Spherical 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,... 

Since each atom in a f.c.c. array of purely metallic atoms is the same as every other atom (except at the surface), only a representative positive ion needs to be considered. Let it interact with a spherical portion (radius = R) of the electron gas which has a density of one electron per ion. This is called the jellium model. 

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

On the practical side, neither the ADA nor the WDA was widely applied in general to many-electron, realistic systems, due to their complicated functional forms.98,136-141,260,314-318,329-345 Only very recently did efficient implementations of the WDA become available.338 342 Even today, the ADA is still a museum artifact, which has been applied only io spherical atomic species and the spherical jellium model.31 316-346 The main obstacle lies in Eq. (110), where in addition to the recursion problem, one needs to do the integration over all space of r for every point r, yielding a numerical cost scaling quadratically, 0( M2), with respect to the integration grid size M. A straightforward application of the... 

This model was, perhaps, first suggested by Pushka and Nieminen [16] in conjunction with the jellium model of C60- Later on, it was used, independently, in another work [47], At a later stage, the idea was greatly extended to numerous studies of various aspects of structure and photoionization of atoms A encaged in various spherical fullerenes [15,20— 22,27-33,42-44]. 

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. 

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. 

Using a tight-binding model together with a Monte Carlo growth method (see Section 2.7), Poteau and Spiegelmann30 studied Na clusters with 4 < N < 21. For the spherical jellium model, magic numbers are found for N... 

The differences in the stable geometries of the AM and AE clusters have been investigated from the electronic structure view point. Ekhardt and Penzar, using a self-consistent jellium model, reported a more stable prolate structure than the spherical one for Na4 (25). The model placed four valence electrons of the Na4 cluster into a spherical potential. Two electrons occupy the I5 shell in the spherical potential and the other two electrons are accommodated in the p shell. Prolate distortion splits the I/7 levels and then the lowered Ip level is filled with two electrons. Therefore, the Na4 cluster prefers the prolate deformation. Using a molecular orbital method, Rao and Jena came to a conclusion which is consistent with the jellium results (13). The Li4 cluster adopts a planar structure while the Be4 cluster has a close packed structure since the latter cluster has eight valence electrons and the molecular orbitals corresponding to the p shell for the Jellium model are completely filled with the electrons. 

The description of the metal is improved considerably if metallic structure is introduced by accounting for the local attractive force of the metal atoms on the free electron gas. This corresponds to the jellium model with pseudopotentials. Each metal atom in the lattice is pictured as being surrounded by a spherical volume Fc in which electrostatic effects may be ignored. Outside of the sphere the metal atom behaves as a point charge of charge number n. Thus, it has a pseudopotential < )(., where... 

A complementary approach is the Tensor Surface Harmonic Theory [19], based on the linear combination of atomic orbitals (LCAO) model, which explicitly incorporates the atomic positions. A set of atomic cores on the surface of a sphere are considered, and a basis set of s atomic orbitals used. If only these s orbitals are used, then the results are identical to the spherical jellium model. The three most stable orbitals are respectively 1,3 and 5 fold degenerate, leading to closed shells at 2, 8 and 18 electrons. 

This model tends to predict more magic numbers than are actually seen in the mass spectrum. The reason for this may have geometric origins since only certain nuclearities can adopt highly spherical structures which coincide with a closed shell of electrons. When the cluster cannot adopt a highly spherical structure, a splitting of the jellium shells occurs, leading to some instability. A further and more crucial limitation is that the spherical jellium model provides no direct information on the structure, even for clusters with closed shells. 

Extensions to the spherical jellium model have been made to incorporate deviations from sphericality. Clemenger [15] replaced the Woods-Saxon potential with a perturbed harmonic oscillator model, which enables the spherical potential well to undergo prolate and oblate distortions. The expansion of a potential field in terms of spherical harmonics has been used in crystal field theory, and these ideas have been extended to the nuclear configuration in a cluster in the structural jellium model [16]. 

The Jahn-Teller theorem indicates that when a degenerate shell is not completely filled, a distortion away from spherical symmetry occurs, leading to a loss of degeneracy. The splitting pattern produced by the Jahn-Teller type distortion is identical to that predicted by the crystal field approach used in the structural jellium model. 

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. 

This interpretation helps to reconcile the different models on the one hand, we have those inspired by quantum chemistry, with clearly defined structures and, on the other, statistical mean field approaches such as the jellium model. Clearly, the two do not apply together, but, by averaging over many isomers, one can arrive at a nearly-spherical shape at finite temperature. 

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... 

An accurate selfconsistent potential can be constructed by applying DFT [3,4,11] within the context of the spherical jellium model (SJM) [12]. In this model the background of positive ions is smeared out over the volume of the... 

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... 

Fig. 1. Self-consistent effective potential for Na2o in the spherical jellium model. The occupied electronic shells are indicated, as well as the lowest unoccupied shell.

Fig. 2. Relative stability djlN) = E(N + 1) + E N - I) - 2 (JV) as a function of cluster of size in the spherical jellium model.

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. 

The first supershell node occurs at iV as 850), Calculations by Nishioka et al. [17], using a nonselfconsistent Woods-Saxon potential (instead of the spherical jellium model) give N 1000. This node has been observed, although the experiments also show some internal discrepancies the first node is located at IV 1000 in Ref. [15] while it is at iV 800 in [16]. The experimental discovery of supershells confirms the predictions of nuclear physicists. However, supershelis have not been observed in nuclei due to an insufficient number of particles. In summary, the existence of supershelis is a rather general property of a system form by a large number of identical fermions in a confining potential. 

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. 

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