Metal clusters free-electron theory

Sharp drops after certain sizes in the abundance spectrum indicate enhanced stability of these clusters compared to neighboring sizes. We will try to understand this phenomenon from the behavior of valence electrons in the clusters by invoking simple quantum mechanical models. The simplest model one uses for valence electrons inside a bulk metal is the free-electron theory valence electrons of all the atoms are free to move over the entire volume occupied by the solid [11]. One can use a similar free electron model in case of metal clusters. As the simplest approximation, shape of the cluster can be taken as spherical, and the electrons strictly confined within the sphere. In this hard sphere model, the Schrbdinger equation describing the valence electrons is... 

A simple description of electrons in a solid is the model of a free electron gas in the lattice of the ions as developed for the description of metals and metal clusters. The interaction of electrons and ions is restricted to Coulomb forces. This model is called a jellium model. Despite its simplicity, the model explains qualitatively several phenomena observed in the bulk and on the surface of metals. For a further development of the description of electrons in solids, the free electron gas can be treated by the rules of quantum mechanics. This treatment leads to the band model. Despite the complexity of the band model, Hoffmann presented a simple description of bands in solids based on the molecular orbital theory of organic molecules that will also be discussed below. 

We showed the existence conditions for volume plasmons in the framework of the free-electron model but a generalization for surface plasmons can also be made in the framework of other models. Since the plasmon theory can be derived for the RPA, the existence of a plasmon can easily be checked by the RPA calculations. In agreement with the general theory presented, the ab initio RPA calculations carried out on alkali metal clusters of the size presented in this work and of considerably larger sizes do not exhibit plasmon-like excitations. 

The photoabsorption spectrum a(co) of a cluster measures the cross-section for electronic excitations induced by an external electromagnetic field oscillating at frequency co. Experimental measurements of a(co) of free clusters in a beam have been reported, most notably for size-selected alkali-metal clusters [4]. Data for size-selected silver aggregates are also available, both for free clusters and for clusters in a frozen argon matrix [94]. The experimental results for the very small species (dimers and trimers) display the variety of excitations that are characteristic of molecular spectra. Beyond these sizes, the spectra are dominated by collective modes, precursors of plasma excitations in the metal. This distinction provides a clear indication of which theoretical method is best suited to analyze the experimental data for the very small systems, standard chemical approaches are required (Cl, coupled clusters), whereas for larger aggregates the many-body perturbation methods developed by the solid-state community provide a computationally more appealing alternative. We briefly sketch two of these approaches, which can be adapted to a DFT framework (1) the random phase approximation (RPA) of Bohm and Pines [95] and the closely related time-dependent density functional theory (TD-DFT) [96], and (2) the GW method of Hedin and Lundqvist [97]. 

The equations must be solved iteratively since the density enters the expressions for Veff. The free electron gas, the Fermi-Thomas, and the DFT theory will be treated in more detail in Chapters 9 and 10. The DFT theory has been used recently in a number of calculations of molecule-surface interaction (sec Table 4.7). Some studies just involves the chemisorption to a small cluster of metal atoms [189], and it is questionable how well this represents the behavior of a bulk metal. 

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