Clusters external potential

Our object of interest is a many electron finite system (such as an atom, molecule, cluster etc.), having, by assumption, a nondegenerate ground state (GS) (this assumption will be removed in Sects. 4.4 and 5). The numter of electrons N and the electron-nuclei potential energy v(r) = Ve (r) (the so-called external potential) are given and common for all schemes to be discussed. The GS energy qs aod the GS wave function Vqs of the system can be found from a variational principle as... 

It is also possible to include in addition to the ionic part an additional external potential Vg t in the Hamiltonian in Eq. (11) in the self-consistent procedure. This could be the long-range static potential from a surrounding crystalline environment, as used in the embedded cluster method. Fig. 7. Clusters are here used to model a small section of an infinite solid or surface. 

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... 

Catalysis and Electrocatalysis at Nanoparticle Surfaces reflects many of the new developments of catalysis, surface science, and electrochemistry. The first three chapters indicate the sophistication of the theory in simulating catalytic processes that occur at the solid-liquid and solid-gas interface in the presence of external potential. The first chapter, by Koper and colleagues, discusses the theory of modeling of catalytic and electrocatalytic reactions. This is followed by studies of simulations of reaction kinetics on nanometer-sized supported catalytic particles by Zhdanov and Kasemo. The final theoretical chapter, by Pacchioni and Illas, deals with the electronic structure and chemisorption properties of supported metal clusters. 

The action of the environment on the cluster can be treated as an external potential due to the external charge density p (r). This external potential consists of three parts... 

A promising extension of the SAPS model has bwn achieved by Schone et al. [109]. These authors expand the external potential of Eq. (57) about the center of the cluster ... 

Fig. 18, A schematic reaction path (RC) energy profile for the molecular - dissociative adsorption rearrangement of water on the 50-atoms (110)-rutile surface cluster. The dN = 0 (P) rearrangement (changing external potential v due to the nuclei) to which the HjO tilt angle coordinate refers, is dominated by the softest and energetically most important (selective) mode S = E, o = 1, which exhibits charge instabiiity, while the dN 0 (CT) processes are dominated by the hardest H (a S3) nonselective background mode. The SINDO SCF MO estimated barrier height, E, = E(2) — E(l), is about S kcal/mol. Note that the structure 4, reached along the - direction, corresponds to V, and dN > 0...

A rigorous quantum-mechanical calculation of some of the energy derivatives is unique to DFT alone [52]. The first and second derivatives with respect to the number of electrons, 0E/6N and 0 E/0N, recognised respectively as measures of chemical electronegativity [60] and hardness [61,62], are amenable to a rigorous calculation [52, 55,63,64]. For a system of N electrons characterised by an external potential v(f) (arising, for example, from the nuclei in an atom, molecule or cluster), the energy density functional can be expressed as... 

In a second study, DFT calculations on cuboidal MIr3S4 type clusters were performed by Tanaka et al, varying the heterometal centers (M=V, Cr, Mn, Fe, Co, Ni, Cu, Mo, Ru, W). This study was conducted in order to determine which of these clusters could potentially be employed for the conversion of N2 to ammonia. As a result of these calculations, the clusters with M=Ru, Mo and W were identified to activate N2, whereas no activation was found with the other metals. In order to obtain further insight into the reactivities of these clusters, calculations on the energetics of proton transfer from an external acid (lutidinium) to the coordinated N2 molecule were performed. These treatments indicated that protonation of the RU-N2 complex is associated with a high-energy barrier, in agreement with the experimental result that for this cluster no protonation of N2 could be achieved. Protonation of the Mo- and W-N2 complexes, in contrast, was predicted to be facile. 

The simple electronic structure of sodium also renders the application of other types of models relatively easy and extendable to relatively large sizes. See e.g. Hiickel calculations and MC structural search (R. Poteau and F. Spiegelmann, Phys. Rev. B 45, 1878 (1992) and J. Chem. Phys. 98, 6540 (1993) Erratum 99, 10089 (1993)) or the so-called spherically averaged pseudopotential (SAPS) model (M. D. Glossman, J. A. Alonso and M. P. Iniguez, Phys. Rev. B 47, 4747 (1993)). This is a simplified atomistic scheme, in which the external potential (written as the sum of the atomic pseudopotentials) acting on the electrons is developed in spherical harmonics around the cluster center of mass, and only the spherical component is retained in the solution of the KS equations. 

A new method based on spontaneous deposition of Pt on Ru has recently been demonstrated for Pt [340] and Pd [341] deposition on a Ru(OOOl) single crystal surface, which involves a reduction of H2PtCl6 coupled with the oxide formation on Ru [340]. A selective Pt deposition on Ru (no deposition on carbon) is attainable without the application of an external potential [342], Spontaneous deposition of Pt on Ru nanoparticles can be used to control the Pt cluster size and to tune the electronic and catalytic properties of PtRu catalysts. In addition, this approach facilitates a considerable reduction of Pt loadings by depositing Pt at the surfaee of Ru nanoparticles rather than having Pt throughout the PtRu... 

The combined quantum mechanics-molecular mechanics approach discussed below includes the external potential and also the polarization of the environment, but uses the ion-pair shell model potential for both the cluster and the surroundings... 

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