Periodic slab

Kruger and Rosch implemented within DFT the Green s matrix approach of Pisani withm an approximate periodic slab enviromnent [180]. They were able to successfiilly extend Pisani s embeddmg approach to metal surfaces by smoothing out the step fiinction that detenuines the occupation numbers near the Fenui level. 

While quantum-chemical calculations related to gas-phase reactions or bulk properties have become now a matter of routine, calculations of local properties and, in particular, surface reactions are still a matter of art. There is no simple and consistent way of adequately constructing a model of a surface impurity or reaction site. We will briefly consider here three main approaches (1) molecular models, (2) cluster models, and (3) periodic slab models. 

As to periodic slab models, their application is rather straightforward, but poses many problems in the cases when the activation energy should be found for a surface reaction at a bulky surface group. On the other hand, this is probably the best approach to determine the structural properties of an active site and the interaction energies between an active site and the reacting species. 

Silvestrelli5 reached a similar conclusion for Si(001) -(2x1) interaction with ethanol in the framework of a periodic slab approach, where the Car - Parrinello method was used, instead of a cluster model. Quantitatively, substantial differences... 

A crucial feature of the metal/oxide interface is that it combines the extended nature of the support with the limited size of the supported metal particles - a feature that is common to the study of defects in solids. Such systems pose special problems for realistic modelling. The requirements of defining computationally tractable, as well as accurate models are of particular importance. Three different approaches are common, namely bare clusters, embedded clusters and periodic slab models. All three are associated with approximations, and the best choice must be defined by the correct compromise between cost and accuracy. 

Detailed models of the (0001) - O terminated polar surfaces were investigated using interatomic potentials within the two-dimensional periodic slab model realized within the MARVIN code. Two typical local environments were identified as terrace termination and interstitial vacancy sites the latter has been used as the active site for the majority of calculations, as illustrated in Figure 5. The atoms highlighted in the figure indicate the surface embedded quantum region. For... 

Atomic and molecular adsorption at vanadium oxide surfaces have been studied theoretically using both periodic slab and cluster models where so far studies are restricted to the pentoxide, V2O5, as a substrate due to its possible importance in catalytic applications as mentioned before. Further, adsorbate species include in all cases atoms (H [122-123, 126, 136-142], O (see below)) or rather small molecules (O2 (see below), H2O [143-144], NH3 [145-147], NO [146, 148], C2H4 [149], propene (CsHg) [140], toluene (CeHsCHj) [140]) that are of catalytic interest but also small enough to make meaningful calculations feasible. 

Up to now, theoretical work on molybdenum oxide surfaces has been restricted to only few periodic slab as well as local cluster studies on the single valence oxide surfaces Mo03(010), (100) and MoO2(011). As for vanadium oxide, there is very little experimental information on quantitative microscopic details of these complex surfaces despite their importance concerning catalytic applications. The following discussion of theoretical results will be based exclusively on cluster studies that have yielded rather detailed microscopic insight. 

Calvo and Balbuena examined the structure and reactivity of Pd-Pt nanoclusters with 10 atoms in the oxygen reduction reaction. In contrast with what is expected in a periodic slab calculation, they found that mixed states with randomly distributed Pd atoms in a Pt7Pd3 cluster was more stable than an ordered cluster structure due to more eflective charge transfer in the mixed state. They found that increasing the concentration of Pd in the surface favors formation of the OOH species in the first step of the reaction, but Pt atoms were needed to promote the second stage of the oxygen reduction reaction. They report that due to charge transfer eflhcts the Pd atoms have an intermediate reactivity between pure Pd and Pt, and in the mixed cluster the Pd atoms the Pd atoms act more similarly to Pt than in the ordered cluster. 

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... 

Both periodic slabs and finite clusters were employed in our DFT calculations. DACAPO (with the new ASE2 python interface) [21] and Amsterdam Density Functional (ADF) [22-24] packages were used for the slab and cluster calculations, respectively. Details of the calculations are presented below. 

FIGURE 10.6 Computational models of Au surfaces. Models are shown with optimized O2 geometries. For Au(m) and Au(211), a single unit cell of the periodic slab is shown [34]. Reprinted with permission from [34]. Copyright (2005) American Chemical Society. 

For touchstone complexes of metal atoms on MgO(OOl), results of DF cluster [96,167] and periodic [191-193] calculations agree. In particular, (i) adsorption sites on-top of O are preferred, (ii) bonding can be rationalized by a polarization of metal atoms in the electrostatic field of the support with modest covalent and only very minor ionic contributions, (iii) adsorption energies are close in these model cluster and periodic slab computational approaches, provided the same xc functional is used. Embedded cluster GGA BP86 calculations on representative systems of Pd and Pt atoms on MgO(OOl) yielded binding energies of 135 kJ/mol [73] and 232 kJ/mol [170], respectively. 

The converged cluster model adsorption energy of CO agreed quantitatively with the values obtained for periodic slab models. Note that only results obtained with the same xc functional can be directly compared [187]. At the BP86 level, the adsorption energy of CO at the hollow position of cluster models, 174 kJ/mol, is by 30 kJ/mol larger than the experimental value for the... 

Some effects that are left out of cluster calculations are illustrated in the work of Vittadini et al. [87]. They have suggested another prepairing mechanism, in which H atoms on the same side of adjacent singly occupied dimers ( interdimer airing) are more stable than independent singly occupied dimers. These are periodic slab calculations using a large unti eell and a local density functional (tests with the nonlocal Becke-Perdew functional do not... 

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