Anderson model

Anisimov V I, Kuiper P and Nordgren J 1994 First-principles calculation of NIG valence spectra in the impurity-Anderson-model approximation Phys. Rev. B 50 8257-65... 

The previous sections have set the stage for describing the essentials of what happens when a molecule approaches the surface of a metal. The most important features of chemisorption are well captured by the Newns-Anderson model [D.M. Newns, Phys. Reu. 178 (1969) 1123 P.W. Anderson, Phys. Rev. 124 (1961) 41], which we describe in Section 6.4.1. Readers who are not particularly fond of quantum mechanics and its somewhat involved use of mathematics, but merely want to learn the outcome of this model, may skip this section and go directly to Section 6.4.2, where we present a summary in qualitative terms. The same readers may also want to consult Roald Hoffmann s Solids and Surfaces [(1988), VCH, Weinheim], abook we warmly recommend. 

By combining the results of the Newns-Andersons model and the considerations from the tight binding model it is now possible to explain a number of trends in surface reactivity. This has been done extensively by Norskov and coworkers and for a thorough review of this work we refer to B. Hammer and J.K. Norskov, Adv. Catal. 45 (2000) 71. We will discuss the adsorption of atoms and molecules in separate sections. 

Figure 6.36. Calculated variation in the heats of adsorption of molecular CO and NO compared with the heats of adsorption of the dissociation products. Open symbols follow from the Newns— Anderson model, closed symbols from density functional theory. [Adapted from B. Hammer and J.K. N0rskov, Adv. Catal. 45 (2000) 71.]...

Equation (89) shows that the allowance for the variation of the charge of the adsorbed atom in the activation-deactivation process in the Anderson model leads to the appearance of a new parameter 2EJ U in the theory. If U — 2Er, the dependence of amn on AFnm becomes very weak as compared to that for the basic model [see Eq. (79)]. In the first papers on chemisorption theory, a U value of 13eV was usually accepted for the process of hydrogen adsorption on tungsten. However, a more refined theory gave values of 6 eV.57 For the adsorption of hydrogen from solution we may expect even smaller values for this quantity due to screening by the dielectric medium. 

In the spirit of the Anderson model, we separate electrons into two subsystems delocalized electrons for which the LDA is assumed to give reasonable results and localized electrons for which it is well known that the LDA can lead to unphysical results. To treat these states in a better way, and to avoid double counting, we exclude the interaction between localized electrons (/ or d) already taken into account in an average way in the LDA-on-site energy... 

Trends in dissociative energies and activation energies for dissociation as a function of the number of d-electrons. The results are calculated in the Newns-Anderson model including the coupling between an adsorbate level epsilon a and the metal d-band. 

Going to the finer details the interaction energy does, for instance, depend on the d-band width, even in the simple Newns-Anderson model. The main effect is that the narrower the band the stronger the interaction. This is an additional reason why, in the calculations described in the previous section, the open surfaces have lower activation energies than the more close packed ones. The surface atoms in an open surface have a lower metal coordination number and since the band width is roughly proportional to the square root of the coordination number, the band width is smaller. 

Figure 2.3. The projected adatom density of states in the Newns-Anderson model in two limiting cases top) when the band width is larger and bottom) when the band width is smaller than the hopping...

The analysis of the Newns-Anderson model becomes particularly elegant by introducing the group orbital ... 

Figure 4.5 shows solutions to the Newns-Anderson model using a semi-elliptical model for the chemisorption function. The solution is shown for different surface projected density of states, nd(e), with increasing d band centers sd. For a given metal the band width and center are coupled because the number of d electrons must be conserved. 

Figure 4.5. Calculated change in the sum of the one-electron energies using the Newns-Anderson model. The parameters are chosen to illustrate an oxygen 2p level interacting with the d states of palladium with a varying d band center, ed. In all cases, the number of d electrons is kept fixed. The corresponding variations in the metal and adsorbate projected densities of states are shown above. Notice that the adsorbate-projected density of states has only a small weight on the antibonding states since it has mostly metal character. Adapted from Ref. [4].

The Ioffe-Regel principle, in our view, is valid always for states in mid-band (in the sense that l cannot be less than a in the Anderson model) and true for elastic collisions in the sense that kl is not less than about n anywhere in a band. 

Fig. 1.20 Density of states in a disordered system (a) in the Anderson model (Fig. 1.17) (i) shows the behaviour without disorder, simple cubic lattice, and (ii) that with disorder ...

Fig. 1.22 Potential energy in the Anderson model, showing behaviour near a band edge. The dotted horizontal line shows the energy E considered. The wells marked A and B have levels below this energy E. The length aE is the mean of distances between wells such as...

In the three-dimensional problem, it will be noticed from (71) that in X oJ2 the density of states and the diffusion coefficient occur in the denominator, as they do also in the expression given by Kawabata (1981). If the disorder broadens the band, as will occur in the Anderson model if V0 > B9 then (75) should be modified to... 

Although chain growth is not a feature relevant to methanation, the initiation and termination steps of the Anderson model for F-T synthesis are believed by at least some workers in the field to be applicable to the mechanism of the highly specific methanation reaction (71). The formation of methane is proposed to follow from the surface bound hydroxycarbene species by (19). 

It is also interesting to consider charge-transfer models developed primarily for metal surfaces. There are clear parallels to the metal oxide case in that there is an interaction between discrete molecular orbitals on one side, and electronic bands on the other side of the interface. The Newns-Anderson model [118] qualitatively accounts for the interactions between adsorbed atoms and metal surfaces. The model is based on resonance of adatom levels with a substrate band. In particular, the model considers an energy shift in the adatom level, as well as a broadening of that level. The width of the level is taken as a measure of the interaction strength with the substrate bands [118]. Also femtosecond electron dynamics have been studied at electrode interfaces, see e.g. [119]. It needs to be established, however, to what extent metal surface models are valid also for organic adsorbates on metal oxides in view of the differences between the metal an the metal oxide band structures. The significance of the band gap, as well as of surface states in it, must in any case be considered [102]. 

Both the cluster and the periodic calculations indicate a similarity to the Newns-Anderson model for metal adsorbates, in that both energy shifts, and broadenings need to be included in models of electron transfer, as shown schematically in Fig. 13. It will be a challenge in the near future to incorporate the increasingly accurate calculations of the crucial electronic coupling-strength parameter in existing dynamical models of the surface electron transfer processes. 

THE PERIODIC ANDERSON MODEL IN THE GENERATING FUNCTIONAL APPROACH... 

Abstract The periodic Anderson model at arbitrary values of the on-site Coulomb... 

Keywords Anderson model, Kondo lattice, magnetic susceptibility, strong correlation. 

Likewise the Hubbard model the periodic Anderson model (PAM) is a basic model in the theory of strongly correlated electron systems. It is destined for the description of the transition metals, lanthanides, actinides and their compositions including the heavy-fermion compounds. The model consists of two groups of electrons itinerant and localized ones (s and d electrons), the hybridization between them is admitted. The model is described by the following parameters the width of the s-electron band W, the energy of the atomic level e, the on-site Coulomb repulsion U of d-electrons with opposite spins, the parameter V of the... 

The periodic Anderson model in the generating functional approach... 

We applied the generating functional approach to the periodic Anderson model. Calculation of the electron GFs gdd, 9ds, 9sd and gss reduces to calculation of only the d-electron GF. For this, an exact matrix equation was derived with the variational derivatives. Iterations with respect to the effective matrix element Aij(to) allow to construct a perturbation theory near the atomic limit. Along with the self-energy, the terminal part of the GF Q is very important. The first order correction for it describes the interaction of d-electrons with spin fluctuations. In the paramagnetic phase this term contains a logarithmic singularity near the Fermi-level and thus produces a Kondo-like resonance peak in the d-electron density of states. The spin susceptibility of d-electrons... 

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