Surface Jellium surfaces

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,... 

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. 

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.]...

Fig. 6.4. Image profile with a Na-atom tip. (a) Geometry of the simulation. Two flat and. structureless jellium surfaces, each with an extra Na atom adsorbed on it, represent the tip and the sample, respectively. The net current from these Na atoms is kept constant while moving the Na atoms across each other. The path is generated numerically, (b) The simulated image (.solid curve) is in good agreement with the contour of the bermi-level LDOS and the total-charge-dcnsity contour. The apparent radius, as determined from the curve, is about 12 A. (After Lang, 1986.)...

A self-consistent calculation of electron-density profiles at strongly charged jellium surfaces, similar to the approach of Halley and co-workers, was made by Gies and Gerhardts [143]. This work was applied by the Patey group... 

Jellium model — Figure. Normalized electronic density n (x) near the jellium surface (full line) is the bulk density... 

Fig. 3. Approximate electron density variation on jellium surfaces with periodic positive charge boundaries. The solid line gives the edge of the uniform ionic charge density. The dashed line indicates the contour where the electron density is equal to one-half its interior value.

Fig. 7. Screening electron charge density (vertical axis not shown) around a unit positive point charge just outside a jellium surface. The surface is at x = 0, and the point charge is at u = 0, X = 4.5 units (the unit of distance is 51 pm, and the positive background density in the jellium corresponds to aluminum). [Reproduced with permission from Smith et al. (28). Copyright 1973 American Physical Society.]...

The theory in which the susceptibility is formally defined for jellium surfaces is the time-dependent density functional theory (TDDFT). In this theory, the susceptibility for interacting electrons (also called screened susceptibility) x(q, z, z ) is related to the susceptibility for non-interacting (independent) electrons Xo(q, ta, q z, z ) via the integral equation... 

The oscillatory behaviour of surface relaxation — inward for Af/12, outward for A 23 — seems to be fairly universal (Fu et al., 1984 Landman et al., 1980 Jiang et al., 1986). It is found not only experimentally and in fully self-consistent calculations, but also in simplified calculations a la Heine-Finnis. If a frozen charge density is used, for example a step density or the Lang -Kohn jellium surface profile, and the ions are relaxed to positions of zero force, oscillatory relaxations are found (Landman et al., 1980). This shows that it is not a consequence of the Fricdel oscillations in the surface charge density. 

Petukhov AVI 995 Sum-frequency generation on isotropic surfaces general phenomenology and microscopic theory for jellium surfaces Phys. Rev. B 52 16 901-11... 

To improve the agreement with experiment, two kinds of corrections have been applied. The first consists in smoothing the discontinuity of the jellium density at the cluster surface. For this purpose, the original step-density is replaced by a continuous function that models a surface with a finite thickness of about 1 a.u. [52]. As a consequence, the electron density is more extended and the polarizability increases in this finite surface jellium model ( column FSJM), improving the agreement with experiment. The FSJM also improves... 

Electron-Density Profiles at Strongly Charged Jellium Surfaces. 

The electronic polarizability of the jel-lium surface, which is the relevant quantity for the interfacial capacity, can be expressed in terms of the effective position of the image plane, which has the meaning that its name suggests a test charge placed in front of the jellium surface experiences an image force as if the image plane were a distance x m in front of the surface (see Fig. 4). In a metal capacitor the effective positions of the plates lie at a distance Xim... 

As already mentioned, the distribution of electrons at the jellium surface entails a surface dipole moment. Therefore, at the pzc a water molecule situated within the electronic tail experiences a positive field that tends to orient the molecule with its oxygen end toward the metal surface. Conversely, the water molecules tend to enhance the electronic spillover. As a result, the work function for jellium in contact with water is lower than in the vacuum. Model calculations for this effect have been performed by several authors [34, 37, 39-41]. While the results depend on the details of the model, it is generally agreed that the larger the overall change in the dipole potential, the higher the electronic density. 

The simplest molecular model for an electrolyte solution is an ensemble of hard spheres treated in the MSA. This can be combined with jellium to obtain a model for the whole interphase [46-48]. The hard sphere model has been solved at the PZC only, so the combined model is restricted to this point. It is natural to consider the jellium surface as a hard wall for the electrolyte and add the contributions of the hard-sphere electrolyte and jellium to... 

Fig. 34a-c. Results of a model calculation of a Xe atom adsorbed on a high electron density jellium surface (e.g. aluminium), (a) Contours of constant electron density in a cut perpendicular to the surface through the center of the Xe atom, (b) Xe valence p-elechon density vs. distance (difference density between metal-adatom system and sum of clean metal plus single Xe atom except 5p level), (c) Effective single particle potential energy contributions due to electrostatic dipole, Ves, and the exchange-correlation interaction, Vxc, respectively, [82Lan],... 

An alternative approach to IL electrochemical windows was put forward by Ballone et al., who implemented a free electron droplet (a 40-electron jellium sphere) as a model electrode in AIMD simulations to investigate the role of the electrode/IL interface on the stability of several ILs [83]. For reference, the predicted stability windows of C4mimTFSI and C4mimPF6 adsorbed on the jellium surface, 3.77 V and 3.48 V, respectively, were in quite good agreement with the corresponding windows, 3.87 V and 3.74 V, reported by Ong et al. using their novel MD-DFT approach [81]. However, as commented and demonstrated by Ong et al. with the use of a different functional, these windows were unduly narrow because of the PBE functional used. 

Rg. 5.2-20 Effective one-electron potential Veff (continuous line) and its electrostatic part (dashed line) near a jellium surface for rs = 5. Notice the characteristic oscillations (Friedel oscillations) [2.29]... 

Armed with these tools, one can now address the jellium surface problem again, this time taking into account the electron-electron interactions in an approximate way. The result is shown in Eigure 5.3. It is qualitatively very similar to the one shown in Figure 5.2, but it allows a more realistic assessment of surface phenomena, which follow from the Eriedel oscillations, such as oscillatory relaxations of interlayer distances close to the surface, the surface dipole layer associated with the Friedel oscillations and the work function. 

Figure 5.3 Charge density distribution at a jellium surface for two different electron densities, r. Is a dimensionless density parameter ( Wigner-Seitz radius ). If n is the electron density, l/ is a volume per electron. Assuming this volume to be spherical.

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