Jellium potential

Ekardt [67] and Beck [68] used this approach and solved the Kohn-Sham equations self-consistently for this type of spherical jellium potential for Na clusters of various sizes. The results of such calculations are visualized for a 40-electron cluster of various metals or choices of rj values in Fig. 2, [62]. Going down in the periodic table from sodium to potassium, the potential becomes shallower and the levels are more loosely bound. The valence electrons in the coinage metal copper has a much higher electron density which will give a smaller r. value and more bound states. 

At z < 0, the jellium potential is shown as determined by At z > 0, the metal potential is zero. The potential is detemincd by the adsorbate. Because of the overlap of atom potential and surface step potential, the potential barrier between... 

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. 

Figure 1.19 Various potential components resulting from Eq. (30) for the ground state of Nag. The left-hand panel shows the spherical component (Eq. (31)) given by the continuous line. For comparison the spherical jellium potential (dashed line) is also given. As one can see the two are almost identical and much larger than the two most important nonspherical potential parts vi=2,m=o(f ) (continuous line) and vi=4,m=o r) (dashed line) displayed in the right-hand panel. Note that these two components are strongly fluctuating compared to the smooth spherical part. These potential parts make the optical response different in the three isomeric states. In the end they are responsible for the transition from the molecule with a point-group symmetry to the solid with a space-group symmetry — a problem that has not yet been solved...

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. 

Clusters. Any attractive interaction between molecules can lead to formation of clusters, so there is no single equation that describes the binding of the particles. One that we don t run into elsewhere, however, is this jellium potential, derived for small clusters of metal atoms ... 

Sketch the energy levels and radial wavefunctions for the Is, 2s, and 3s states of the jellium potential energy model. Point out the qualitative differences when compared to the same states of the one-electron atom. THINKING AHEAD [What is the shape of the potential energy curve, and how will this affect the energies and wavefunctions ]... 

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.

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

We have returned to a former notation, denoting the electrostatic potential by , so that (0) is the potential at the jellium surface and bulk property and the right member a surface property. Another proof of Eq. (32) was given by Vannimenus and Budd.72 They also derived a related important theorem,... 

Another model which combined a model for the solvent with a jellium-type model for the metal electrons was given by Badiali et a/.83 The metal electrons were supposed to be in the potential of a jellium background, plus a repulsive pseudopotential averaged over the jellium profile. The solvent was modeled as a collection of equal-sized hard spheres, charged and dipolar. In this model, the distance of closest approach of ions and molecules to the metal surface at z = 0 is fixed in terms of the molecular and ionic radii. The effect of the metal on the solution is thus that of an infinitely smooth, infinitely high barrier, as well as charged surface. The solution species are also under the influence of the electronic tail of the metal, represented by an exponential profile. 

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. 

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 = 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. 

The fact that evaporated potassium arrives at the surface as a neutral atom, whereas in real life it is applied as KOH, is not a real drawback, because atomically dispersed potassium is almost a K+ ion. The reason is that alkali metals have a low ionization potential (see Table A.3). Consequently, they tend to charge positively on many metal surfaces, as explained in the Appendix. A density-of-state calculation of a potassium atom adsorbed on the model metal jellium (see Appendix) reveals that the 4s orbital of adsorbed K, occupied with one electron in the free atom, falls largely above the Fermi level of the metal, such that it is about 80% empty. Thus adsorbed potassium is present as K, with 8close to one [35]. Calculations with a more realistic substrate such as nickel show a similar result. The K 4s orbital shifts largely above the Fermi level of the substrate and potassium becomes positive [36], Table 9.2 shows the charge of K on several metals. 

Figure 9.10 Up work function of alkali-promoted metals as a function of alkali coverage (see also Table 9.2). Down electrostatic potential around a single alkali atom adsorbed on jellium. The effective local work function at each position is the sum of the substrate work function and the value of the electrostatic potential in the figure (from Lang el at. [39]).

A good starting point for discussing chemisorption is the resonant level model. The substrate metal is jellium, implying that we look at metals without d-electrons, and the adsorbate is an atom. We focus on only two electron levels of the atom. Level 1 is occupied and has ionization potential / level 2 is empty and... 

Figure A.9 A potential energy diagram of an atom chemisorbed on the model metal jellium shows the broadening of the adsorbate orbitals in the resonant level model.

Figure A.l 1 shows the change in density of states due to chemisorption of Cl and Li. Note that the zero of energy has been chosen at the vacuum level and that all levels below the Fermi level are filled. For lithium, we are looking at the broadened 2s level with an ionization potential in the free atom of 5.4 eV. The density functional calculation tells us that chemisorption has shifted this level above the Fermi level so that it is largely empty. Thus, lithium atoms on jellium are present as Li, with 8 almost equal to 1. Chemisorption of chlorine involves the initially unoccupied 3p level, which has the high electron affinity of 3.8 eV. This level has shifted down in energy upon adsorption and ended up below the Fermi level, where it has become occupied. Hence the charge on the chlorine atom is about-1.

Figure A.13 Potential energy diagram for the adsorption of a molecule on jellium. In situation b) the antibonding orbital of the adsorbate is partially occupied, implying that the intramolecular bond is weakened.

The intimate relationship between double layer emersion and parameters fundamental to electrochemical interfaces is shown. The surface dipole layer (xs) of 80SS sat. KC1 electrolyte is measured as the difference in outer potentials of an emersed oxide-coated Au electrode and the electrolyte. The value of +0.050 V compares favorably with previous determinations of g. Emersion of Au is discussed in terms of UHV work function measurements and the relationship between emersed electrodes and absolute half-cell potentials. Results show that either the accepted work function value of Hg in N2 is off by 0.4 eV, or the dipole contribution to the double layer (perhaps the "jellium" surface dipole layer of noble metal electrodes) changes by 0.4 V between solution and UHV. 

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