Jellium model, metal cluster potentials

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. 

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. 

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. 

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... 

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... 

Experiments on noble metal clusters (Cun, AgN, Aun) indicate the existence of shell-effects, similar to those observed in alkali clusters. These are reflected in the mass spectrum [10] and in the variations of the ionization potential with N. The shell-closing numbers are the same as for alkali metals, that is N = 2,S,20,40, etc. Cu, Ag and Au atoms have an electronic configuration of the type nd °(n + l)s so the DFT jellium model explains the magic numbers if we assume that the s electrons (one per atom) move within the self-consistent, spherically symmetric, effective jellium potential. 

It was mentioned in Sect. 4 that electronic-shell effects appear in the mass abundance [10,43], ionization potentials [88], and electron affinities [89] of noble metal clusters that are very similar to those observed for alkalis. These can be readily interpreted within the spherical jellium model if we treat the noble metal atoms as monovalent, that is, each atom contributes its external s-el tron only. Even more, odd-even effects are also observed for small N in the properties mentioned above, and have been explained by Penzar and Ekardt [32] within the context of the spheroidally deformed jellium model. 

The isolated potassium atom has an ionization potential (IP) of 4.44 eV while the work function of the bulk metal, the energy required to extract an electron with zero kinetic energy, is 2.3 eV. The experimental measm-ements of IP as a function of cluster size show evidence for magic numbers so that to discuss this figinre realistically, one needs a model at least as sophisticated as the jellium model discussed later in Sec. 5. Here we just want to concentrate on the mean smooth behavior of the ionization curve. The energy necessary to remove an electron from a neutral sphere of radius R to infinity is j lR. The radius of a spherical cluster should scale with the number n of atoms as /2 It is therefore suggestive to take the ionization potential to scale with size as W (eV) = 2.3 + Here W is the work function or... 

Apparently the detailed configuration of the atomic arrangement in simple metal clusters does not seem to play an important role in the study of their physical properties. The spherical jellium model is very successful in correlating the prominent features of the ionization potential and also the main features of the mass spectra. However, there is also evidence of some features that the spherical assumption is unable to explain. Whenever a top-shell is not completely filled (N 2, 8, 20, 40, 58, 92,...) the electronic density becomes non-spherical, which in turn leads to an ellipsoidal distortion of the ionic background. This Jahn-Teller-type distortion, similar to those observed for molecules and nuclei, leads to a splitting of all spherical shells into spheroidal sub-shells [41]. Ellipsoidal clusters are prevalent for open-shell configurations. 

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... 

The next physical quantities of interest are the size dependence of the ionization potential and of the electron affinity because these quantities can be related to the chemical reactivity of metal clusters. Within the DFT jellium model the size dependence of the ionization potential is easily obtained from two total energy calculations ... 

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