Clusters Jellium model

Data taken in another mass range also showed the inertness of AlJ7. While it was not produced by reaction of larger clusters, of which there are few, it was, however, found to be unreactive over a wide oxygen concentration range. In general, our results support the electron droplet Jellium model, although some anomalies (for... 

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... 

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,... 

G. Gerber By applying two-photon ionization spectroscopy with tunable femtosecond laser pulses we recorded the absorption through intermediate resonances in cluster sizes Na with n = 3,. 21. The fragmentation channels and decay pattern vary not only for different cluster sizes but also for different resonances corresponding to a particular size n. This variation of r and the fragmentation channels cannot be explained by collective type processes (jellium model with surface plasmon excitation) but rather require molecular structure type calculations and considerations. 

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. ... 

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 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. 

FIG. 11. Energy level diagram for the cluster modelling the Cu(lOO) surface, the bare clusters Cut, Cus and Cug evaluated with the standard MO-LCAO method to the left and the calculations for jellium to the right were done using the spherical jeUium model [72,74]. The MO-LCAO results include contribution from the 3d levels while the jellium model only include the free 4s electrons. The one-electron orbitals are characterized by the symmetry notation corresponding to C v symmetry. Since the calculation for jellium were performed within the LSD scheme the one electron levels are split. 

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