Ensemble of atoms

Class 111-type behavior is the consequence of this impossibihty to create step-edge-type sites on smaller particles. Larger particles wiU also support the step-edge sites. Details may vary. Surface step directions can have a different orientation and so does the coordinative unsaturation of the atoms that participate in the ensemble of atoms that form the reactive center. This wiU enhance the activation barrier compared to that on the smaller clusters. Recombination as well as dissociation reactions of tt molecular bonds will show Class 111-type behavior. 

Ammonia activation by Pt, to be discussed in the next section, is an interesting example, because it illustrates the basic principle that provides chemical direction to the identification of surface topologies that give low reaction barriers in surface reactions. This holds specifically for elementary reactions that require a surface ensemble of atoms. 

However, if the particles are indistinguishable, as the atoms in a gas, the number of possible configurations is significantly reduced and the partition function for an ensemble of atoms or molecules is therefore... 

The Holy Grail of catalysis has been to identify what Taylor described as the active site that is, that ensemble of atoms which is responsible for the surface reactions involved in catalytic turnover. With the advent of atomically resolving techniques such as scanning tunnelling microscopy it is now possible to identify reaction centres on planar surfaces. This gives a greater insight also into reaction kinetics and mechanisms in catalysis. In this paper two examples of such work are described, namely CO oxidation on a Rh(llO) crystal and methanol selective oxidation to formaldehyde on Cu(llO). 

The main idea behind classical molecular (or atom) dynamics (MD) is fairly simple. To illustrate this for the relatively simple case of an ensemble of atoms, let us consider a system of N particles, each having mass m, with Cartesian coordinates r,. The motion of this system of particles can be described by solving a set of equations of the type... 

Multicenter bonding is the key to understanding carboranes. Classical multicenter n bonding gives rise to electron-precise structures characteristic of Hiickel aromatics, which are planar and have 4n + 2 n electrons. Clusters are defined here as ensembles of atoms connected by non-classical multicenter bonding , i.e., all... 

The adsorbed species discussed above are simplified representations of the actual surface complexes since the number of participating surface metal atoms and the number of bonds between the molecule and the surface are not known. Hydrogenolysis reactions, however, are believed to take place on multiple surface sites, specifically, on a certain ensemble of atoms. For example, on the basis of kinetic data, Martin concluded273 that at least 12 neighboring nickel atoms free from adsorbed hydrogen are required in the complete cleavage of ethane ... 

The theory of spectral moments and line shape is based on time-dependent perturbation theory, Eqs. 2.85 and 2.86, applied to ensembles of atoms, or equivalently on the Heisenberg formalism involving dipole autocorrelation functions, Eq. 2.90. 

In fact these calculations did not treat the time scales correctly because they generally fixed most features of the atomic structure of solvent and then calculated the resulting electronic structure, for fixed potential drop across the interface. (A recent calculation [34] that takes more detailed account of the electronic structure of the electrode than these early calculations also suffers from this defect.) In fact, of course, in the Bom-Oppenheimer approximation, the electronic structure should be recalculated for each atomic configuration in an ensemble of atomic configurations that follow the Bom-Oppenheimer surface. This became possible with Car-Parrinello... 

A second important feature of reactions on metals is that a metal surface is a locus of very high density of active atoms. A molecule can thus be very easily bound to several contiguous atoms simultaneously or, as one says, to an ensemble of atoms [62,63]. For example, it is known that CO can be coordinated to 1, 2 or 3 (perhaps also 4) metal atoms, whereby each form shows a different frequency of CO stretching vibration v(C=0). 

Also the mode of adsorption (e.g. of CO, hydrocarbons, etc.) can depend on the available ensemble size or given composition of the surface [64—68]. It appears that the heat of adsorption of various modes of CO adsorption is only marginally influenced when the required ensemble (1,2 or 3 and more) is transferred from a pure metal into a matrix of another metal (for instance alloys with Cu, Au and Ag). When a CO molecule, monitored by IR spectroscopy, is taken as a probe of the local electronic structure of atoms (or ensembles of atoms), no pronounced effects of alloying are found 69-71]. 

Since dissociation of a molecule requires a certain minimum ensemble of surface atoms, the reactivity of very small metal clusters tends to increase with particle size. Once a cluster with the proper ensemble of atoms has become available the reactivity decreases, because the ensemble acquires an increasing number of metal atom neighbours. These two basic features, illustrated in Fig. 4.50, may explain the frequently observed maximum in cluster reactivity as a function of the number of metal atoms. 

Small metal particles form multiple bonds to the adsorbed species less easily than large particles or continuous planes of single crystals [32]. This is probably the reason why the very small metal particles are less active in FTS of hydrocarbons than the large ones FTS is initiated by CO dissociation and the products of the dissociation have to be multiply bound. The multiple bonding is probably easier with an ensemble of atoms rather than with single atoms. 

As stated in Section I, the equilibrium configuration of an ensemble of atoms or molecules corresponds to a minimum of the free energy ... 

Let the system under discussion be a specific ensemble of atoms with the possible number of states equal to Q. While the system exists actually in only one of the possible states, the number of state provided by the atom assemblage appears lower and equal to fimicro- Therefore, the difference As between the maximal and actual entropy of the system equals... 

Each of the N atoms or molecules in an MD simulation is treated as a point mass, and Newton s equations are integrated to compute the motion of these particles. A range of useful information, both microscopic and macroscopic, can be obtained from the motion of the ensemble of atoms, including transport coefficients, phase diagrams, and structural or conformational properties. MD simulations for complex physical systems often involve large system... 

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