Infinite solids modeling with finite cluster

A physical approach to the electronic structure problems of solids contrasts sharply with the notion that local interactions dominate the structure and properties of molecular systems. Hence, it is very appealing to replace the infinite solid, which is difficult to treat quantum-chemically, by finite sites that can model considered species. Intuitively, cutouts from the bulk or the surface are made and treated like molecules. This type of method is called the cluster approach, and the models made as cutouts of the periodic structure are called cluster models [22]. 

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 molecular DFT calculations, it is natural to include all electrons in the calculations and hence no further subtleties than the ones described arise in the calculation of the isomer shift. However, there are situations where other approaches are advantageous. The most prominent situation is met in the case of solids. Here, it is difficult to capture the effects of an infinite system with a finite size cluster model and one should resort to dedicated solid state techniques. It appears that very efficient solid state DFT implementations are possible on the basis of plane wave basis sets. However, it is difficult to describe the core region with plane wave basis sets. Hence, the core electrons need to be replaced by pseudopotentials, which precludes a direct calculation of the electron density at the Mossbauer absorber atom. However, there are workarounds and the subtleties involved in this subject are discussed in a complementary chapter by Blaha (see CD-ROM, Part HI). 

The precise features of real catalysts at a microscopic scale are rather unknown but in all cases the main interactions occur through a surface. Two different theoretical models are often used to describe the electronic and other microscopic features of a surface. On the one hand, there is the solid state physics approach in which a surface is considered as a slab of a given thickness, finite in the direction perpendicular to the surface and infinite in the two other dimensions with perfect two-dimensional periodical symmetry. On the other hand, one has the cluster model approach which represents the surface with a finite number of atoms and the surface-adsorbate interaction as a supermolecule this is essentially a quantum chemical approach. It is important to realize that both approaches are crude representations of physical reality because real surfaces are far from being perfect, usually... 

The theoretical chemistry community developed density functional theory for finite molecular systems which involve molecules and cluster models that describe the catalytic systems. They use the same constructs used in many ab initio wavefunction methods, i.e. Gaussian or Slater basis sets. The solid-state physics community, on the other hand, developed density functional theory to describe bulk solid-state systems and infinite surfaces by using a supercell approach along with periodic basis functions, i.e. plane waves . Nearly all of our discussion has focused on finite molecular systems. In the next section we will describe in more detail infinite periodic systems. 

To examine the effective interaction of urea with specific surfaces of KCl, an approach similar to surface docking developed to predict the influence of additives on the crystal morphology has been employed [21-27], The basis of this approach is to analyze the effect of additives on the individual crystal faces, which are cleaved from a crystal. If the additive has a preferred interaction on a special face, the growth of this face will be slower. As a result, the other fast-growing surfaces will disappear, and eventually, the slow-growing surface will control the morphology. In this way, the additive influences the morphology of crystals. For simulations of surfaces of crystalline solids, slab, and cluster models are nevertheless by far more popular because they are feasible from the computational point of view [28]. However, the cluster models came under scrutiny due to their finite size representation. Slab models rather mimic the infinite surface of solids and are considered to be a better approach than the cluster models. In this study, a conventional array of these alkali... 

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