Sodium clusters electronic properties

The first depletion spectra obtained for neutral sodium clusters N = 2-40 were characterized by structureless broad features containing one or two bands. The results were interpreted in terms of collective resonances of valence electrons (plasmons) for all clusters larger than tetramers [2, 52-55]. The analogies between findings for metallic clusters and observations of giant dipole resonances in nuclei have attracted a large attention. Therefore the methods employed in nuclear physics, such as different versions of RPA in connection with the jellium model, have also been applied for studying the optical properties of small clusters. Another aspect was the onset of conductivity in metal-insulator transitions. 

Early LSDA static pseudopotential approaches to sodium microclusters date back approximately 20 years [122], see Appendix C. It would be misleading to consider LDA calculations as the natural extension of jellium models. However, the global validity of the latter cannot but anticipate the success of the former. Clearly, these should also clarify the role of the atomic structure in determining the electronic behavior of the clusters and the extent to which the inhomogeneity of the electron distribution is reflected in the measurable properties. Many structural determinations are by now available for the smaller aggregates, made at different levels of approximation and of accuracy (e.g. [110, 111], see Appendix C). The most extensive investigation of sodium clusters so far is the LDA-CP study of Ref. [123] (see Appendix C), which makes use of all the features of the CP method. Namely, it uses dynamical SA to explore the potential-energy surface, MD to simulate clusters at different temperatures, and detailed analysis of the one-electron properties, which can be compared to the predictions of jellium-based models. 

The glass-coloring experiments have been performed with gold, silver, nickel and other metals, which are much more difficult to handle theoretically than the alkalis. Among the latter, sodium is the best representative of the nearly free electron gas or jellium model which forms the basic assumption of some of the articles found here. Therefore this review is restricted to sodium clusters, and more specifically to their optical and thermal properties. 

It is also meaningful to look at some of the other electronic properties such as AIP and AEA of sodium clusters as well. These electronic properties of sodium clusters are calculated at B3LYP/6-31 + G(d) level of theory and are reported in Table 11.5. Generally, the AIP of sodium is found to decrease and AEA to increase with increasing cluster size. A probable justification for the existence of magic clusters is offered on the basis of DFT hardness descriptor [35]. A species with high IP and low EA is... 

Most of the electronic properties of molecules and clusters depend on their shape and size. Even though there are varied definitions for shape and size of a species, a widely used definition is the MED topological one. Boyd has used the MED contour to discuss the relative sizes of atoms [49]. On the other hand for molecules, Bader et al. [32] proposed the surface of constant electron density (0.002 a.u.) to describe shape and size of diatomic molecules. Here in the present paper, for justifying the variation in polarizability and to quantify the delocalization of the valence electrons in these clusters, the volume enclosed within the 0.0001 a.u. MED isosurface is quantified. The MED contours at 0.01,0.001, and 0.0001 a.u. of the Lij, Li4, Na2, and Na4 are depicted in Eigure 11.8. The respective volume enclosed within the 0.0001 a.u. MED isosurfaces of lithium and sodium clusters with sizes ranging from 2 through 40 is reported in Table 11.7. 

Electron configuration of metal clusters with itinerant electrons is represented in terms of the phenomenological shell model (PSM). The main assumption of this model is that the itinerant electrons are confined in a box according to the cluster shape, and these determine the properties of the given cluster to a great extent. This model was developed to explain the observed stability patterns of sodium clusters and has been successfully applied in other elements (such as Li, Al, Cu) and properties (such as ionization energy, electronaffinity). Furthermore, it was formulated for different cluster shapes and also for doped metal clusters. In this chapter, we aim to demonstrate that the aromaticity of metal clusters can be interpreted in terms of the PSM, which can be used to formulate the criteria to obtain a closed electronic structure in different cluster shapes. Therefore, the PSM provides the different electron... 

A precise theoretical and experimental determination of polarizability would provide an important probe of the electronic structure of clusters, as a is very sensitive to the presence of low-energy optical excitations. Accurate experimental data for a wide range of size-selected clusters are available only for sodium, potassium [104] and aluminum [105, 106]. Theoretical predictions based on DFT and realistic models do not cover even this limited sample of experimental data. The reason for this scarcity is that the evaluation of polarizability by the sum rule (46) requires the preliminary computation of S(co), which, with the exception of Ref. [101], is available only for idealized models. Two additional routes exist to the evaluation of a, in close analogy with the computation of vibrational properties static second-order perturbation theory and finite differences [107]. Again, the first approach has been used exclusively for the spherical jellium model. In this case, the equations to be solved are very similar to those introduced in Ref. [108] for the computation of atomic polarizabilities. Applications of this formalism to simple metal clusters are reported, for instance, in Ref. [109]. 

Clusters of alkali metals and especially of sodium are the most studied of all. From the theoretical point of view, sodium is the one most amenable to treatments with simple models. The free-electron behavior known for the bulk phase has suggested that jellium-like models could also be suitable for small-size aggregates. By means of these models, in fact, a large variety of measurable properties have been calculated. This in turn has allowed the approximations used to be tested at several levels [121]. Two comprehensive and very instructive reviews have been dedicated to both experimental and theoretical approaches to simple metal clusters with an emphasis on phenomenological aspects and jellium or jellium-derived models [4, 5]. Here we shall report on DFT calculations that go beyond the assumption of a homogeneous, positively charged background. 

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