Jellium calculations

Two other jellium calculations are of interest. One (27) is a study of the local density of states in the spill-out region. It has been found that the farther out from the surface, the larger the contribution of electrons at the Fermi level to the total charge density (Fig. 6). The latter quantity decreases with distance it decreases exponentially, with a characteristic length that is inversely proportional to the square root of the work function. The other... 

The results of spherical jellium calculations performed by Genzken for Na clusters [18] are displayed in Fig, 3. After the cluster energy has been calculated as a function of N, it can be conveniently separated into a smooth and an oscillating part ... 

As compared to experiment, all spherical jellium calculations yield an insufficient redshift of the Mie resonance. This is connected to the low polarizability. Therefore, a jellium density with a smooth surface [52], or other corrections found to improve the polarizability, also improve the position of the dipole resonance. Replacing the LDA by a nonlocal description of exchange and... 

A contraction of the cluster volume with respect to that of an equivalent piece of bulk metal has also been predicted [96]. The calculated cluster radius is smaller than the radius assumed in the spherical jellium model, where the volume is the same as that of an equivalent piece cut out of a macroscopic metal. This global contraction seems to be a general feature of small metallic clusters, and is well documented experimentally [97]. The volume contraction explains the discrepancies between experimentally determined static polarizabilities of small aluminium clusters and those obtained from jellium calculations [98]. The measured polarizabilities of Aljv clusters with N < 40 are smaller than those predicted by a SJM calculation. The classical static polarizability (per atom) for... 

Although very useful for certain classes of clusters, the SAPS model has its limitations for small or medium size clusters, because many of these are deformed. Nevertheless, deformed-jellium calculations indicate that most clusters still maintain axial symmetry (see Sect. 12) and that truly triaxial deformations are rare. These are good reasons to assume that the valence electron cloud is nearly axially symmetric, even if the ionic structure is fully three-dimensional. Taking up this idea. Montag and Reinhard [129] have developed the cylindrically averaged pseudopotential scheme (CAPS), which is the extension of the SAPS to axial symmetry. The essential approximation is to reduce the treatment of the electrons to axial symmetry and using cylindrical coordinates (p, z). In this way, only the cylindrical average of the pseudopotential... 

Figure 4 Illustration of the geometries used in jellium calculations (a) a semi-infinite metal, where the metal is infinite in the z direction (perpendicular to the surface) and (b) a metal slab, where the metal has a finite width in the z direction. In both cases the metal is infinite in the x and y directions (parallel to the surface).

The electron density of the model Hg-water interface obtained from jellium calculations is shown in Figure 10. Within the metal, the density oscillates with a wavelength rr/fep, where fep = and quickly decays to the... 

Fig. 9 Surface electronic profile of a neutral and negatively charged jellium calculated for the bulk electron density of Hg.

In addition to the present shape parameterization [43], other two-center shape param-eterizations (mainly in connection with KS-LDA jellium calculations) have been used [81-83] in studies of metal-cluster fission. They can be grouped into two categories, namely, the two-intersected-spheres jellium [81, 84], and the variable-necking-in parame-terizations [82, 83]. In the latter group. Ref. [82] accounts for various necking-in situations by using the funny-hills parameterization [85], while Ref. [83] describes the necking-in... 

From a jellium calculation one gets exactly / = 1 per 3s electron for sodium. Experimentally, one sees something very similar, showing how good the nearly free electron assumption is for sodium. Also the experimental /-values are nearly independent of temperature, as could be expected. 

In O Fig. 26-13 we show the resulting stability function and in O Fig. 26-14 the radial distances of the atoms. The stability functions show only a marginal similarity and only the DFTB calculations find particularly stable clusters for sizes that agree with those of the mass abundance experiments and the jellium calculations (2,8,18,20,34,40,58,...) (Knight et al. 1984). On the other hand, the radial distances indicate that the structures found in the EAM calculations are somewhat more symmetric than are those of the DFTB calculations. In fact, based on a shape analysis (not shown), we find that none of the DFTB structures can be considered as being roughly spherical, in marked contrast to the inherent assumptions behind the jellium calculations. 

According to Vitanov et a/.,61,151 C,- varies in the order Ag(100) < Ag(lll), i.e., in the reverse order with respect to that of Valette and Hamelin.24 63 67 150 383-390 The order of electrolytically grown planes clashes with the results of quantum-chemical calculations,436 439 as well as with the results of the jellium/hard sphere model for the metal/electro-lyte interface.428 429 435 A comparison of C, values for quasi-perfect Ag planes with the data of real Ag planes shows that for quasi-perfect Ag planes, the values of Cf 0 are remarkably higher than those for real Ag planes. A definite difference between real and quasi-perfect Ag electrodes may be the higher number of defects expected for a real Ag crystal. 15 32 i25 401407 10-416-422 since the defects seem to be the sites of stronger adsorption, one would expect that quasi-perfect surfaces would have a smaller surface activity toward H20 molecules and so lower Cf"0 values. The influence of the surface defects on H20 adsorption at Ag from a gas phase has been demonstrated by Klaua and Madey.445... 

Figure 6.26. Density functional calculations show the change in the density of states induced by adsorption of Cl, Si and Li on jellium. Lithium charges positively and chlorine negatively. [From N.D. Lang and A.R. Williams,...

In this contribution we will deal with electron-electron correlation in solids and how to learn about these by means of inelastic X-ray scattering both in the regime of small and large momentum transfer. We will compare the predictions of simple models (free electron gas, jellium model) and more sophisticated ones (calculations using the self-energy influenced spectral weight function) to experimental results. In a last step, lattice effects will be included in the theoretical treatment. 

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. 

Perhaps of greater interest to us are results derived by the same authors71 that relate surface and bulk electronic properties of jellium. Considering two jellium slabs, one extending from —L to -D and the other from D to L, they calculated the force per unit area exerted by one on the other. According to the Hellmann-Feynman theorem, this is just the sum of the electric fields acting... 

The ionic profile of the metal was modeled as a step function, since it was anticipated that it would be much narrower than the electronic profile, and the distance dx from this step to the beginning of the water monolayer, which reflects the interaction of metal ions and solvent molecules, was taken as the crystallographic radius of the metal ions, Rc. Inside the metal, and out to dl9 the relative dielectric constant was taken as unity. (It may be noted that these calculations, and subsequent ones83 which couple this model for the metal with a model for the interface, take the position of the outer layer of metal ion cores to be on the jellium edge, which is at variance with the usual interpretation in terms of Wigner-Seitz... 

This was averaged over the total distribution of ionic and dipolar spheres in the solution phase. Parameters in the calculations were chosen to simulate the Hg/DMSO and Ga/DMSO interfaces, since the mean-spherical approximation, used for the charge and dipole distributions in the solution, is not suited to describe hydrogen-bonded solvents. Some parameters still had to be chosen arbitrarily. It was found that the calculated capacitance depended crucially on d, the metal-solution distance. However, the capacitance was always greater for Ga than for Hg, partly because of the different electron densities on the two metals and partly because d depends on the crystallographic radius. The importance of d is specific to these models, because the solution is supposed (perhaps incorrectly see above) to begin at some distance away from the jellium edge. 

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. 

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. 

---