Self-consistent 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).)...

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. 

Brack, M. (1993). The physics of simple metal clusters Self-consistent jellium model and semiclas-sical approaches. Reviews of Modern Physics, 65, 677-732. 

Fig. 4.2. Charge distribution and surface potential in a jellium model, (a) Distribution of the positive charge (a uniform background abruptly drops to zero at the boundary) and the negative charge density, determined by a self-consistent field calculation. (b) Potential energy as seen by an electron. By including all the many-body effects, including the exchange potential and the correlation potential, the classical image potential provides an adequate approximation. (After Bardeen, 1936 see Herring, 1992.)...

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. 1 Asymptotic structure coefficients as(j8), c ks,xG ) crw03), a CS), and aKS gOS) as fimction of barrier height parameter )S =VW/eF, where W is the barrier height and eF the Fermi energy. Corresponding values of the Wigner-Seitz radius rs for jellium and structureless-pseudopotential models over the metallic range of densities are also given. The relationship between rs and ff is via self-consistent calculations in the local density approximation for exchange-correlation.

The methodology focuses, as many density-functional schemes do, on the key role of the electron density. The Schrodinger equation is then solved self-consistently in the Kohn-Sham scheme.86 Initial approaches dealt with a jellium-adatom system, which would at first sight seem rather unchemical, lacking microscopic detail. But there is much physics in such an effective medium theory, and with time the atomic details at the surface have come to be modeled with greater accuracy. 

One of the earliest treatments of a metal surface was based upon a jellium model (Bardeen, 19.36). If the electron gas terminated abruptly at the surface of the jellium there would be no net potential to contain the electrons in the metal. Therefore the electron gas extends beyond the metal, giving a dipole layer, as illustrated in Fig. 17-5. Bardeen attempted the self-consistent calculation of the resulting potential. It should be mentioned that the Fermi-Thomas approximation is not adequate for this task and was not used by Bardeen it is not difficult to see that it would predict the Fermi energy to be at the vacuum level, corresponding to a vanishing work function. 

Fig. 3. Stopping power as a function of the distance to the top-most layer for V = 1 a.u. protons traveling parallel to the Cu (111) surface. The solid line is the result of the model in which the surface band structure of the target is considered, the short-dashed line is obtained neglecting the surface state in the calculation, the long-dashed line is the result of using the jellium model and constructing XoCG, z, z , within the RPA using the self-consistent solutions for a finite step potential (see text), and the dot-dashed line is obtained in the jellium model within the SRM.

Fig. 6. Self-consistent surface charge density in the jellium model for K (solid line) and A1 (dashed line) (Lang and Kohn, 1 y70). Distance is measured in Fermi wavelengths from the positive background edge charge density is measured relative to the bulk density pa-...

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. 

Ekardt W 1984 Work function of small metal particles self-consistent spherical jellium-background model Phys. Rev. B 29 1558... 

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.

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. 

As in the simpler jellium model, we retain the simple description of independent electrons, each moving in a confining potential U(r). Here however, that potential is not an arbitrary, made-up model potential chosen to fit data or to make a calculation convenient this potential includes such effects as the exchange interaction with the other electrons. In this, the present approach is quite reminiscent of the Hartree-Fock self-consistent procedure, which will be described next. There is one essential difference. Unlike the Hartree-Fock procedure, here the exchange term is approximated as a local function, depending only on the one-electron density. This approximation yields fast convergence to a self-consistent density. As in the Hartree-Fock... 

Clemenger s model is, however, non-selfconsistent. Ekardt and Penzar [41,4.3] have extended the jellium mode to account for spheroidal deformations. In this model the ionic background is represented by a distribution of positive charge with constant density and a distorted, spheroidal, shape. The advantage with respect to Clemenger s model is that the spheroidal jellium model is parameter-free and that the calculation of the electronic wave functions is performed self-consistently using the density functional formalism. The distortion parameter is determined by solving the Kohn-Sham equations for different... 

The properties of metal clusters within the jellium model were first studied within the local density approximation (LDA) to the density functional theory (DFT) [3]. This means the following set of equations has to be solved self-consistently, starting from a proper initial density (for details see [3]). The total electronic density p(r) obeys the subsidiary condition... 

Figure 1.10 Experimental ionization potential (IP) of K clusters [27]. Note that the total IP shows strong discontinuities at magic numbers, whereas the predicted rise of the IP between two magic numbers is missing. As has been shown by Penzar and Ekardt [31] this can be reproduced within the self-consistent deformed Jellium model. Reproduced by permission of Academic Press...

It should be stressed that another successful approach applied to metal clusters is the jellium model [14], which is often mistakenly considered as the PSM. An important difference between the two models is that although the jellium model treats the electron-electron interaction self-consistently in a positive background potential, the PSM is a one-electron approximation using a (not necessarily homogeneous) confining potential. Thus in order to use the PSM, we solve the Schrbdinger equation for the one-electron in a box problem using different box shapes. 

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