D band model

Figure 2. The structural energy difference (a) and the magnetic moment (b) as a function of the occupation of the canonical d-band n corresponding to the Fe-Co alloy. The same lines as in Fig. 1 are used for the different structures. In (b) the concentration dependence of the Stoner exchange integral Id used for the spin-polarized canonical d-band model calculations is shown as a thin dashed line with the solid circles. The value of Id for pure Fe and Co, calculated from LSDA and scaled to canonical units, are also shown in (b) as solid squares.

Hammer-Nprskov d-band model, 70, 272-273, 327 Heme-copper oxidase, 610 High Throughput Synthesis of Nanoparticles, 572-574 Hydrogen (underpotential) adsorption, 60-63,254, 471-484, 526 Hydrogen evolution reaction (HER), 31, 79-87... 

Within the rectangular d band model the bond energy is proportional to the bandwidth, IF, through eqn (7.33). We may relate the bandwidth to... 

The heats of formation of equiatomic AB transition-metal alloys may be predicted by generalizing the rectangular d band model for the elements to the case of disordered binary systems, as illustrated in the lower panel of Fig. 7.13. Assuming that the A and transition elements are characterized by bands of width WA and WB, respectively, then they will mix together in the disordered AB alloy to create a common band with some new width, WAB. The alloy bandwidth, WAB may be related to the elemental bond integrals, hAA and , and the atomic energy level mismatch, AE — EB — EAt by evaluating the second moment of the total alloy density of states per atom ab( ), namely... 

Hence, within the rectangular d band model for the AB alloy density of states, from eqs (7.33) the bond energy becomes... 

We see that the simple rectangular d band model reproduces the behaviour found by experiment and predicted by Miedema s semi-empirical scheme. However, we must stress that the model does not give credence to any theory that bases the heat of formation of transition-metal alloys on ionic Madelung contributions that arise from electronegativity differences between the constituent atoms because in the metallic state the atoms are perfectly screened and, hence, locally charge neutral. Instead, the model supports... 

Fig. 8.12 The rectangular d band model of the (a) nonmagnetic, (b) ferromagnetic, and (c) antiferromagnetic states. (From Pettifor (1980).)...

This is the rectangular d-band model criterion equivalent to the exact second-order result, namely... 

These systems represent bonding to surfaces where the adsorbate atoms have unpaired electrons available for covalent interaction with unsaturated electronic states on the metal surface. We denote this bonding mechanism as radical adsorption where the open-shell electrons on the adsorbate atom can form electron pairs with the metal atoms at the surface. These radical atoms have in most cases been obtained through the dissociation of molecules on the surface. Let us make a simple picture of the electronic structure when a simple atomic adsorbate interacts with a transition metal, denoted the d-band model [31,32]. A similar description can also be found in Chapter 4. 

One of the basic assumptions of the d band model is that E0 is independent of the metal. This is not a rigorous approximation. It will for instance fail when metal particles get small enough that the sp levels do not form a continuous (on the scale of the metal-adsorbate coupling strength) spectrum. It will also fail for metals where the d-states do not contribute to the bonding at all. The other basic assumption is that we can estimate the d contribution as the non-self-consistent one-electron energy change as derived above ... 

As a first example of the use of the d band model, consider the trends in dissociative chemisorption energies for atomic oxygen on a series of 4d transition metals (Figure 4.6). Both experiment and DFT calculations show that the bonding becomes... 

Figure 4.6. Variations in the adsorption energy along the 4d transition metal series. The results of full DFT calculations are compared to those from the simple d band model and to experiments. Below the same data are plotted as a function of the d band center. Adapted from Ref. [4].

The d band model, including Pauli repulsion, can therefore be used to understand variations in oxygen binding energies in the periodic table. It turns out that a similar description can be used for a number of other adsorbates [4,18]. 

Pt surfaces tend to restructure into overlayers with an even higher density of Pt atoms than the close-packed (111) surface [21]. The Pt atoms are closer to each other on the reconstructed surfaces than in the (111) surface. The overlap matrix elements and hence the bandwidth are therefore larger, the d bands are lower and consequently these reconstructed surfaces bind CO even weaker than the (111) surface. The reconstructed Pt surfaces are examples of strained overlayers. The effect of strain can be studied theoretically by simply straining a slab. Examples of continuous changes in the d band center and in the stability of adsorbed CO due to strain are included in Figure 4.10. The effect due to variations in the number of layers of a thin film of one metal on another can also be described in the d band model [22,23]. 

For atomic chemisorption, similar structural effects are found (see the middle panel of Figure 4.10). As for molecular chemisorption, low-coordinated atoms at steps bind adsorbates stronger and have lower barriers for dissociation than surfaces with high coordination numbers and lower d band centers. The d band model thus explains the many observations that steps form stronger chemisorption bonds than flat surfaces [1,20,24-28]. The finding that the correlation with the d band center is independent of the adsorbate illustrates the generality of the d band model. 

While the interpolation model is far from perfect it gives a fast way of estimating the adsorption energies for alloys. Given the simplicity of the model it is surprising how well it works. The d band model can be used to indicate why this is the case. 

The arguments behind the d band model are quite general and should apply to the interactions in the transition state as well as in the initial and final (adsorbed) states of the process. We therefore expect correlations between the d band center and transition state energies to be the same as for chemisorption energies. This is illustrated in the bottom panel of Figure 4.10. Figure 4.16 shows in detail how the activation energy for methane on different Ni surfaces scales with the center of the d bands projected onto the appropriate metal states to which the transition state couples. 

Figure 4.16 also illustrates a case where the indirect interaction of one adsorbate with another can be described by the d band model (the effects of pre-adsorbed C atoms). If two adsorbates interact with the same metal atom, they are often found... 

Figure 4.16 also shows that there are additional effects due to direct interactions between adsorbates that are not described using the d band model. A large adsorbate like S, will have a sizable overlap to the valence orbitals of the incoming molecule, giving rise to a repulsion which is larger than what can be readily explained by the indirect interaction through d band shifts. 

We note that other systems not resembling the simple diatomic molecules considered here may follow a different relationship [86]. There may be other classes of reactions, dehydrogenation or —C bond breaking that may follow other similar relationships and thus form another universality class. We also note that there are exceptions to the relations, most notably for H2 dissociation on near-surface alloys [87]. These deviations from the rules are still describable within the d band model, though [87]. 

Figure 6.16. Illustration of the d-band model governing surface chemical bonding on transition metal surfaces. As the d-band center of a catalytic surface shifts downward more antibonding orbitals become occupied and the surface bond energy of an adsorbate (here an oxygen atom) decreases. An upward shift in the d-band center predicts strengthening of the surface bond.

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