Metal clusters jellium model

With respect to the thermodynamic stability of metal clusters, there is a plethora of results which support the spherical Jellium model for the alkalis as well as for other metals, like copper. This appears to be the case for cluster reactivity, at least for etching reactions, where electronic structure dominates reactivity and minor anomalies are attributable to geometric influence. These cases, however, illustrate a situation where significant addition or diminution of valence electron density occurs via loss or gain of metal atoms. A small molecule, like carbon monoxide,... 

Abstract This chapter reviews the methods that are useful for understanding the structure and bonding in Zintl ions and related bare post-transition element clusters in approximate historical order. After briefly discussing the Zintl-Klemm model the Wade-Mingos rules and related ideas are discussed. The chapter concludes with a discussion of the jellium model and special methods pertaining to bare metal clusters with interstitial atoms. 

Keywords Jellium model Metal clusters Wade-Mingos rules Zintl ions Contents... 

The jellium model of the free-electron gas can account for the increased abundance of alkali metal clusters of a certain size which are observed in mass spectroscopy experiments. This occurrence of so-called magic numbers is related directly to the electronic shell structure of the atomic clusters. Rather than solving the Schrodinger equation self-consistently for jellium clusters, we first consider the two simpler problems of a free-electron gas that is confined either within a sphere of radius, R, or within a cubic box of edge length, L (cf. problem 28 of Sutton (1993)). This corresponds to imposing hard-wall boundary conditions on the electrons, namely... 

Various refinements of the above model have been proposed for example, using alternative spherical potentials or allowing for nonspherical perturbations,and these can improve the agreement of the model with the abundance peaks observed in different experimental spectra. For small alkali metal clusters, the results are essentially equivalent to those obtained by TSH theory, for the simple reason that both approaches start from an assumption of zeroth-order spherical symmetry. This connection has been emphasized in two reviews,and also holds to some extent when considerations of symmetry breaking are applied. This aspect is discussed further below. The same shell structure is also observed in simple Hiickel calculations for alkali metals, again basically due to the symmetry of the systems considered. However, the developments of TSH theory, below, and the assumptions made in the jellium model itself, should make it clear that the latter approach is only likely to be successful for alkali and perhaps alkali earth metals. For example, recent results for aluminium clusters have led to the suggestion that symmetry-breaking effects are more important in these systems. ... 

The linear photoresponse of metal clusters was successfully calculated for spherical [158-160, 163] as well as for spheroidal clusters [164] within the jellium model [188] using the LDA. The results are improved considerably by the use of self-interaction corrected functionals. In the context of response calculations, self-interaction effects occur at three different levels First of all, the static KS orbitals, which enter the response function, have a self-interaction error if calculated within LDA. This is because the LDA xc potential of finite systems shows an exponential rather than the correct — 1/r behaviour in the asymptotic region. As a consequence, the valence electrons of finite systems are too weakly bound and the effective (ground-state) potential does not support high-lying unoccupied states. Apart from the response function Xs, the xc kernel /xc[ o] no matter which approximation is used for it, also has a self-interaction error. This is because /ic[no] is evaluated at the unperturbed ground-state density no(r), and this density exhibits self-interaction errors if the KS orbitals were calculated in LDA. Finally the ALDA form of /,c itself carries another self-interaction error. 

Particularly exciting in cluster science has been how the simple jellium model has been able to in first order explain many quantities for clusters of alkali and noble metals as the appearance of magic numbers. Examples were given in Fig. 3. We have in connection with our studies of the reactivity of... 

Clusters are studied in several forms. A study of the ionization energy and electron affinity of a metal cluster in the stabilized jellium model was recently performed by Sidl et al. [83]. A strictly variational procedure for cluster embedding, based on the extended subspace approach, has been presented by Gutdeutsch, Birkenheuer, and R6sch[84]. Initially used with the tight-binding model Hamiltonians, it has the potential to be extended to real Hamiltonians. 

Although the jellium model has been successful in explaining very many properties of clusters of simple metals (see, e.g., 62,63), the clusters are after all made up of discrete atoms and, therefore, a highly relevant question is whether the predictions of the jellium model will hold when explicitly including the positions of the atoms in the calculations. 

Banerjee and Harbola [69] have worked out a variation perturbation method within the hydrodynamic approach to the time-dependent density functional theory (TDDFT) in order to evaluate the linear and nonlinear responses of alkali metal clusters. They employed the spherical jellium background model to determine the static and degenerate four-wave mixing (DFWM) y and showed that y evolves almost linearly with the number of atoms in the cluster. 

A more complex situation was found for Na Pb. The unusual stability of this cluster was explained by comparison with the analogous Mg compound (Albert et al. 1995) and was found to originate in a larger charge transfer to the more electronegative lead and a larger polarizability of the Pb atom. In a combined experimental and theoretical study it was shown for the clusters Na Au and Cs Au that certain properties of the bulk are qualitatively present at the level of small clusters (Heiz et al. 1995). While the Na compounds show metallic behaviour, and the electronic structure can be described by means of the jellium model, in the Cs-Au clusters an ionic bond is most prominent. 

When such clusters are formed in beams, essentially by many slow collisions of alkali atoms within a nozzle, it is found experimentally (by analysing the mass distribution using a time of flight spectrometer) that there are discontinuities in the intensity distribution of the peaks from which abundances can be computed for each cluster size. These discontinuities correspond to enhanced stability of metallic clusters around specific sizes (8, 20, 40, 58, etc). They are the same for all the different alkali metals, and are therefore referred to as magic numbers see fig. 12.5. They also turn out to be the same (at least, for the first few magic numbers) as those observed in nuclear physics. This similarity has led to an explanation based on the shell model and to the suggestion that the jellium model can be used to account for the properties of metallic clusters [683]. 

One can develop a particularly simple scheme by using the assumption of spherical symmetry together with the jellium model of solid state or nuclear physics to compute the effective potential for clusters of different sizes. In this model, the electrons are treated as free particles by analogy with the conduction band of the solid and the ionic structure within the cluster is completely neglected. This obviously results in a great simplification of the problem, especially if the system is spherical, and might be thought too drastic an approximation. In fact, the jellium model only applies to a specific class of clusters (which we call metallic), but was of enormous importance to the history of the field as it revolutionised cluster physics. 

The electronic charge distribution turns out (see fig.12.7) to exhibit oscillations, called Friedel oscillations, due to the sharp edge in the positive charge distribution at the surface of the cluster. It is possible to take the jellium model somewhat further than described here by allowing for the back reaction of the negative charge distribution on the positive charges in the self-consistent procedure. Note, however, that the jellium model is not appropriate for all clusters made from metal atoms. 

Within the jellium model for metal clusters [95,53] as described in the introduction, the positive background potential is in a first approximation normally chosen as a spherical shape of the following form... 

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