Anderson-Newns model

Initially we consider a simple atom with one valence electron of energy and wave function which adsorbs on a solid in which the electrons occupy a set of continuous states Tj, with energies Ej. When the adsorbate approaches the surface we need to describe the complete system by a Hamiltonian H, including both systems and their interaction. The latter comes into play through matrix elements of the form Vai = / We assume that the solutions T j to this eigen value problem 

In the following will also assume that the basis set is orthogonal, i.e. matrix elements of the type J 03 vanish. The solution is found from the eigen value problem 

As we already have seen are there infinitely many k states (n °o) in the metal and it is rather problematic to keep track of all of them. It is better to consider the projection of the new states onto the original adsorbate state 0. This maps out the development of the adsorbates state as the atom approaches the surface, and is technically carried out by determining the quantity 

To proceed we write the product in the left-hand of Eq. (62) in matrix form and set it equal to the unit matrix 

Remember that the quantity of interest is njfi). It now becomes evident why it was worth introducing the Greens function as 

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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.]...

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].

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. 

Figure 12.3 shows a schematic illustration of the resulting electron density of states projected onto the adatom in the Newns-Anderson model [17, 18] for two different cases. In this model, the interaction strength between the adatom wave function of one specific electronic level and the metal states is often denoted the hopping matrix element. When the hopping matrix element is much smaller than the bandwidth of the metal states, in this case the i-electrons, the interaction leads... 

Figure 2.56. Restricted Ilartree Pock solution of the Newns>Anderson model,...

Figure 2.57. Unrestricted Hartree-Fock solutions of the Newns-Anderson model, (schematic).

The chemisorption is due to an interaction between the states of the H2 molecule and the electrons in the metal. The simple picture is given by the Newns-Anderson model where an adsorbate state is lowered and broadened when interacting with a sea of valence electrons in a metal, as indicated in Figure 4.15. Basically, aU metals have a broad band of sp-electrons that are populated up to the Fermi level. Since these metals can be considered more or less as free electrons their density of states (DOS) is usually assumed to be proportional to v, as indicated in Figure... 

We will formulate the Newns-Anderson model within the language of molecular orbital theory. When the adsorbate is represented with a single s-atomic orbital, the Newns-Anderson model can be understood as sketched in Figure 10.1a and b [17,18], We consider the case where the highest occupied metal valence orbital, the Fermi level, has the same energy as an electron in the isolated adsorbate orbital. Figure 10.1a illustrates the situation when the adsorbate only weakly interacts with the surface. 

One concludes that bonding is covalent When surface atom coordination changes, the increased localization of the electrons is responsible for increased adsorption energies. The actual position of is not of dominating importance, it merely reflects the decrease in d-valence band width W. The Newns-Anderson model applies and the chemical bond in the surface complex approximates the surface molecule limit... 

Figure 10.6 Calculated variation in the adsorption energy of molecular CO compared with atomically adsorbed C and O for close-packed surfaces of the 4d transition metals. Solid symbols are DFT calculations open symbols are Newns—Anderson model effective medium calculations. (Adapted from Ref [24]).

Figure 3.20. (Left) Calculated and model estimates of the variation in the adsorpion energy of molecular CO compared with atomically adsorbed C and O for the most close-packed surface of the 4d transition metals. (Right) Calculated molecular and dissociative chemisorption of NO. Solid symbols are DFT calculations open symbols are Newns—Anderson model effective mediumt ] calculations. For CO, dissociative chemisorption appears to the left of rhodium. For NO, dissociative chemisorption appears further to the right, i.e., also on rhodium, adapted from Hammer and NprskovW.

So far, we have focused on describing the structure of the materials upon which our catalytic reactions are supposed to take place. The next step is to look at what happens when a molecule approaches the surface of such a material and begins to interact with the material and hence forms a chemical bond. The Newns-Anderson model is a model that describes the hybridization of a single adsorption state on an atom or a molecule with the large continuum of states at the surface. 

In the following, we will describe the Newns-Anderson model in detail. 

The main result from the Newns-Anderson model describes how a single adsorbate state Ifl) with energy develops as it approaches a surface with a large number of states Ifc) fcs 1,2,..., . Here, we again use the bra and ket vector notation to describe a specific state of the system. 

Let us look at the Newns-Anderson model in more detail. We do that by varying the parameters in the model separately. There are three terms that define the bonding between an adsorbate state and a surface in the Newns-Anderson model (1) the structure of the local projection of the surface DOS around the adsorbate A(c), (2) the coupling strength V, and (3) the energy position of the adsorbate level relative to the Fermi level. 

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