The jellium model

The simplest model of a metal surface is the jellium model, which is a Sommerfeld metal with an abrupt boundary. In provides a useful semiquanti-tative description of the work function and the surface potential (Bardeen, 1936). It validates the independent-electron picture of surface electronic structure Essentially all the quantum mechanical many-body effects can be represented by the classical image force, which has been discussed briefly in Section 

The simplest model of metals is the Sommerfeld theory of free-electron metals (Ashcroft and Mermin 1985, Chapter 2), where a metal is described by a single parameter, the conduction electron density n. A widely used measure of 

The value of n is always given in the atomic unit of length, bohr (1 bohr = 0.529 A). For almost all the metals used in STM, 2-3 bohr. 

According to the Pauli exclusion principle, the conduction electrons occupy the states from the bottom of the conduction band up to an energy level where the metal becomes neutral. This highest energy level occupied by an electron is the Fermi level, /.. In the Sommerfeld theory of metals, a natural reference point of the energy level is the bottom of the conduction band. The Fermi level with respect to that reference point is 

In STM experiments, we are only interested in the electrons near the Fermi level. An important quantity is the number of electrons per unit volume per unit energy near the Fermi level, or the Fermi-level local density of states (LDOS). Inside a Sommerfeld metal, it is a constant 

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. 

For cluster physics to exist as a coherent subject, it is necessary that the properties of a cluster of one given size should not be totally different from those of another of a different size made up from the same atoms (this is 

Clearly, the validity of this approach will depend on the nature of the atoms involved. From the standpoint of a chemist, the obvious flaw of the jellium model is its total neglect of ionic structure. The model requires that the valence electrons should be strongly delocalised. This can only be true for certain metals which are very good conductors. It is also favoured if the ionic background is easily perturbed, in which case electronic single particle energies determine the structure. Finally, it tends to apply better when the wavefunctions have s character and when binding is non- 

The assumption of spherical symmetry is only reasonable for closed-shell systems, to which our discussion will apply. For open-shell systems, departures from sphericity occur due to the Jahn-Teller effect, and can be described by analogy with the deformed-shell model of nuclear physics [687], but lie beyond the scope of the simple theory described here. 

Under a closed-shell assumption, only the radial charge density needs to be determined (cf chapter 1). To fix the ionic background, we suppose that the density of positive charges follows a Heaviside step function 0 thus  

In order to estimate the magnitude of the surface dipole potential and its variation with the charge density, we require a detailed model of the metal. Here we will explore the jellium model further, which was briefly mentioned in Chapter 3. 

Jellium is a good model for sp metals. This group of metals comprises, amongst others, the elements Hg, Cd, Zn, Tl, In, Ga and Pb, all of which are important as electrode materials in aqueous solutions. They possess wide conduction bands with delocalized electrons, which form a quasi-free-electron gas. The jellium model cannot be applied to transition metals, which have narrow d bands with a localized character. The sd metals Cu, Ag and Au are borderline cases. Cu and Ag have been successfully treated by a modified version of jellium [3], because their d orbitals are sufficiently low in energy. This is not possible for gold, whose characteristic color is caused by a d band near the Fermi level. 

The basic features of this model have already been stated. The positive charge on the ion cores is smeared out into a constant background charge, which drops abruptly to zero at the metal surface. Using the 

For quantitative considerations it is convenient to use atomic units (a.u.), in which h = eo = me = 1 (me is the electronic mass) by definition. They are based on the electrostatic system of units so Coulomb s law for the potential of a point charge is f = q/r. Conversion factors to SI units are given in Appendix B here we note that 1 a.u. of length is 0.529 A, and 1 a.u. of energy, also called a hartree, is 27.211 eV. Practically all publications on jellium use atomic units, since they avoid cluttering equations with constants, and simplify calculations. This more than compensates for the labor of changing back and forth between two systems of units. 

According to a theorem by Hohenberg, Kohn and Sham [4], the total energy E of an electron gas can be written as a functional of the electronic density n(r) in the following form  

2 Electronic Structures of Metal Clusters and Cluster Compounds 

FigHre 2-2. Jellium potential for a spherical Na2o cluster. Filled circles indicate electrons occupying the lowest levels the open c e marks the level where an extra electron should go. 

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Figure 2. Sketch of an uncharged metal surface (simulated by the jellium model) covered by a macroscopic solvent layer, showing the components of the electric potential drop. 8%M is the surface potential of the metal modified by the solvent layer %s + 6%s is the surface potential of the solvent modified by the contact with the metal %s is the unmodified surface potential of the solvent layer at the external surface.

Schmidder and Henderson282 have studied several solvents and metals, using the jellium model for the metal and the MSA for the solution. Deviations of the Parsons-Zobel plot from linearity in the experimental results72,286-288 at the highest concentration have been attributed to the onset of ion-specific adsorption. However, data at other electrode charges... 

The C, values for Sb faces are noticeably lower than those for Bi. Just as for Bi, the closest-packed faces show the lowest values of C, [except Bi(lll) and Sb(lll)].28,152,153 This result is in good agreement with the theory428,429 based on the jellium model for the metal and the simple hard sphere model for the electrolyte solution. The adsorption of organic compounds at Sb and Bi single-crystal face electrodes28,152,726 shows that the surface activity of Bi(lll) and Sb(lll) is lower than for the other planes. Thus the anomalous position of Sb(lll) as well as Bi(lll) is probably caused by a more pronounced influence of the capacitance of the metal phase compared with other Sb and Bi faces28... 

Figure 5.7. Schematic representation of the definitions of work function O, chemical potential of electrons i, electrochemical potential of electrons or Fermi level p = EF, surface potential %, Galvani (or inner) potential

A simple metal like lithium or aluminum should best reveal the properties of the jellium model. To be sure, all long range order influence has been switched off, we measured S(q, co) of liquid A1 (T = 1000K). Figure 6 shows the result of a measurement for q = 1.5 a.u. together with theoretical calculations. 

Figure 11 shows the influence of correlation and lattice effects on the shape of n (k) for the case of lithium. The short dashed line shows (k) according to the jellium model with no electron-electron interaction included. Inclusion of correlation effects can be described using a model-w(k) ... 

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. 

The latter effect can be understood within a simple model for metals the jellium model, which is based on the following ideas As is... 

Figure 3.4 Distribution of the electronic density in the jellium model the metal occupies the region x < 0. The unmarked curve is for an uncharged surface, the other two curves are for the indicated surface-charge densities. The distance along the x axis is measured in atomic units (a.u.), where 1 a.u. of length = 0.529 A.

Figure 17.3 Electronic density and charge distribution in the jellium model (schematic).

As with the jellium model, the main significance of these calculations lies in the physical insight that they give into the structure of the solution at the interface, and the origin of the Helmholtz capacity. 

To explain where the surface contribution to the work function comes from, we need a model for the electron distribution in the surface region of a metal. One of the simplest is the jellium model [18]. 

The role of the metal electrons in determining properties of the metal/electrolyte solution interface was reviewed by Komyshev, who presented and discussed an alternative approach to the jellium model. As stated by Komyshev, the jellium model, which is correct for the interpretation of capacity measurements, appears to be too rough to explain the data obtained with modem techniques, such as many spectroscopic and miaoscopic techniques (see later discussion). 

In this chapter, hydrogen adsorption, particularly observed on Pt electrodes, was not treated as an underpotential deposition phenomenon. However, from a theoretical point of view, it may provide a breakthrough insight into underpotential deposition. Since the underpotential deposition ofM on M is quite characteristic among different combinations of M and M, together with a change in the kind of anions in the solution, a theoretical approach, which requires simplification, is still limited, and more experimental clarification is needed for theoretical work. However, the jellium model was successfully used to describe the lattice contraction of adsorbate T1 and Pb on Ag(lll) as underpotential deposition. ... 

Fig. 2-10. Profile of electron density and electronic potential energy across a metal/vacuum interface calculated by using the jellium model of metals MS = jellium surface of metals Xf = Fermi wave length p. - average positive charge density P- s negative charge density V = electron exchange and correlation energy V, - kinetic energy of electrons. [From Lange-Kohn, 1970.]...

Kg. 2-11. Work function, 4>, observed and calculated by using the jellium model as a function of Wigner-Seitz radius, rs, for various metals rs = 3 / (4 n n, = electron... 

Fig. 6-21. Charge distribution profile across a metal/aqueous solution interface (M/S) (a) the hard sphere model of aqueous solution and the jellium model of metal (the jellium-sphere model), (b) the effective image plane (IMP) and the effective excess charge plane x, (c) reduction in distance /lxd,p to the closest approach of water molecules due to electrostatic pressure, o, = differential excess charge on the solution side og = total excess charge on the solution side Oy = total excess charge on the metal side.

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. 

Beyond the Wade-Mingos Rules The Jellium Model. 12... 

The order of filling of the free electronic states in a jellium sphere is quite different from that of a free atom (Fig. 6). Although in a free atom the 2s level immediately follows the Is level, in a jellium sphere the p and lfree atom only the 2p level follows the 2s level before the 3s level is reached. However, in the jellium model the 2s level is followed by the 1/, 2p, Ig, and 2d levels before the 3s level is reached. 

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