Surfaces modeling with finite cluster

Table 3. Electric dipole polarizabilities in units of R, of neutral sodium clusters in the spherical jellium model (SJM) and in a jellium model with finite surface thickness (FSJM)...

We have studied above a model for the surface reaction A + 5B2 -> 0 on a disordered surface. For the case when the density of active sites S is smaller than the kinetically defined percolation threshold So, a system has no reactive state, the production rate is zero and all sites are covered by A or B particles. This is quite understandable because the active sites form finite clusters which can be completely covered by one-kind species. Due to the natural boundaries of the clusters of active sites and the irreversible character of the studied system (no desorption) the system cannot escape from this case. If one allows desorption of the A particles a reactive state arises, it exists also for the case S > Sq. Here an infinite cluster of active sites exists from which a reactive state of the system can be obtained. If S approaches So from above we observe a smooth change of the values of the phase-transition points which approach each other. At S = So the phase transition points coincide (y 1 = t/2) and no reactive state occurs. This condition defines kinetically the percolation threshold for the present reaction (which is found to be 0.63). The difference with the percolation threshold of Sc = 0.59275 is attributed to the reduced adsorption probability of the B2 particles on percolation clusters compared to the square lattice arising from the two site requirement for adsorption, to balance this effect more compact clusters are needed which means So exceeds Sc. The correlation functions reveal the strong correlations in the reactive state as well as segregation effects. 

We emphasize two natural limitations of the finite cluster model. It does not allow to make a statement about the dependence of essential parameters such as adsorption and transition energies on the level of surface coverage, and it does not account adequately for charge delocalization or surface relaxation phenomena. Further, it excludes by definition any information about the modification of the surface band structure as a consequence of the organic molecule adsorption. The following case study of 1-propanol on Si(001) - (2 x 1) is intended to clarify how these elements can be consistently incorporated into the description of the Si surface interaction with organic species. 

The theoretical treatment of Si surfaces with organic adsorbates has proceeded along two main avenues approaches relying on a finite cluster model are to be distinguished from those employing density functional theory in conjunction with periodic boundary conditions. The principal virtue of the first methodology is that... 

The basic approach of chemical theory to surface science is to model a surface with a cluster of a finite number of atoms, with one or more adsorbate atoms or molecules bonded to various sites on the cluster. In parallel with the chemical theory there is also the solid state physics approach. This starts from an extended surface surface model, where an array of atoms perfectly periodic in two dimensions represents both the substrate and any adsorbates. Many theoretical techniques have been developed for the extended-surface model. We can only refer the interested reader to the literature/87,88,89,90,91,92,93,94/ and remark that the relative merits of the cluster and extended-surface approaches are still very much under active debate. It is clear that certain properties, such as bonding, are very localized in character and are well represented in a cluster. On the other hand, there are properties that have a delocalized nature, such as adsorbate-adsorbate interactions and electrostatic effects, for which an extended surface model is more appropriate. 

Another standing topic during the last two decades has been to evaluate the electronic structure of solids, surfaces and adsorbates on surfaces. This can be done using standard band structure methods [107] or in more recent years slab codes for studies of surfaces. An alternative and very popular approach has been to model the infinite solid or surface with a finite cluster, where the choice of the form and size of the cluster has been determined by the local geometry. These clusters have in more advanced calculations been embedded in some type of external potential as discussed above. It should be noted that these types of cluster have in general quite different geometries compared with... 

In this chapter we review the field of electronic structure calculations on metal clusters and nano aggregates deposited on oxide surfaces. This topic can be addressed theoretically either with periodic calculations or with embedded cluster models. The two techniques are presented and discussed underlying the advantages and limitations of each approach. Once the model to represent the system is defined (periodic slab or finite cluster), possible ways of solving the Schrddinger equation are discussed. In particular, wave function based methods making use of explicit inclusion of correlation effects are compared to methods based on functionals of the... 

Figure 14. The density profiles at different bubble radii for ( He)jy clusters with iV= 1.88x10 reflecting on the formation of a helium balloon with a finite thickness (AR = R — Rb) in the cluster. The profile thicknesses for the bubble and for the cluster surface obtained from this simple model are t, t2 = 6-lOA (see text). The exterior surface profile of the cluster was characterized by the 90-10% fall-off width W2, while the interior bubble profile was characterized by a 10-90% rise width wj. For N = 1.88 x 10 clusters wi = 6.2 A for Rt = 0 (no bubble), W2 = 7.8 A and wi = 6.2 A for Rj = 10.3 A, while W2 = 6.8 A and Wi = 12.3 A for Rb = 19.2 A. These results demonstrate that the cluster surface profile width n>2 remains nearly independent of the bubble size, while Wi increases with increasing the bubble radius.

Adsorption energies on metals calculated in a cluster approach often show considerable oscillations with size and shape of the cluster models because such (finite) clusters describe the surface electronic structure insufficiently [257-260]. These models may yield rather different results for the Pauli repulsion between adsorbate and substrate, depending on whether pertinent cluster orbitals localized at the adsorption site are occupied or empty. The discrete density of states is an inherent feature of clusters that may prevent a correct description of the polarizability of a metal surface and thus hinders cluster size convergence of adsorption energies [257]. Even embedding of metal clusters does not offer an easy way out of this dilemma [260,261]. Anyway, the form of conventional moderately large cluster models may be particularly crucial. Such models are inherently two-dimensional with substrate atoms from two or three crystal layers usually taken into accormt thus, a large fraction of atoms at the cluster boundaries lacks proper coordination. 

An alternative to these simple, more or less unspecific, models is the use of ab initio electronic structure calculations, in a similar spirit as in the corresponding water-metal models [40, 49]. Most studies so far have been performed with the cluster model approach [114]. In this approach, a finite number of atoms is chosen to model a local site on the crystal surface. Usually, the geometry of the substrate is kept fixed. The size of the substrate cluster varies between 4 and 20 metal atoms. The geometrical arrangement of the surface atoms is usually taken from the unreconstructed surface. 

Finally, one may consider a finite cluster as a model system for the surface without or with adsorbants. In this case there may be dangling bonds in all directions, although one may be interested in only those of one of the surfaces. The other dangling bonds may be saturated by, e.g., hydrogen atoms. 

Instead we consider a special case of a surface, i.e., the inner surfaces of a zeolite. The channels and cavities of zeolites are often so large that smaller molecules can be accommodated there and, in some cases, interact with the zeolite host. An example of this was studied by Broclawik et alP They studied the dissociation of methane on a gallium site in the so-called ZSM-5 zeolite. They considered a finite cluster as a model system, and by performing density-functional calculations for that system they were able to identify the reaction mechanism, including educts, products, and transition states. In principle, the calculations are very similar to those discussed above in Section 6, except for differences in the systems. 17... 

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