The Formation of Neutral Copper Clusters from Experimental Binding Energies and Reactivity Descriptors Pablo Jaque and Alejandro Toro-Labbe ´ * Laboratorio de Quı ´mica Teo ´ rica Computacional (QTC), Facultad de Quı ´mica, Pontificia UniVersidad Cato ´ lica de Chile, Casilla 306, Correo 22, Santiago, Chile ReceiVed: August 1, 2003; In Final Form: NoVember 26, 2003 In this paper we study and rationalize the formation of neutral copper clusters from dimer to nonamer using the available experimental data of binding energies and electronic properties. A complete and consistent picture of the formation of copper clusters in terms of the changes in chemical potential and hardness emerges indicating that the one-atom growth reactions are mainly driven by changes in hardness. An analytic expression for the binding energy as a function of the cluster size is proposed and used to predict the growth pattern of copper clusters. 1. Introduction The interest in the study of small metal clusters has grown considerably in recent years because new experimental and theoretical techniques have been developed allowing detailed characterizations of these types of systems. 1,2 Much information is now available concerning clusters’ spectroscopy, structure, and their chemical reactivity toward small molecules. Transition metal clusters are particularly interesting for their potential use in many processes such as heterogeneous catalysis, organome- tallic chemistry, or new electronic materials. 1,3 An interesting topic in clusters science is the study of the evolution of clusters’ properties when increasing the number of constituent atoms. Neutral even-numbered copper clusters are closed-shell systems, whereas odd-numbered copper clusters are open-shell systems. 4-10 In investigations on ion abun- dances, 11-13 ionization potentials, 14-18 electron affinities, 19 and binding energies of neutral and charged copper clusters (Cu n ), 20,21 a typical behavior known as even-odd alternation has been found. In this paper our main goal is to provide a new viewpoint to understand the formation reaction Cu n-1 + Cu f Cu n (n ) 2-9); in doing so, the observed even-odd alternation of binding energies will be analyzed in detail. Our approach consists of the rationalization of the available experimental data of binding energies through the use of reactivity descriptors calculated from experimental data of ionization potentials and electron affin- ities. 14-21 As a result of this approach, correlations between energetic and electronic properties can emerge; such correlations are expected to be useful in understanding the formation reaction of copper clusters and may provides new elements to rationalize growth reactions of metal clusters. In a recent paper 10 we have characterized neutral copper clusters (Cu n ; n ) 1-9) in terms of calculated chemical reactivity descriptors defined within the framework of density- functional theory (DFT), usually used to study the reactivity pattern of molecules and molecular aggregates. 10,22,23 The set of reactivity descriptors, electronic chemical potential (µ), chemical hardness (η), and electrophilicity index (ω), together with chemical reactivity principles such as the principle of maximum hardness, define quite powerful tools to analyze and rationalizedifferentprocessesexperiencedbycomplexsystems. 10,23-26 A formation reaction can be seen as resulting from the combination and redistribution of atom’s and fragment’s electron densities, giving rise to a new electronic distribution from which the electronic descriptors of the new aggregate are derived. DFT 27 is quite well suited to describe such electronic reorga- nization processes as it provides the basis for rigorous math- ematical definitions of reactivity descriptors such as chemical potential, electronegativity, 28 chemical hardness, 29 softness (S ) 1/η), and so forth.; all these are well-established global quantities in chemical reactivity studies and will be used in this paper to study the formation process of copper clusters Cu n (n ) 2-9) in one-atom growth reactions. The chemical potential characterizes the escaping tendency of electrons from equilibrium; on the one hand it is the Lagrange multiplier associated with the normalization constraint in DFT, on the other hand it is the negative of the electronegativity ( )-µ). 28 Hardness represents the resistance to charge transfer, whereas softness is a measure of the propensity of the system to change its electronic distribution and has been qualitatively related to the polarizability R. 27,29-31 The electrophilicity index measures the stabilization of a system when it acquires electronic charge from the surroundings. 32 A major focus of attention and discussion in the application of DFT to chemical reactivity is the principle of maximum hardness (PMH) proposed by Pearson. 30 The PMH asserts that molecular systems at equilibrium present the highest value of hardness; the PMH is a widely accepted electronic structure principle that in most cases complements the minimum energy criterion for molecular stability. The PMH provides an inde- pendent criterion to rationalize the behavior of chemical reactions. In the next section we present a summary of the theoretical elements we use here; section 3 contains the results and discussion, and in section 4 we draw our conclusions. 2. Theoretical Background In DFT, the chemical potential and molecular hardness for an N-electron system with total energy E and external potential υ(r b ) are defined as the following first and second derivatives * Author to whom correspondence should be addressed. E-mail: atola@puc.cl. 2568 J. Phys. Chem. B 2004, 108, 2568-2574 10.1021/jp036260v CCC: $27.50 © 2004 American Chemical Society Published on Web 01/31/2004