Sodium clusters jellium model

Fig. 5.3 The total energy per atom of sodium clusters versus the number of atoms in the cluster, evaluated within the self-consistent jellium model. (From Cohen (1987).)...

Using the spherical jellium model explain the expected special stability of sodium clusters containing the magic number of atoms 2, 8, 18, 20,... 

Before leaving the jellium model, it seems appropriate to mention some results for sodium clusters of up to 22000 atoms. The abundances observed can be explained in terms of the extra stability associated with both completed jellium shells (for less than about 3000 atoms) and with completed icosahedral or cuboctahedral geometric shells for larger sizes. The results are especially interesting because of the observation of supershells that occur because of an interference effect, which has been explained using semiclassical arguments. ... 

The spherical jellium model has been applied to alkali metal clusters by many authors (see Ref. [6]). Fig. (1) shows the self-consistent effective potential for a sodium cluster with twenty atoms. The degenerate levels are filled up to electron number = 20. In a spherical cluster with 21 electrons, the last electron will have to occupy the If level above (dashed line). This electron is less... 

The effect of finite temperature on the shells and supershelis has been analyzed by Genzken for sodium clusters. For this purpose, calculations of the cluster free energy were performed by treating the valence electrons as a canonical ensemble in the heat bath of the ions [23]. (The spherical jellium model is even better at finite temperature.) Finite temperature leads to decreasing amplitudes of shell and supershell oscillations with increasing T. This is particularly important in the region of the first supershell node at N 850, which is smeared out already at a quite moderate temperature of T = 600 K. However, temperature does not shift the positions of the magic numbers. 

The strong fluctuation of IP or of the mass abundance is an electronic-structure effect, reflecting the global shape of the cluster, but not necessarily its detailed ionic structure. This is demonstrated in Fig. 6, where the ionization potentials of sodium clusters obtained by the spheroidal jellium model [32] are compared with their experimental values [46]. The odd-even oscillation of IP for low N is reproduced well. The amplitude of these oscillations is exaggerated, but this is corrected by using the spin-dependent LSDA, instead of the simple LDA [47]. The same occurs for the staggering of d2 N) [48]. 

Fig. S. Comparison of the ionization potential of sodium clusters, obtained with the spheroidal jellium model [32] and experiment [46]. Redrawn from data in Ref. [32].

Table 3. Electric dipole polarizabilities in units of R, of neutral sodium clusters in the spherical jellium model (SJM) and in a jellium model with finite surface thickness (FSJM)...

A splitting of the dipole resonance into three peaks has been observed in some sodium clusters [64], This observation is interpreted as corresponding to collective vibrations of the valence electrons in the directions of the principal axes of a triaxially deformed cluster, and has motivated an extension of the deformed jellium model to fully triaxial shapes. Lauritsch et al. [65] have applied this model to Nai2 and Nai4. 

Within the shell-model of the electronic structure of clusters of monovalent metals, the ionization potential drops to a low value between sizes and N(, -I-1, where N. indicates a closed-shells cluster. The electron affinity, on the other hand, drops between Nj — 1 and Nc, since the cluster with size N — 1 easily accepts an extra electron to close its nearly-filled external shell. Consequently, the cluster of size N has a large ionization potential and a low electron affinity and will be inert towards reaction. One then expects peaks in a plot of 1 — A versus N for closed shell clusters. The shell effects arc clearly displayed in a Kohn-Sham density functional calculation. Figure 10 shows the results of such a calculation for jellium-like Sodium clusters using the non-local WDA description of exchange and correlation. This calculation employed the Przybylski-Borstel version of the WDA see reference 30 for details). The peaks in I — A occur at the familiar magic clusters with N = 2, 8,18, 20,34,40 and 58. It is well... 

Fig. 11. Fukui function (see Eq. (46)) for two sodium clusters (Na4o and Naioo) in the spherical jellium model. The maximum in the Fukui function occurs at the cluster surface. This is, evidently the region of the cluster most susceptible to radical attack...

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. 

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]. 

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. 

Early interest in heteroatom clusters having alkali metals as the host was academic rather than dictated by precise observations. The main question regarded the extent to which the jellium-derived shell model retained its validity. However, this question was approached on the basis of oversimplified structural models in which the heteroatom (typically a closed-shell alkali-earth such as Mg) was located at the center of the cluster [235, 236]. In this hypothetical scheme, the perturbation of the electronic structure relative to that of the isoelectronic alkali cluster is somewhat trivial for instance, in the Na Mg system the presence of Mg would only alter the sequence of levels of the shell jellium model from Is, Ip, Is, 2s,. .. (appropriate to sodium clusters) to Is, Ip, 2s, Id,. .. (see also [236]). This would lead to the prediction that Na6Mg and NasMg are MNs. 

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. 

The data discussed here give a fairly complete map of the optical response of sodium cluster ions. The response has been studied as a function of three parameters temperature, size and charge of the cluster. The overall result is that good agreement with the jellium model is obtained. Exceptions are the splitting of the resonances at low temperature and the lifetime/width problem, both of which are influenced by the detailed arrangement of the atoms. The thermodynamic experiments are too new, so that no satisfactory understanding has yet evolved. 

The calculated finite temperature sodium cluster polarizabilities show characteristic minima at the dimer and octamer as expected from the jellium model. 

Sodium can be considered the prototype of metal clusters for which the undirectional bonding due to the 3s valence electrons guarantees that the arguments above provide a good starting point for rationalizing the properties of Na clusters. Thus, packing effects, that can be modeled accurately with a simple potential that does not include electronic orbitals, will be responsible for the structures, and the particularly stable structures can be identified with a jellium model for delocalized electrons. 

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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