two different electronic states

Summary

The combination of a DZ basis set, 11-RECP pseudopotential, and the MP2 method, to include electron correlation effects, has proven to be reliable for a qualitative and quantitative analysis of the lowest energy structures of Aun=2–6 clusters. This methodology was used to study non-additive effects in the binding energy of such systems. For Au3 we reproduced the almost degeneracy of the 2A1 and 2B2 electronic states, for which more accurate calculations confirmed the latter as the more stable,

Acknowledgements

The authors thank I.G. Kaplan for useful discussions. This work was supported by the DGSCA-UNAM Supercomputer Center, DGAPA-UNAM under Project IN101297, and CONACYT-Mexico under Grant 25083-E.

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