Finite cluster

First-principles models of solid surfaces and adsorption and reaction of atoms and molecules on those surfaces range from ab initio quantum chemistry (HF configuration interaction (Cl), perturbation theory (PT), etc for details see chapter B3.1 ) on small, finite clusters of atoms to HF or DFT on two-dimensionally infinite slabs. In between these... 

B3.2.4.3 FINITE-CLUSTER MODEL IN CONTACT WITH A CLASSICAL BACKGROUND... 

Pisani [169] has used the density of states from periodic FIP (see B3.2.2.4) slab calculations to describe the host in which the cluster is embedded, where the applications have been primarily to ionic crystals such as LiE. The original calculation to derive the external Coulomb and exchange fields is usually done on a finite cluster and at a low level of ab initio theory (typically minimum basis set FIP, one electron only per atom treated explicitly). 

The main drawback of the chister-m-chister methods is that the embedding operators are derived from a wavefunction that does not reflect the proper periodicity of the crystal a two-dimensionally infinite wavefiinction/density with a proper band structure would be preferable. Indeed, Rosch and co-workers pointed out recently a series of problems with such chister-m-chister embedding approaches. These include the lack of marked improvement of the results over finite clusters of the same size, problems with the orbital space partitioning such that charge conservation is violated, spurious mixing of virtual orbitals into the density matrix [170], the inlierent delocalized nature of metallic orbitals [171], etc. 

Head J D and Silva S J 1996 A localized orbitals based embedded cluster procedure for modeling chemisorption on large finite clusters and infinitely extended surfaces J. Chem. Phys. 104 3244... 

Figure 3 Periodic boundary conditions realized as the limit of finite clusters of replicated simulation cells. The limit depends in general on the asymptotic shape of the clusters here it is spherical. Cations are presented as shaded circles anions as open circles.

In the CVM, the free energy of a given alloy is approximated in terms of probabihties for a selected set of finite clusters. The largest cluster explicitly considered in the free energy functional specifies the level of the approximation. The common practice for an fcc-based system is the tetrahedron approximation [26] in which nearest neighbor tetrahedron cluster is taken as the largest cluster. Hence, within the tetrahedron approximation, the free energy expression, F,is symbolically expressed as... 

One of the m jor attractions in the metal-atom synthesis of dimer and cluster species is the ability to isolate highly unsaturated species, M Lm, that may then be considered to be models for chemisorption of the ligand, L, on either a bare, or a supported, metal surface (,100). It is quite informative to compare the spectral properties of these finite cluster-complexes to those of the corresponding, adsorbed surface-layers (100), in an effort to test localized-bonding aspects of chemisorption, and for deciphering UPS data and vibrational-energy-loss data for the chemisorbed state. At times, the similarities are quite striking. 

After having described molybdenum trioxide, we intend to specify the best finite clusters allowing to represent each of the (010), (001) and (100) faces in order to study surface properties such as energy and electronic distribution. For this purpose, the evolution of the electronic properties will be studied as a function of the cluster size and referred to the results of an EHT - band calculation [12] all calculations have been made with QCPE programs [13,14] and Hoffmann parameters [15],... 

The chemical gel point defines the instant of LST of chemically crosslinking polymers. Before the crosslinking polymer has reached its gel point it consists of a distribution of finite clusters. It is called a sol since it is soluble in good solvents. Beyond the gel point, it is called a gel . The gel is an infinitely large... 

While in principle all of the methods discussed here are Hartree-Fock, that name is commonly reserved for specific techniques that are based on quantum-chemical approaches and involve a finite cluster of atoms. Typically one uses a standard technique such as GAUSSIAN-82 (Binkley et al., 1982). In its simplest form GAUSSIAN-82 utilizes single Slater determinants. A basis set of LCAO-MOs is used, which for computational purposes is expanded in Gaussian orbitals about each atom. Exchange and Coulomb integrals are treated exactly. In practice the quality of the atomic basis sets may be varied, in some cases even including d-type orbitals. Core states are included explicitly in these calculations. 

When applied to silicon, both approaches suffer from problems common to small-basis-set techniques, namely they do not treat the conduction bands accurately. They can be parametrized to yield the proper band gap, and the defect properties seem not to be extremely sensitive to this factor. These approaches can be modified into supercell or cyclic-cluster forms, although most applications to date have involved finite clusters with hydrogen terminators. 

When reacting Ba dissolved in liquid Na with nitrogen at temperatures not too far from the melting point of the solvent, there was evidence for the formation of Ba-rich nitride clusters prior to the deposition of solid Ba2N [8]. Using both Na as well as a K-Na alloy as solvents, an extensive class of new subnitrides can be prepared. They contain [M6N] octahedra, either discrete or condensed into finite clusters and infinite arrays, respectively. 

In the cellular multiple scattering model , finite clusters of atoms are subjected to condensed matter boundary conditions in such a manner that a continuous spectrum is allowed. They are therefore not molecular calculations. An X type of exchange was used to create a local potential and different potentials for up and down spin-states could be constructed. For uranium pnictides and chalcogenides compounds the clusters were of 8 atoms (4 metal, 4 non-metal). The local density of states was calculated directly from the imaginary part of the Green function. The major features of the results are ... 

We study here the A + 5B2 —> 0 reaction upon a disordered square lattice on which only a certain fraction S of lattice sites can be accessed by the particles (the so-called active sites). We study the system behaviour as a function of the mole fractions of A and B in the gas phase and as a function of a new parameter S. We obtain reactive states for S > Sq where Sq is the kinetically defined percolation threshold which means existence of an infinite cluster of active sites. For S < Sq we obtain only finite clusters of active sites exist. On such a lattice all active sites are covered by A and B and no reaction takes place as t —> 00. 

We have studied the system (9.1.39) to (9.1.41) by means of the Monte Carlo method on a disordered surface where the active sites form a percolation cluster built at the percolation threshold and also above this threshold [25]. Finite clusters of active sites were removed from the surface to study only the effect of the ramification of the infinite cluster. The phase transition points show strong dependence on the fraction of active sites and on the... 

We study the system for different densities S of lattice sites which are accessible (active) to the reacting particles. For the case of 0 < S < 1 we obtain a lattice in which the active sites form smaller or larger islands. These islands are separated on the lattice in such a way that the particles occupying one island cannot interact with particles sitting on another island. In contrast to the computer simulations performed for such a system [25], we do not remove here finite clusters from the surface. 

If S is smaller than the site percolation threshold for the square lattice Sc = 0.59275 we obtain a system which consists only of finite clusters. In principle these clusters can be completely occupied by one-kind species. For the case that no desorption is allowed this represents a poisoned state for which the production rate Rco2 goes to zero as t —> oo. Then the whole system consists of finite clusters poisoned by particles A or B. For this state the condition Ca + Cb = S holds where C is the density of particles of type A (C a + Cb + Co = S). 

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. 

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